Math - EBJ

# Pre-Calculus ### Things to Study 1. Index Laws 2. Root Laws 3. Quartiles & IQR 4. Standard Deviation 5. Rearranging Equations to make different variables the subject 6. Completing the Square ### Introduction & Basics ###### Origins of Math Mathematics has been used to describe and explain the world for thousands of years. The regular marks engraved on a piece of bone found in Africa, known as the Ishango bone, appear to indicate that people were counting and thinking mathematically over 20 000 years ago. More sophisticated mathematical activities such as measuring size and using money were also adopted by many civilisations thousands of years ago. The Ishango bone, discovered at the "Fisherman Settlement" of Ishango in the Democratic Republic of the Congo, is a bone tool and possible mathematical device that dates to the Upper Paleolithic era. The curved bone is dark brown in color, about 10 centimeters in length, and features a sharp piece of quartz affixed to one end, perhaps for engraving ###### BIDMAS In the UK, the acronym BIDMAS is preferred over PEMDAS. BIDMAS stands for Brackets, Indices, (Division, Multiplication), (Addition, and Subtraction). Operations within parentheses have the same precedence, so you must work from left to right for these. Note that indices refers to powers and roots. ###### Metric in the UK In 1969 the UK government set up the Metrication Board, with the aim of ensuring substantial adoption of metric units in the UK by 1975. In particular, it was planned that road sign conversion would take place in 1973. However, in 1970 the conversion programme was put on hold indefinitely, and successive British governments negotiated with the European Union to opt out of using metric units on road signs. ###### Fraud via Rounding Errors There have been a few reports of fraudsters in the US using rounding to become rich. They used computer programs to remove tiny amounts of money from lots of bank accounts, by rounding down to the nearest cent and putting the remaining fractions of cents into other accounts. These small amounts were not noticed as missing from any particular account, but they built up into huge sums of money. This is an example of so-called salami-slicing or penny-shaving fraud, in which thin slivers of money are removed from many accounts. Modern-day banking systems have built-in checks to prevent this type of fraud. ###### Significant Figures Another way of specifying where a number should be rounded involves looking at its significant figures. The first significant figure of a number is its first non-zero digit (from the left) When rounding sig figs, make sure to leave zeros in spots where the digit is still significant after rounding. For example, rounding 0.0198 to two sig figs will be 0.020. The last zero is still significant and cannot be dropped. If you are told that a distance is 3700 metres, say, without any information about how the number has been rounded, then you cannot tell whether the zeros are significant. The number 3700 could be the result of rounding 3684 to two significant figures, 3697 to three significant figures, or 3700 to four significant figures, for example. In the first case neither of the zeros is significant, in the second case only the first zero is significant, and in the third case both zeros are significant. This is one reason why it is important to state how a number has been rounded When you see a measurement such as 3700 metres with no information about whether or how the number has been rounded, you can usually assume that any zeros at the end are not significant When you are rounding answers, you should round to no more significant figures than the number of significant figures in the least precise number in the calculation ###### Fractions Simplifying a fraction can also be called cancelling a fraction. A proper fraction is when the numerator is smaller than the denominator. Otherwise, it's an improper fraction. An improper fraction can also be called a top-heavy fraction A mixed number is an integer and a fraction combined. ###### Percentages To convert a percentage to a fraction, put it over 100, and then simplify. In the opposite direction, just multiply by 100. Thus $\frac{2}{5} = \frac{2}{5} \times 100 = 40\%$ Increases in percentages If your rent increases by 5%, you can also do calculate the new total by multiplying the current price by 1.05 / 105% ###### Adding Odd Numbers If you add up the the first $n$ odd numbers in order, then the sum is always $n^2$. For example: | Number of Odd Numbers, $n$ | Adding Together | Sum | Square of $n$ | | ---- | ---- | ---- | ---- | | 1 | 1 | 1 | 1 | | 2 | 1 + 3 | 4 | 4 | | 3 | 1 + 3 + 5 | 9 | 9 | | 4 | 1 + 3 + 5 + 7 | 16 | 16 | Conjecture A conjecture is an informed guess about what might be true from considering a few real case Theorem A mathematical statement that has been proved is called a theorem or a result ###### Heighway Dragons & Fractals The Heighway dragon is sometimes called the Jurassic Park dragon, as it was printed in copies of Michael Crichton’s novel Jurassic Park. It is also sometimes known as the Harter–Heighway dragon, or even just the dragon curve. It was first investigated by NASA physicists John Heighway, William Harter and Bruce Banks and was described by Martin Gardner in Scientific American in 1967. The word ‘fractal’ was coined by the French mathematician Benoit Mandelbrot to describe a shape that is irregular at all scales, no matter how closely you look. Many fractals can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole shape. A shape that has this property is said to be self-similar. Real Life Examples of Fractals Mathematicians have carried out a great deal of research into the properties of fractals in recent years. Fractals are also used in many practical situations, from modelling internet traffic and fluctuations in world stock markets, to medical research. They also abound in nature, for example in the structure of clouds and snowflakes, and in the patterns of lightning. Our own bodies contain a myriad of self-similar fractal systems: for instance, our circulatory systems have this structure – the branching of tiny capillaries is similar to the branching of major arteries and veins. ### Unit 2 ###### Mathematical Models This unit is primarily concerned with a central theme of the module – how you can use mathematics to help investigate and solve practical problems. The aims of a mathematical model are to: 1. describe the important features of a real situation mathematically – for example, by using numbers, formulas or graphs 2. allow you to make predictions about the situation. ###### Algorithms An algorithm is a set of instructions to solve a problem step by step, which ###### Dijkstra’s algorithm One algorithm used in many route planners is called Dijkstra’s algorithm. It was developed in 1959 by a Dutch computer scientist, Edsger Dijkstra. In the algorithm, places are represented by dots, and the lines connecting the dots show the time (or distance) between the two places. The algorithm then systematically searches for the shortest time or shortest distance between two points on the diagram. ###### Maps & Scales Roads in the UK Note that A-roads in the UK can also be called principle roads Scale Factor On an example map 1 cm on said map represents 500 000 cm on the ground, and 1 mm on the map represents 500 000 mm on the ground, and so on. The number 500 000 is called the scale factor of the map. A map scale is given in words as ‘1 cm represents 20 km’. What is the scale factor? Convert 20 km to centimetres, by using 1 km = 1000 m and 1m = 100cm. So, 1 cm on the map represents 2 000 000 cm on the ground. That is, the map scale is 1 : 2 000 000, so the scale factor is 2 000 000 ###### The modeling cycle The modeling cycle can be used as a framework for solving many practical problems involving both basic and advanced mathematical skills. Its steps are as follows 1. Describe the problem concisely so that you are clear about what you are trying to do. In real life, and particularly if you are working within a team, this may involve discussing the problem with others. 2. Make assumptions to simplify the problem, so that you retain the essential features but will be able to describe it mathematically. At this stage, it is also useful to sort out what you already know about the problem, by collecting data and other information. 3. Describe the problem mathematically using numbers, formulas or graphs, and use these to obtain new results. 4. Consider what these new results mean practically, and check that the predictions seem reasonable. If the predictions do not match reality, then you may need to refine the assumptions, collect further information and go round the cycle again. Your conclusions are only as good as the data you have used and the assumptions you have made! ###### Conversion Graphs A conversion graph is a type of graph that can be used to convert between units of measurement ###### Formula Subjects The subject of a formula is the variable that is being solved for on its own side ###### Naismith’s Rule William Naismith was a Scottish climber who, in 1892, developed a rule for estimating walking times. Naismith’s Rule estimates that the time taken for a walk up a hill is given by the formula: $\Huge T = \frac{D}{5} + \frac{H}{600}$ where T is the time for the walk in hours, D is the horizontal distance walked in kilometres, H is the height climbed in metres. Naismith’s original rule has since been updated for the metric system. The rule is based on the assumptions that someone can walk at a speed of 5 km/h on flat ground and also needs to allow an extra minute to climb a height of 10 metres. Example: Estimate how long a walk will take if the horizontal distance is 20 km and the height is 1200 m. $\Huge T = \frac{20\ km}{5} + \frac{1200\ m}{600}$ $\Huge T = 6\ hours$ ###### Hooper's Rule Hooper's rule (named for Dr. Max Hooper) is based on ecological data obtained from hedges of known age, and suggests that the age of a hedge can be roughly estimated by counting the number of woody species in a thirty-yard section and multiplying by 110 years. Max Hooper published his original formula in the book Hedges in 1974. This method is only a rule of thumb, and can be off by a couple of centuries; it should always be backed up by documentary evidence, if possible, and take into account other factors. Caveats include the fact that planted hedgerows, hedgerows with elm, and hedgerows in the north of England tend not to follow the rule as closely. The formula also does not work on hedges more than a thousand years old. The formula is as follows: $\Huge A = 110 \times m + 30$ where A is the age in years, and m is the mean number of tree and shrub species in a 30-yard section. ###### Finding the Volume or Area of a Box $\Huge V = l \times w \times h$ $\Huge A = l \times w$ ###### Finding the Volume of a Cylinder Straight and Oblique Cylinders $\Huge V = h \pi r^2$ Where $\qquad$h is the height $\qquad$\pi$ is pi $\qquad$r is the radius ###### Gabriel Daniel Fahrenheit Gabriel Daniel Fahrenheit (1686–1736) was the inventor of the mercury thermometer. For the zero point of his temperature scale, he used the lowest measurable temperature that he could reach in his laboratory. He did this so that no everyday temperature would have a negative value. ### Unit 3 ###### Common Multiples & Factors A **common multiple** of two or more numbers is a number that is a multiple of all of them. The lowest common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of them. An alternative name for lowest common multiple is least common multiple. A natural number that divides exactly into a second natural number is called a factor or **divisor** of the second number. For example, 2 is a factor of 10, since 2 divides exactly into 10. "Because the factors of a number form pairs, most numbers have an even number of factors. The only exceptions are the square numbers, each of which has an odd number of factors. Remember that a square number is the result of multiplying a whole number by itself. For example, 25 is a square number because 25 = 5 × 5. Square numbers have an odd number of factors because one of the factors of a square number pairs with itself" ###### Divisibility tests A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8 3 if its digits add up to a multiple of 3 5 if it ends in 0 or 5 9 if its digits add up to a multiple of 9. If a number does not satisfy a test above, then it is not divisible by the specified number. ###### Common Factors A common factor of two or more numbers is a number that is a factor of all of them. The highest common factor (HCF) of two or more numbers is the largest number that is a factor of all of them. An alternative name for highest common factor is greatest common divisor (GCD). ###### Prime Numbers A natural number that has exactly two factors is called a prime number. Note that every prime number except 2 and 5 ends in 1, 3, 7 or 9 The prime numbers under 100 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 ###### Sieve of Eratosthenes There is a simple algorithm for finding all the prime numbers up to a certain number, which is attributed to the Greek mathematician Eratosthenes (c. 276 BCE – c. 197 BCE), and known as the Sieve of Eratosthenes. Eratosthenes was a librarian at the famous library at Alexandria. Eratosthenes was a man of many talents: he was a mathematician, geographer, historian and literary critic. He was the first person to make a good measurement of the circumference of the Earth, and he came up with the idea of the leap year, which stops the calendar drifting out of step with the seasons. The algorithm for finding prime numbers works well provided that the certain number is not too large, and modified forms of it are still used by mathematicians today. ###### Remainder when dividing Prime Numbers by 4 | Prime Number | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | | Remainder | 3 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 1 | ###### Prime Numbers that are the sum of two squares | Prime | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | ---- | | Sum of two squares? | | 1 + 4 | | | 9 + 4 | 1 + 16 | | | 25 + 4 | ###### Fermat's theorem on sums of two squares The two charts above show that when a prime number has a remainder of 1, it can also be expressed as the sum of two squares. In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: $\Large p = x^2 + y^2$ with x and y integers, iff $\Large p \equiv 1 \mod 4$ The prime numbers for which this is true are called Pythagorean primes ###### Mersenne Primes In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form $M_n = 2^n − 1$ for some integer n. Thousands of prime number enthusiasts throughout the world are interested in breaking the record for the largest known prime number. Most of them search for Mersenne primes – primes of the form $2^n - 1$ for some natural number n. For example, 7 and 31 are Mersenne primes because $7 = 2^3 − 1$ and $31 = 2^5 − 1$. Mersenne primes are named after the French mathematician Marin Mersenne (1588–1648), who investigated them. It can be proved that if $2^n - 1$ is prime, then n must be prime. At the time of writing, fifty-one Mersenne primes are known. The latest one was discovered in December 2018. Before that, in September 2008, the discoverer of the first Mersenne prime with more than ten million digits won a $100 000 prize. These prime numbers (and 15 other large Mersenne primes) were found using software provided by the Great Internet Mersenne Prime Search, a scheme in which enthusiasts can download free software that uses the spare processing power of their computers. At the time of writing, a $3000 prize is available for each new Mersenne prime discovered, and there is a $150 000 prize for the first Mersenne prime with more than 100 million digits. ###### Prime factors A natural number greater than 1 that is not a prime number is called a composite number. Unlike a prime number, a composite number can be written as a product of two factors, neither of which is 1. So 360 can be written as a product of four factors, none of which is 1: 360 = 4×5×3×6. The process here can be set out as a factor tree ![[PrimeFactorTree.png]] ###### The fundamental theorem of arithmetic Every natural number greater than 1 can be written as a product of prime numbers in just one way (except that the order of the primes in the product can be changed). ###### Powers 'Raising a number to a power’ means multiplying the number by itself a specified number of times. For example, raising 2 to the power 3 gives 23 = 2×2×2. Here the number 2 is called the base number or just base, and the superscript 3 is called the **power**, **index** or **exponent**. The word ‘power’ is also used to refer to the result of raising a number to a power – for example, we say that $2^3$ is a power of 2. When we write expressions like $2^3$, we say that we are using **index form** or **index notation**. For this definition of the word, index's plural is indices. ###### Inventor of Superscript Notation for Powers The superscript notation for powers was introduced by the French philosopher and mathematician René Descartes (1596–1650). ###### Long & Short Number Scales The word ‘billion’ meant $10^{12}$ rather than $10^9$ in the UK until 1974, when the British government decided to switch to the American meaning to avoid confusion in financial markets. Similarly, the word ‘trillion’ has traditionally meant $10^{18}$ in the UK, but there has recently been a switch to the American meaning, $10^{12}$. Many European countries still use these alternative meanings of ‘billion’ and ‘trillion’. ###### Googol A googol is 10100, but this number is of limited use, as it is greater than the number of atoms in the observable universe! The word was invented by a child, nine-year-old Milton Sirotta, in 1938. He was asked by his uncle, the American mathematician Edward Kasner (1878–1955), what name he would give to a really large number. The word ‘googol’ gave rise, via a playful misspelling, to the name of the internet search engine Google. Multiply & Dividing Bases with Exponents To multiply numbers in index form that have the same base number, add the indices: $\Large a^m \times a^n = a^{m+n}$ Correlatively, you would subtract if these bases were being divided ###### Index Laws **Multiplying & Dividing numbers with powers of the same base** If the base is the same, you can add and subtract the powers as shown here: $\Large a^n \times a^m = a^{n+m}$ $\Large \frac{a^n}{a^m} = a^{n-m}$ **Raising a Base with a Power to another Power** To raise a number in index form to a power, multiply the indices: $\Large (a^m)^n = a^{mn}$ **Raising a product or quotient to a power** A power of a product is the same as a product of powers. A power of a quotient is the same as a quotient of powers. $\Large (a \times b)^n = a^n \times b^n$ $\Large (\frac{a}{b})^n = \frac{a^n}{b^n}$ We can use this fact to solve difficult prime factorization problems. For example, find the prime factors of: $21^4$ These would be $3^4 \quad and \quad 7^4$ because $21^4 = 3^4 \times 7^4$ ###### Rational Numbers A rational number is a number that can be written in the form: $\Large \frac{integer}{integer}$ Examples of rational numbers: Any fraction Any whole number Any decimal number with a finite number of digits after the decimal point Some decimal numbers with infinitely many digits after the decimal point may be a rational number, but they may not be. ###### Decimals You might get a decimal number that has only a finite number of digits after the decimal point. This is called a terminating decimal Alternatively, you might get a decimal number with a block of one or more digits after the decimal point that repeats indefinitely. For example: 0.666666 or 0.123123123. A decimal like this is called a recurring decimal. There are two alternative notations for indicating a recurring decimal. You can either put a dot above the first and last digit of the repeating block, or you can put a line above the whole repeating block. For example: $\Large 0.\dot{6}$ $\Large 0.\overline{6}$ $\Large 0.\dot{1}29\dot{6}$ $\Large 0.\overline{1296}$ ###### Adding & Subtracting Fractions In order to add or subtract fractions, you need to find the Lowest Common Multiple of the divisor. 1. Find the LCM 2. Multiply each fraction by its multiple 3. Add or subtract **Example** $\Large \frac{1}{4} + \frac{2}{5}$ Its LCMs are 5 and 4 respectively, so you will multiply each by its multiple to get to a common denominator. $\Large \frac{5}{20} + \frac{8}{20} = \frac{13}{20}$ ###### Multiplying Fractions Multiply the numerators together and multiply the denominators together. ###### Dividing Fractions To divide two fractions, first take the reciprocal of the second fraction or the fraction on bottom, the divisor, and then multiply. See the following: $\Large \frac{a}{b} \div \frac{c}{d} \quad = \quad \frac{a}{b} \times \frac{d}{c} \quad = \quad \frac{a \times d}{b \times c}$ The first equation is thus equivalent to the following: $\Large \frac{\frac{a}{b}}{\frac{c}{d}}$ Simplify, and you have your answer. Notice that this process also allows us to simplify more complex fractions ###### Negative Powers A negative power is equal to its positive reciprocal, as follows: $\Large 2^{-3} = \frac{1}{2^3}$ ###### Zeroth Power A non-zero number to the zeroth power is 1. ###### Scientific Notation To convert a number to scientific notation: 1. Put a decimal between the first and second sig digits to create a number between 1 and 10. 2. Count the number of digits that come after the decimal to create the index Example: $\Large 2500 = 2.5 \times 10^3$ Use Index Laws to carry out operations on numbers in Scientific Notation. For example: $\Large (4 \times 10^9) \times (6 \times 10^{-7}) = 24 \times 10^{9-7} = 2.4 \times 10^3$ ###### Irrational Numbers & Real Numbers Rational numbers by themselves do not form a workable system of numbers. We must include many more numbers to obtain such a system, and the new numbers that we must include are the ones with decimal forms that are not terminating or recurring – that is, we must include the decimals with an infinite number of digits after the decimal point but no repeating block of digits. These numbers are called the irrational numbers – they are the numbers that are not rational. ###### Roots The square root symbol was introduce by René Descartes. Note that the square root symbol means the positive square root. Unless otherwise specified, do not answer a square root with two answers. Asking for both would look like: $\Large \pm \sqrt{4}$ Roots other than square roots have to be explicitly written: $\Large \sqrt[3]{4}$ ###### Root Laws A square root of a product is the same as a product of square roots; a square root of a quotient is the same as a quotient of square roots: $\Large \sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ $\Large \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ For example, to simplify: $\Large \frac{2}{\sqrt{2}}$ You have to first know that $\Large 2 = \sqrt{2} \times \sqrt{2}$ As a result, we can substitute this in for the numerator, and then cancel out one set in the numerator and denominator. Thereby leaving just $\sqrt{2}$ ###### Powers within Roots Note that a power underneath a root can be brought outside of it, as follows: $\Large \sqrt{b^n} = (\sqrt{b})^n$ ###### Surds A surd is an expression with an irrational root. We usually leave these as roots since they cannot be written as a fraction or with a terminating decimal. Example $\Large \sqrt{2}$ ###### Simplifying Roots in Expressions or in the Denominator Let's take an example: $\Large \frac{3 \sqrt{11}}{\sqrt{33}} + \sqrt{48}$ We can first split the denominator: $\Large \frac{3 \sqrt{11}}{\sqrt{3} \sqrt{11}} + \sqrt{48}$ This causes the $\sqrt{11}$ to cancel out, and then we can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$ This makes: $\Large \frac{3 \sqrt{3}}{3} + \sqrt{48}$ Notice that the 3s will cancel out now, leaving just the $\sqrt{3}$ We now need to simplify the $\sqrt{48}$ $\Large \sqrt{16} \sqrt{3}$ which equals $\Large \sqrt{3} + 4 \sqrt{3}$ Since there is an implicit 1 in front of $\sqrt{3}$, we can add these two together to get: $\Large 5 \sqrt{3}$ ###### Fractional Powers A fractional power is equivalent it a root in the following way: $\Large a^{\frac{1}{n}} = \sqrt[n]{a}$ The denominator of the fraction is the base of the root. The numerator is the power of the number under the root. Note that decimals are equivalent to fractions in this regard. Example: $\Large b^{1.7} = b^{\frac{17}{10}} = \sqrt[10]{b^{17}} = (\sqrt[10]{b})^{17}$ ###### Ratios A ratio is in its simplest form when each number in the ratio is a whole number, and these numbers are cancelled down as much as possible – that is, they have no common factors. For example, the ratio 0.5:1.25 would be written in its simplest form as 2:5. To simply a ratio, just divide all numbers in the ratio by a common factor until it can't be reduced further. ###### Determining parts of a ratio Three flatmates, Amy, Becky and Carol, have agreed to contribute to their joint budget in the ratio 5 : 2 : 3. (For every £5 Amy contributes, Becky contributes £2 and Carol contributes £3.) The flatmates’ expenses amount to £1250 per month. How much does each flatmate contribute to this? We can figure this out by determining how much of the total each contributes and multiplying it by the total contribution. 5 + 2 + 3 = 10, so each individual contribution can be expressed as a fraction of ten, which can be multiplied by the total contribution to determine the individual contribution in pounds. 5/10 times 1250 = 625 ###### ISO 216 Paper, AKA A-Series The ISO 216 paper sizes were developed in Germany in the early 1900s. They were adopted as a standard there in 1922, and soon spread to other European countries. The UK adopted them in 1959, and they were adopted by the International Organization for Standardization in 1975. They are now used throughout the world: at the time of writing the only exceptions are the USA, Canada, Peru, Colombia and the Dominican Republic. The largest ISO 216 paper size, A0, has an area of $1m^2$. ### Unit 4 ###### Primary and Secondary Data Data that you collect yourself are called primary data. Secondary data are data that already exist and can be used or adapted for your purpose. A dataset is a collection of data, usually presented in tabular form, or as a single row, or sometimes as a single column Discrete data are data that can take one of a particular set of values; such data typically, though not necessarily, take integer values. Here are some examples of discrete data: • the number of days in a week on which one takes exercise • the number of times a particular website is visited in a day • the quality of a person’s recovery after a serious accident when coded 0 for full recovery, 1 for partial recovery, 2 for failure to recover. A widely-occurring example of this is when there are just two categories such as yes/no, pass/fail or true/false. Such data, coded by two numerical values, are said to be binary data. Unlike discrete data, continuous data can take all the in-between values on a number scale. ###### Mass vs Weight The mass of an object is a measure of the amount of matter that it contains, whereas its weight is a measure of the gravitational force acting on it. Weight, being a force, is measured in newtons. However, you will often see weights quoted in kilograms in everyday life To convert weight as described in kilograms for objects on earth, you can use the following formula to convert to Newtons: $F = m \times g$ where: F is the force in newtons (N), m is the mass in kilograms (kg), and g is the acceleration due to gravity, approximately $9.81\ m/s^2$ on the surface of the Earth. Thus, my current weight, 75 kilograms, converted to Newtons would be expressed as: $735\ N = 75\ kg \times 9.81\ m/s^2$ ###### Outliers One or more data values that are considerably smaller or larger than the other values in the same dataset are called outliers ###### Single and Paired Data Column H of the backache dataset in Table 2 contains 33 values of the weights of babies (in kg). This constitutes a single sample of weights of babies from mothers, many of whom suffered from backache in pregnancy. For the moment, ignore the remainder of the data in the table. Looking just at column H, these values are all based on a single measure (weight) and can be described as single data. Suppose now that there was a second sample, of birth weights from a different set of mothers, the babies in this second sample all being classed as premature. This is now a two-sample, as opposed to one-sample, dataset. Now return to the babies recorded in the backache dataset in Table 2. A statistical question of interest might be how the babies’ weights (in column H) relate to the weights of their mothers at the start of their pregnancy (column F). This question links two pieces of information for each of the people in the study – a case of paired data. ###### Dataset Location The location of a dataset is a single number that represents an ‘average’, ‘typical’, or ‘central’ value. There is no single and universally most appropriate measure of location, but there are various useful measures that can be chosen, depending on the situation and on the nature of the data. Each measure has its own pros and cons. The three most common measures of location in statistics textbooks are the mean, the mode and the median. ###### Mean, Median, and Mode **Pythagorean Mean** There are three main types of means. 1. Arithmetic Mean 2. Geometric Mean 3. Harmonic Mean **Arithmetic Mean (AM)** is best used when the data can be summed up. For instance, If I earn 5, 7, 8, and 10 dollars each day for four days, I will have earned 30 dollars. Because this can be expressed by addition, the arithmetic mean is the right choice for averaging, and we get an average of $7.50 per day. $\Large AM(x_1,\ ...,\ x_n) = \frac{x_1\ ...\ x_n}{n}$ **Geometric Mean (GM)** is best used with ratio scales, such as growth rates of the human population or interest rates of a financial investment over time. It is so called because exponential growth can also be called geometric growth. It also applies to benchmarking, where it is particularly useful for computing means of speedup ratios: since the mean of 0.5x (half as fast) and 2x (twice as fast) will be 1. $\Large GM(x_1,\ ...,\ x_n) = \sqrt[n]{\prod_{i=1}^{n} x_i} = \sqrt[n]{a_1 a_2 ⋯ a_n}$ What this equation tells us is that the geometric mean is equal to the $n$th root of the product of numbers. Suppose for example a person invests 1000 dollars in shares and achieves annual returns of +10%, -12%, +90%, -30% and +25% over 5 consecutive years to give a final investment value of 1,609 dollars. The arithmetic mean of the annual percent changes is 16.6%. However, this value is unrepresentative, i.e. it would be incorrect to do this. If the initial investment grew by 16.6% per annum, it would be worth 2155 dollars after 5 years. To find the true mean, you would use the geometric mean equation as follows: Make sure to convert your percentages to decimal values first: $\Large = \sqrt[5]{1.1 \times 0.88 \times 1.9 \times 0.7 \times 1.25}$ Then you can convert back to a percentage after: $\Large = 1.099 \quad = 110 \%$ total change from initial investment Thus, the average yearly interest is 10%. **Harmonic Mean (HM)** is best used when the data represents things like speeds, densities, or cost-efficiency, and the reciprocal of values is more meaningful. It is so called because the harmonic series, an infinite series of sums, also uses reciprocals. $\Large HM(x_1,\ ...,\ x_n) = \frac{n}{\sum^n_{i=1} \frac{1}{x_i}} = \frac{n}{\frac{1}{x_1} + ⋯ + \frac{1}{x_n}}$ Example problem: A cyclist travels from City A to City B at an average speed of 20 km/h. On the return trip from City B to City A, the cyclist travels at an average speed of 30 km/h. If the distance between City A and City B is the same both ways, what is the cyclist's average speed for the entire trip? First, plug the values into the formula $\Large = \frac{n}{\frac{1}{20} + \frac{1}{30}}$ And then solve to get the average speed: $\Large = 24$ km/h **Quadratic Mean (QM)** is not a Pythagorean Mean, but it is here for comparison. The quadratic mean, also known as the root mean square (RMS), is primarily used when dealing with data that involves both positive and negative values, or when the magnitude of numbers is more important than their sign. It's especially useful in situations where the squares of values are significant, such as in physical measurements. Note that the quadratic mean is similar to Standard Deviation. $\Large QM(x_1,\ ...,\ x_n) = \sqrt{\frac{1}{n} \sum^n_{i=1} x_i^2} = \sqrt{\frac{1}{n} \left( x_1^2 + ⋯ + x_n^2 \right)}$ Example problem: An electrical engineer is measuring the voltage fluctuations in a circuit over time. The voltage at five different times is recorded as: 3V, -5V, 2V, -4V, and 6V. What is the effective voltage of the circuit, which is the RMS (root mean square) value of the voltage readings? First I can plug the values into the formula: $\Large = \sqrt{\frac{1}{5} \left( 3^2 + -5^2 + 2^2 + -4^2 + 6^2\right)}$ Then I can solve to get the average effective voltage of a circuit: $\Large = \frac{2 \sqrt{10}}{5} \quad = 1.26\ V$ **Median** - Sort the data into increasing or decreasing order. - If there is an odd number of data values, the median is the middle value. - If there is an even number of data values, the median is defined as the mean of the middle two values. (If the middle two values are equal, then the median is just this common value.) **Mode** The most common of number in a set ###### Range As you have just seen, a simple way of estimating spread is to scan along the data to find the smallest (minimum, or ‘min’) and largest (maximum, or ‘max’) values. The range is the difference between these two values. In other words, range = max − min. ###### Quartiles and Interquartile Range To find the quartiles and the interquartile range of a dataset 1. Arrange the dataset in increasing order. 2. Next: 1. If there is an even number of data values, then the lower quartile (Q1) is the median of the lower half of the dataset, and the upper quartile (Q3) is the median of the upper half of the dataset. 2. If there is an odd number of data values, throw out the middle data point (which of course has the median value of the dataset). Then the lower quartile (Q1) is the median of the lower half of the new dataset, and the upper quartile (Q3) is the median of the upper half of the new dataset. 3. The interquartile range (IQR) is Q3 − Q1. Quartiles and IQR are used for box-and-whisker plots. ###### Standard Deviation The best known measure of spread is the standard deviation, or SD. An alternative name for the standard deviation is the RMS deviation – in full, the root mean squared deviation To find the standard deviation of a dataset 1. Find the mean of the dataset. 2. Find the difference of each value from the mean – these are the ‘deviations’, often labelled as the d values. 3. Square each deviation – this gives the $d^2$ values. 4. Find the mean of these squared deviations – this number is the ‘mean squared deviation’, better known as the variance. 5. Find the square root of the variance to get the ‘root mean squared deviation’ – that is, the standard deviation. $\Large \sigma = \sqrt \frac{\Sigma \left( x_i - \mu \right)^2}{N} $ where: $\sigma$ is the population standard deviation $N$ is the size of the population $x_i$ is each value from the population $\mu$ is the population mean Example: find the SD of 1, 2, 4, 6, 7 The arithmetic mean of the dataset is 4, which will be plugged into $\mu$ Then we need to find the deviation from the mean, what's in the parentheses above: -3, -2, 0, 2, 3 Then we square each number: 9, 4, 0, 4, 9 Then we get the mean of the variance: 5.2 Then we take the square root of this number, and the SD is: $\sigma = 2.3$ (to 1 d.p.) Reasons for using the standard deviation as a measure of spread The standard deviation is the best known and most commonly used measure of spread. All the values in a dataset are included in its calculation. However, unlike the interquartile range, its value can be to some extent distorted by outliers. ###### Accuracy vs. Precision For a set of (repeated) measurements: • Accuracy describes how close the average is to the true value. • Precision describes how close the measurements are to each other. ### Unit 5 ###### Etymology of Algebra The word ‘algebra’ is derived from the title of the treatise Al-kitab-al mukhtasar fi hisab al-jabr (Compendium on calculation by completion and reduction), written by the Central Asian mathematician Muh.ammad ibn Mu ̄s ̄a al-Khw ̄arizm ̄i in around 825. Muh.ammad ibn Mu ̄s ̄a al-Khw ̄arizm ̄i was a member of the House of Wisdom in Baghdad, a research centre established by the Caliph al-Ma’mu ̄n. His name is the origin of the word ‘algorithm’. As you saw in Unit 2, an algorithm is a procedure for solving a problem or doing a calculation. The treatise deals with solving linear and quadratic equations, which you’ll learn about in this module, starting with linear equations in this unit. The treatise doesn’t use algebra in the modern sense, as no letters or other symbols are used to represent numbers. Modern algebra developed gradually over time, and it was not until the sixteenth and seventeenth centuries that it emerged in the forms that we recognise and use today. ###### Algebraic Expressions An algebraic expression, or just expression for short, is a collection of letters, numbers and/or mathematical symbols (such as +, −, ×, ÷, brackets, and so on), arranged in such a way that if numbers are substituted for the letters, then you can work out the value of the expression. ###### Invention of Letters in Algebra The first person to systematically use letters to represent numbers was the French mathematician François Viète. His treatise In artem analyticem isagoge (Introduction to the analytic art) of 1591 gives methods for solving equations, including ones more complicated than those in this module. Viète represented unknowns by vowels and known numbers by consonants (he represented known numbers by letters to help him describe the methods). However, he used words for connectives such as plus, equals and so on, and also to indicate powers. For example, he wrote ‘=’ as ‘aequatur’, $a^2$ as ‘a quadratus’ and $a^3$ as ‘a cubus’. So his algebra was still far from symbolic. Viète also wrote books on astronomy, geometry and trigonometry, but he was never employed as a professional mathematician. He was trained in law, and followed a legal career for a few years before leaving the profession to oversee the education of the daughter of a local aristocratic family. His later career was spent in high public office, apart from a period of five years when he was banished from the court in Paris for political and religious reasons. Throughout his life, the only time he could devote to mathematics was when he was free from official duties. Terms Terms in an expression are any segments being operated on; for example, in: $-2xy + 3z - y^2$ $-2xy$, $3z$, $y^2$ ###### Proofs Let's create an example proof to learn more about doing them. Prove that the sum of any three consecutive integers is divisible by 3 First, let's represent the sum of any three consecutive integers algebraically. $n + (n + 1) + (n + 2)$ $= n + n + 1 + n + 2$ If we collect like terms: $= 3n + 3$ We can then factor out the 3 and get: $3(n + 1)$ This shows that $n$ will always be a multiple of 3, and as a result, it has to be divisible by 3. ###### Linear Equations Linear equations in one unknown means equations with one singular variable and that variable is only to the first power, e.g. $x^1$. ###### Age Word Problem Mary is four times the age she was 63 years ago. How old is Mary? Let Mary's current age be $m$ $m = 4(m - 63)$ Expand the parenthetical expression $m = 4m - 252$ Subtract the LHS $m$ $0 = 3m - 252$ Add 252 $252 = 3m$ Divide by 3 $84 = m$ Mary's current age is 84. ### Unit 6 ###### The Cartesian Coordinate System The French mathematician René Descartes developed the way of specifying the position of a point in 1637. It is known as the Cartesian coordinate system, after him. Each point that can be plotted on a graph is represented by a pair of numbers called the coordinates of the point. ###### Scatterplots A graph on which data pairs are plotted is called a scatterplot, and the points plottedare referred to as data points. The data points on the scatterplot in do not lie exactly in a straight line, so the relationship between the selling price of the tomatoes and the quantity sold does not correspond exactly to a straight line. However, the data points lie approximately in a straight line, so the relationship can be modelled by a straight line ###### Slope In England, this is also called the gradient. The increase in the x-coordinate is known as the run, and the increase in the y-coordinate is known as the rise. The gradient of the line can be calculated by dividing the rise by the run Since coordinates are written in the form (x, y), and there are two points, it is convenient to represent the coordinates of the left-hand point by $(x_1,\ y_1)$ and those of the right-hand point by $(x_2,\ y_2)$ $\Large \frac{y_2 - y_1}{x_2 - x_1}$ The straight line that passes through the origin and has gradient m has equation $y = mx$. It is traditional to use the letter m to represent gradient, though the reason is no longer known! The earliest known use of the letter m for gradient is by the Italian mathematician Vincenzo Riccati in 1757. ###### Direct proportion relationships If two quantities x and y are directly proportional to each other, then the relationship between them is described by an equation of the form $y = kx$ where k is a non-zero number, known as the constant of proportionality. The statement ‘y is directly proportional to x’ is sometimes written as $\Large y \propto x$ ###### Equation of a Straight Line $\Large y = mx + b$ ###### Linear Regression & Interpolation You can often use a model based on data to estimate values that are not given in the data. If the estimated value is within the range of the data, then this process is known as interpolation. As long as the model fits the data reasonably well, interpolation can provide reasonable estimates. You can use a model based on a dataset to estimate a value that lies outside the range of the dataset. This is called extrapolation, and it can be unreliable. It is important to know how accurate a prediction provided by a regression line is likely to be, so it is useful to have some indication of how well a regression line fits its data points. To provide this, statisticians have developed a measure known as the correlation coefficient, which is a number calculated from the data pairs, and is often denoted by r. The value of the correlation coefficient indicates how well the regression line fits the data pairs. The correlation coefficient of a set of paired data is always between −1 and 1, inclusive. If the correlation coefficient is positive, then the regression line for the data has a positive gradient, which means that one of the quantities tends to increase as the other increases. In this case, the quantities are said to have a positive correlation. On the other hand, if the correlation coefficient is negative, then the regression line has a negative gradient, which means that one of the quantities tends to decrease as the other increases. In this case, the quantities are said to have a negative correlation. The correlation coefficient is exactly 1 or exactly −1 when all the data points lie exactly on the regression line. When this happens, the quantities are said to have a perfect correlation ### Unit 7 ###### Drawing a Line from an Equation It's often easiest to start with the X and Y intercepts, thus solve the equation twice, each time alternating when one of the two variables is equal to zero to get the intercept points. ###### Simultaneous Linear Equations AKA, a set of linear equations in two unknowns or a System of Equations Let's say we have the following two equations: $d = 30t$ $d = 90t -135$ Set the right sides equal to each other and solve $30t = 90t - 135$ $30t + 135 = 90t$ $135 = 60t$ $\frac{135}{60} = t$ We can use this to solve the original equations for D at the point of intersection ###### Systems of Equations in more than Two Unknowns When you have multiple equations with many unknowns, you can use the substitution and addition to solve just as before. Start by reducing one variable at a time. An efficient formalization of this process of solving simultaneous equations in many unknowns by reducing them to equations in progressively fewer unknowns is known as Gaussian Elimination after the great eighteenth-century mathematician and physicist Johann Carl Friedrich Gauss. In practice, these days most systems of several equations in several unknowns are solved by computer. The analysis of stress patterns in bridges and buildings, for instance, often requires systems of several thousand simultaneous equations. ###### Simplifying and Solving Inequalities The rules for doing inequalities are almost identical to those of equations; however, if you switch the sides of data, don't forget to reverse the inequality sign! ### Unit 8 ###### Euclid Euclid was a Greek mathematician who worked in the city of Alexandria in Egypt in the third century BCE. Little is known about his life other than that he produced ten works, of which five have survived to the present day. His reputation as a mathematician is based on his main work, the Elements, which was the standard introduction to mathematics for over two thousand years. Much of the content of Euclid’s Elements was the core of school geometry in the UK until about 1970. It is claimed to be the best-selling mathematics book of all time, and is one of the most frequently printed books ever. One reason why the Elements was so important was that it introduced generations of mathematicians to ideas of rigorous proof. It starts with a small number of axioms (truths that were taken as self-evident), such as the fact that a straight line can be drawn between any two points. It then proceeds to prove theorems, such as Pythagoras’ Theorem, using nothing more than the axioms and previously proven theorems. This approach was very influential in the development of mathematics. ###### William Jones In 1706 the English mathematician William Jones introduced the use of π to mean the ratio of the circumference of a circle to its diameter, but the symbol didn’t come into general use until it was popularized by Leonhard Euler in his Introductio in analysin infinitorum in 1748. ###### Hipparchus of Nicea The mathematician and astronomer Hipparchus of Nicea (ca. 180–125 bce) is thought to have chosen 360 for the number of degrees in a full turn. Earlier, Babylonian astronomers had divided the day into 360 parts. ###### Angles Angles greater than 180 degrees but less than 360 degrees are called reflex angles Angles that are along a straight line will add up to 180 degrees. Opposite Angles, Corresponding Angles, and Alternate Angles are equal. ![[OppositeAngles.png]] ![[CorrespondingAndAlternateAngles.png]] Look at Figure 8. The angles α and γ are known as alternate angles, because they are on alternate sides of the line that crosses the pair of parallel lines. They are also known informally as Z angles, because there is a pair of such angles in a capital Z. ###### Triangles If a triangle has just two sides that are the same length, then it also has two equal angles, known as the base angles, and the triangle is called an isosceles triangle. A triangle that is neither equilateral nor isosceles is known as a scalene triangle. All its sides are of different lengths. The angles of a triangle always add up to 180º. ###### Rectangles The angles of a rectangle always add up to 360º. ###### Polygons A polygon is a plane shape with straight sides – so triangles and squares are examples of polygons. In particular: • A quadrilateral is a polygon with four sides. • A pentagon is a polygon with five sides. • A hexagon is a polygon with six sides. Similarly, heptagons, octagons, nonagons and decagons are polygons with, respectively, 7, 8, 9 and 10 sides. ###### Rotational Symmetry Some shapes have the property that if you rotate them through a fixed angle (less than a full turn) about a fixed point, then the rotated shape looks the same as the original shape. Such a shape is said to have rotational symmetry, and the fixed point is called the centre of rotation. For example, if an equilateral triangle is rotated through one third of a full turn (120°) about its centre, then the rotated triangle looks the same as the original triangle Since there are three positions in which the rotated triangle looks the same, it is said to have rotational symmetry of order 3, or three-fold rotational symmetry. Another way to think about the three-fold rotational symmetry of the triangle is that three rotations are needed to return it to its starting position All regular polygons have rotational symmetry. The order of the rotational symmetry is the same as the number of sides. ###### Line Symmetry All regular polygons have rotational symmetry. The order of the rotational symmetry is the same as the number of sides. Some shapes exhibit a different kind of symmetry, known as line symmetry, mirror symmetry or reflectional symmetry. This imaginary line is known as a line of symmetry (or a reflection line or mirror line). Shapes can have more than one line of symmetry. ###### Congruency Geometric figures with the same size and shape are said to be congruent. So two shapes are congruent if you can pick one of the shapes up and place it exactly on top of the other shape, rotating or flipping over the first shape if necessary. Geometric figures that have the same shape, but not necessarily the same size, are said to be similar. The symbol $\cong$ (read as ‘is congruent to’) is used to indicate that two shapes are congruent ###### Pythagoras’ Theorem For a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Pythagoras’ Theorem has long been attributed to Pythagoras, a Greek of the sixth century BCE, who gave his name to a sect called the Pythagoreans. (Pythagoras’ existence is disputed by many historians, however.) The Pythagoreans believed that numbers and number patterns were the key to understanding the world. However, it is clear from clay tablets dating from about 2000 BCE that early Babylonian scribes knew about the theorem, and the result is also found in ancient Chinese manuscripts. Pythagoras’ Theorem is Proposition 47 of Book 1 of Euclid’s Elements, and this is where the first rigorous proof appears. There are many ways of proving Pythagoras’ Theorem – the book The Pythagorean Proposition by E.S. Loomis (published in 1968) collects and classifies 370 proofs. See page 50 of Unit 8 for the book's proof. ###### Pythagorean Triples Three whole numbers (like 3, 4 and 5) such that the square of one of them is equal to the sum of the squares of the other two are said to form a **Pythagorean triple**. The Italian mathematician Leonardo Fibonacci (1170–1250) gave the following method for finding Pythagorean triples in one of his books. Take any odd square number. Add up all the odd numbers that are smaller than this number; this will give another square number. When you add the two square numbers together, you always get a third square number. That is, you’ve found a Pythagorean triple. There is strong evidence that the 3, 4, 5 Pythagorean triple was known to the ancient Egyptians and Babylonians long before Pythagoras. There is also evidence that the ancient Egyptians knew that the converse of Pythagoras’ Theorem is true and used it in practical situations to construct right angles; it is thought that they constructed 3, 4, 5 triangles using knots on a string in order to obtain accurate right angles. ###### Perimeters and Areas You can convert between units of area by considering how many of the smaller units fit into one of the larger units. For example, a square of side 1 m contains 100 × 100 = 10 000 squares of side 1 cm; thus $1m^2 = 10,000 m^2$ **Rectangles & Parallelograms** The formula for the area of a rectangle can be used to find a formula for the area of a parallelogram. The base of a parallelogram can be taken to be any of its sides, and its perpendicular height is then its height measured at right angles to the base **Triangles** $area = \frac{1}{2}bh$ **Circles** A line segment starting and ending on the circumference is called a chord. A chord that passes through the centre of the circle is called a diameter. Any unbroken section of the circumference is called an arc. The shape enclosed by an arc of a circle together with the two radii from the endpoints of the arc is called a sector. A segment is the shape enclosed by an arc and the chord joining the ends of the arc. The circumference of a circle of radius r is 2πr The area of a circle of radius r is $\pi r^2$. The circle is the shape with minimum perimeter for a certain area. ###### Pi & 祖冲之 Another approximation by a fraction that is worth mentioning is 355/113, which was discovered by Chinese mathematician Zu Chongzhi in about 480 CE. It is the closest approximation with the denominator below 1000, and is memorable because the sequence 113 355 appears when you read it from bottom to top. ###### 2D & 3D shapes 2D shapes are also known as plane shapes 3D shapes are also known as solid shapes ###### Prisms A solid geometric figure whose two end faces are similar, equal, and parallel rectilinear figures, and whose sides are parallelograms. A cuboid, or more informally a box, is a prism whose cross-section is a rectangle. So a cube is a special case of a cuboid. ###### Volume **Cylinder** $\Large V = \pi r^2 h$ **Sphere** $\Large V = \frac{4}{3} \pi r^3$ The sphere is the shape with minimum surface area for a certain volume **Cube** $\Large V = a^3$ **Triangular Prism** $\Large V = \frac{1}{2} a b h$ where $a$ is the height of the triangle $b$ is the length of the base of the triangle, and $h$ is the length of the prism Cone $\Large V = \frac{1}{3} \pi r^2 h$ ###### Surface Area Sphere $\Large A = 4 \pi r^2$ Cone $\Large A = \pi r^2 + \pi r l$ ###### Hypervolume Hypersphere (3-sphere) $\Large V = \frac{1}{2} \pi^2 r^4$ ### Unit 9 ###### Arithmetic Sequences An arithmetic sequence is a sequence with the property that the difference between consecutive terms is constant. Any list of numbers is called a sequence. For example, 1, 2, 3, . . . , 100 is an example of a finite sequence, whereas 1, 2, 3, . . . is an example of an infinite sequence. The numbers in a sequence are called the terms of the sequence. Many sequences of numbers can be pithily expressed. For instance: the sum of a sequence of odd numbers is equal to the square of the number of numbers being summed. If you sum 1, 3, 5, you get 9. You have three numbers and the square of 3 is also 9. What about for the sequence of natural numbers 1, 2, 3, 4 ... ? This one is harder to figure out, but creating a triangle of dots and then putting two of them into a square visually helps. The formula for this sequence is: $\Large 1 + 2 + 3\ ... +\ n = \frac{n(n + 1)}{2}$ The numbers given by the expression are called triangular numbers since they occur as the numbers of dots in a triangular array An arithmetic sequence is a sequence with the property that the difference between consecutive terms is constant. An arithmetic sequence is sometimes called an arithmetic progression, and the difference is often called the common difference. If the sequence is finite, we also give the number of terms, denoted by n. The first term a and the difference d can be any number, positive, negative or zero, but the number of terms n is always a positive integer. ###### Thomas Harriot Thomas Harriot was an English astronomer and mathematician. He worked for Sir Walter Raleigh (also spelt Ralegh), providing mathematical information on practical matters such as navigation and the optimal method of stacking cannonballs, and later for Henry Percy, Duke of Northumberland, who had a great interest in scientific questions. Unfortunately, Harriot published very little, but he is now credited with being the first person to view the Moon through a telescope and make a drawing of it, the first to view sunspots, and the first to state Snell’s law of refraction in optics. Harriot taught courses on navigation (based on ‘spherical’ trigonometry) to Raleigh’s seamen, and in mathematics he established many of the basics of algebra needed to solve quadratic equations and equations involving higher powers of the unknown x, such as $x^3$, drawing on the work of the French mathematician Viète. Harriot introduced a simplified notation for doing algebra (though he still wrote $a^2$ as aa, $a^3$ as aaa, and so on) and he understood the idea of factorisation of quadratics and expressions involving higher powers of the unknown x. He also used negative solutions of equations and even solutions that involved square roots of negative numbers. In this way, Harriot anticipated the much later development of complex numbers. ### Unit 10 ###### Parabolas The word ‘parabola’ was first used for curves like those in Figure 5 by the Greek geometer and astronomer Apollonius, in around 200 BCE, though the shape itself was discovered even earlier. The word means ‘juxtaposition’ or ‘application’ in Greek. Later writers thought this word appropriate because the plane shown in Figure 7 can be thought of as being ‘juxtaposed to’ the cone – parallel to its side. Quadratics, and thus parabolas, will take the following form as we know: $y=ax^2+bx+c$ The parabola will open upward if $a > 0$, and downward if not. ###### Projectiles & Ballistics A projectile is an object that is propelled through space by a force that ceases after launch, such as a ball that is thrown, or a cannonball that is fired from a cannon. The trajectory of a projectile is the path that it follows. The science of projectiles, especially those fired from firearms, is called ballistics. > [!tip] Galileo > Galileo, a Florentine, was appointed to a professorship of mathematics at the University of Pisa in 1589, by Fernando, Duke of Tuscany. Much of Galileo’s work on the military applications of mathematics was dedicated to him. The Duke’s eldest son, Cosimo II, was taught mathematics by Galileo. ###### Stopping Distance when Driving Different countries have different equations for determining stopping distance. In the UK, they use the equations in the Highway Code. It defines to equations: Thinking Distance $T = s$ where $T$ is the distance covered in feet just to think about stopping, and $s$ is the speed in mph. Thus, if you're going 50 mph, you will cover 50 feet before applying the brakes. Braking Distance $B = \frac{1}{20} s^2$ where $B$ is the distance covered in feet to come to a stop after applying the brakes, and $s$ is the speed in mph. Thus, if you're going 50 mph, the braking distance is 125 feet Stopping Distance $D_S = T + B$ Using the previous two calculations, we can show that the total stopping distance in this scenario would be 175 feet according to the UK Highway Code. ###### The Golden Rectangle There is a story that the ancient Greeks believed that the most aesthetically pleasing shape of rectangle was the shape such that if you cut it into a square and a smaller rectangle, then the smaller rectangle has the same shape – that is, the same aspect ratio. This shape is known as the golden rectangle. ###### The Quadratic Formula The first person to give a formula for solving quadratic equations was the Indian mathematician Brahmagupta, in 628. He described the formula in words, but it was essentially the same as the modern quadratic formula. $\Large x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ The value $b^2 − 4ac$ is called the **discriminant** of the quadratic expression $ax^2 +bx+c$, and it can be represented as $\Delta$, thus: $\Large \Delta = b^2 - 4ac$ A quadratic equation has: 1. Two real solutions if the discriminant is positive, thus: 1. $\Large \frac{-b + \sqrt{ \Delta}}{2a}$, and $\Large \frac{-b - \sqrt{ \Delta}}{2a}$ 2. One real solution if the discriminant is zero, sometimes called a repeated or double root or two equal roots, thus: 1. $\Large -\frac{b}{2a}$ 3. No real solutions if the discriminant is negative, but two complex solutions exist, which are complex conjugates of each other. 1. $\Large \frac{-b}{2a} + i \frac{\sqrt{\Delta}}{2a}$, and $\Large \frac{-b}{2a} - i \frac{\sqrt{\Delta}}{2a}$ Example: $9x^2 +30x+25=0$ Setting it up as a discriminant function: $30^2 - 4(9)(25)$ We get 0, so the quadratic equation has one real solution. There are no complex solutions because the discriminant is zero, indicating a perfect square trinomial and a double root. ###### Complex Solutions to the Quadratic Formula Although some quadratic equations have no solutions among the real numbers, all quadratic equations have either one or two solutions among the complex numbers. Remember that the domain of complex numbers includes real numbers. The complex numbers consist of all the usual real numbers, together with many ‘imaginary’ numbers, including the square roots of negative numbers. Finding the imaginary solutions of a quadratic equation can be more useful than you might think! For example, they are used in many engineering mathematical models, such as those used to design car suspensions. You can learn about the complex solutions of quadratic equations if you go on to study more mathematics. The first extensive discussion of complex numbers, including the formulation of how to add, subtract, multiply and divide them, was provided by the sixteenth-century Italian mathematician and engineer Rafael Bombelli in his book Algebra of 1572. Example of a quadratic equation with only non-real solutions $\Large x^2 + 4x + 5 = 0$ Its solutions will take the form: $\Large \frac{-b}{2a} + i \frac{\sqrt{\Delta}}{2a}$, and $\Large \frac{-b}{2a} - i \frac{\sqrt{\Delta}}{2a}$ Let's start with the discriminant: $\Large \Delta = 4^2 - 4(5)$ $\Large \Delta = 16 - 20$ $\Large \Delta = -4$ We can put this back into the two solutions, and then we just need to simplify: $\Large -2 + \frac{\sqrt{-4}}{2}$ If we simplify the square root, we get $2i$ because $i = \sqrt{-1}$. Since it's divided by 2, the 2's cancel out, and we're left with $i$ Thus, we get $\Large x = -2 + i$, and $\Large x = -2 - i$ ###### Completing the Square to solve Quadratics The Babylonians in about 1850–1650 BCE were able to solve problems equivalent to quadratic equations. Their method was essentially one of completing the square, but they found only positive solutions, as their problems involved positive quantities such as length. Steps to complete the square 1. Move constant to the other side 2. Add $(\frac{b}{2})^2$ to both sides 3. Factor Completing the square is used in 1. solving quadratic equations, 2. deriving the quadratic formula, 3. graphing quadratic functions, 4. evaluating integrals in calculus, such as Gaussian integrals with a linear term in the exponent, 5. finding Laplace transforms. Example: $\Large x^2+6x+8=0$ This quadratic is not a perfect square, since it is not a perfect square trinomial. So how do we know if the polynomial is a perfect square trinomial? The second coefficient must be equal to two times the square root of the product of the coefficients of the first and third terms. Just the coefficients! $\Large b = 2 \sqrt{ac}$ Since this is not the case, use the formula to complete the square, thus: First move the constant $\Large x^2+6x=-8$ Find the value of $(\frac{b}{2})^2$ $\Large (\frac{b}{2})^2 = (\frac{6}{2})^2 = 9$ So add 9 to both sides $\Large x^2+6x + 9= 1$ Now you can factor as we have a perfect square trinomial $\Large (x + 3)^2 = \pm 1$ We can go further to find the roots of the quadratic by solving this. !! Note that the above method is the "easy way", and the hard way is to use the FULL formula as seen below. ###### Completed Square Form The format of the completed square form is as follows: $\Large a\left( x - h \right)^2 + k$ where $h$ and $k$ are the coordinates of the parabola's vertex, $(h, k)$ We can substitute values here using the formula as follows: $\Large a\left( x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}$ Example: $\Large x^2+6x+8=0$ Filling out the formula gives us: $\Large \left( x + \frac{6}{2}\right)^2 + 8 - \frac{6^2}{4}$ Simplify: $\Large (x + 3)^2 - 1$ This is the completed square form. We can move the constant to the other side to solve for its roots now. Now, take the square root of both sides: $\sqrt{(x + 3)^2} = \sqrt{1}$ This simplifies to: $x + 3 = \pm 1$ Now, we could solve for both solutions. ###### Finding the maximum value of a parabola If you have a quadratic equation, such as the following: $\Large y = x^2+6x+8$ You can find its maximum value, its vertex, by using the following formula to find the vertex's x value: $\Large x = -\frac{b}{2a}$ Which allows us to find the y value by substituting this back into the original equation. $\Large x = -\frac{6}{2}$ Thus, $x = -3$ $\Large y = -3^2+6(-3)+8$ $= 9 -18 + 8$ $= -1$ Thus $\Large (-3, -1)$ ## Unit 11 ##### Boxplots The boxplot was invented by the American statistician John Tukey (1915–2000). Tukey was also partly responsible for the word ‘software’, referring to computer programs as opposed to the machines that they run on (the hardware) Adjacent values In a boxplot, the whiskers are drawn outwards as far as observations called adjacent values. The lower adjacent value is the lowest data value that is within one and a half times the IQR of the lower quartile (Q1). The upper adjacent value is the highest data value that is within one and a half times the IQR of the upper quartile (Q3). ##### Histograms Histograms are usually used to represent continuous data, such as measurements. However, if the data values in a dataset are discrete but take a large number of different values, then they are often treated as continuous and can be represented by a histogram. For example, you could draw a histogram of examination scores, which may take any whole-number value from 0 to 100. ##### Bar Charts If the data values in a dataset are discrete and take only a small number of values, then it is usual to represent them not by a histogram, but by a similar-looking type of statistical chart called a bar chart. When a dataset is depicted by a bar chart, each bar corresponds to a single possible data value, rather than an interval of possible data values. The bars in a bar chart are drawn with gaps between them to reflect this fact. ##### Skewness To estimate a distribution’s skewness from a boxplot, look at the relative positions of the median, the quartiles, and the whiskers: 1. **Median Position Within the Box** - If the median (the line inside the box) is closer to the top (Q3) than to the bottom (Q1), it often indicates a **left (negative) skew**, because a longer tail extends toward smaller values. - If the median is closer to the bottom (Q1) than to the top (Q3), it often indicates a **right (positive) skew**, because a longer tail extends toward larger values. 2. **Length of Whiskers** - If the upper whisker is much longer than the lower whisker, it suggests a right (positive) skew. - If the lower whisker is much longer than the upper whisker, it suggests a left (negative) skew. 3. **Outliers** - Outliers on one side of the box can accentuate the tail on that side and thus signal skewness in that direction. In many practical cases, you look at both the median’s shift inside the box and the relative whisker lengths to gain a sense of which side (if any) is elongated, revealing the skew of the distribution. ## Unit 12 ##### Angle of elevation The angle of elevation is the angle between the horizontal line and the line of sight which is above the horizontal line. It is formed when an observer looks upwards. Suppose you are standing at the terrace of a building and looking upwards at the sky or at the sun or moon. The angle thus formed between your height from the ground level and the line of sight formed is called the angle of elevation. ##### Trigonometric Functions are Ratios The base trig functions return a ratio when given an angle, which is also a specific point. SOH CAH TOA $\Large \sin \Theta = \frac{opp}{adj} \quad \therefore \quad \frac{y}{r}$, and it returns $y$ $\Large \cos \Theta = \frac{adj}{hyp} \quad \therefore \quad \frac{x}{r}$, and it returns $x$ $\Large \tan \Theta = \frac{opp}{adj} \quad \therefore \quad \frac{y}{x}$, and it returns the slope > [!tip] Note > It's important to remember that trigonometric functions thus return ratios, so the value they return is simply the quotient of these sides, i.e. how long one is to another. > [!faq] Is it really just a ratio? > > While yes they return a ratio, this ratio is also a specific value as well. > > The sine function returns the y coordinate of the corner of the triangle where it intersects the circle’s perimeter. > > Conversely, cosine takes the angle of the triangle and returns the x coordinate. > > The tangent function returns the slope of the line passing through the origin and the point (cosθ, sinθ) on the unit circle. ##### Origins of Functions The word sine has its origins in Sanskrit, coming to us through Arabic and Latin. The Latin word sinus appeared in European mathematics texts in the twelfth century. Arguably the earliest table of trigonometric values is the table of lengths of chords in a circle in Claudius Ptolemy’s Almagest, a book on mathematical astronomy, written in about 150 CE and known to us through later Arabic translations. In the Almagest, Ptolemy tabulated the lengths of the sides of many right-angled triangles, taking the hypotenuse to be of length 60. For centuries, values of sines, cosines and tangents of acute angles were available only in mathematical tables. Ptolemy’s work became widely known in Europe after the publication in 1496 of the Epitome of the Almagest, by the German astronomers Georg Peurbach and Johannes Müller von Königsberg, known as Regiomontanus; for example, it was used by Copernicus and Galileo. The first person to relate sines and cosines directly to angles in a triangle as we do today was the Austrian astronomer and mathematician Georg Joachim Rheticus, in his pamphlet Canon doctrinae triangulorum of 1551. Rheticus’ masterwork, his immense Opus palatinum de triangulis of 1596, which ran to some 1500 pages, contained tables calculated to ten decimal places, which were of such accuracy that they were considered the standard until the early twentieth century. ##### Inverse Trig Functions Inverse trig functions return an angle when given the ratio of two sides. $\Large \arcsin (\frac{opp}{hyp}) = \Theta$ $\Large \arccos (\frac{adj}{hyp}) = \Theta$ $\Large \arcsin (\frac{opp}{adj}) = \Theta$ > [!tip] Notation > arc notation is best and least ambiguous, but it's important to remember that these inverse trig functions can also be written as: > > $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$ > ##### Trig Identities The trig functions can also be expressed being equal to other trig functions, thus we can say the following: $\Large \tan \Theta = \frac{\sin \Theta}{\cos \Theta}$ ##### Pythagorean Identity with Trig Functions $\Large \sin^2 \Theta + \cos^2 \Theta = 1$ > [!faq] Why does this work? > > This works because $\sin^2 = y^2$, $\cos^2 = x^2$, and the radius, $c^2$, equals $1^2$. ##### The Law of Sines **Oblique triangles** are triangles that do not have a right angle. Note that you **cannot** use the Pythagorean Theorem for oblique triangles. We must use trigonometry to determine the lengths of its sides. $\Large \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ > [!tip] Important > The Law of Sines can only be used when you have SAA or ASS where the angle is non-included. > > **A non-included angle** is when the angle is not between the two known sides. ##### Law of Cosines The Law of Cosines can only be used when you either known all three sides, SSS, or when you know two sides and an included angle, SAS. There are three formulas to use, and which one you use depends on what information you have. $\Large \cos A = \frac{b^2 + c^2 - a^2}{2bc}$ $\Large \cos B = \frac{a^2 + c^2 - b^2}{2ac}$ $\Large \cos C = \frac{a^2 + b^2 - c^2}{2ab}$ > [!Note] > Once you use the Law of Cosines, it is very likely your next step will be to use the Law of Sines to gather the remaining information > [!tip] >Book II of Euclid’s Elements contains a geometric theorem equivalent to the Cosine Rule. It was put into its present useful form by the fourteenth-century Persian mathematician Jamshid Al-Kashi, and is still known in France as ‘Le Théorème d’Al-Kashi’. ##### Heron's Formula In many cases calculating the area of a triangle would be easier if there were a simple formula for the area of a triangle in terms of just the lengths of the three sides. Such a formula was derived by the Greek mathematician Heron around 62 ce and later found independently by Chinese mathematicians. Heron of Alexandria taught physics and engineering, as well as mathematics. Heron’s Formula is derived in Book 1 of his textbook Metrica. $\Large Area = \sqrt{s(s- a)(s - b)(s - c)}$ where $s = \frac{1}{2}(a + b + c)$ ##### Length of an arc of a circle $\Large LoA = r\Theta$ where r is the radius of the circle and θ is the angle subtended by the arc, measured in radians. ##### Area of a Sector $\Large AoS = \frac{1}{2} r^2 \Theta$ ![[Sector.png]] ### Exponentials and Logarithms ##### Exponential Growth & Decay Exponential growth is growth that arises from repeated multiplication by the same number. The number that you start with is called the starting number, and the number that you multiply by is called the scale factor (or multiplication factor). If the starting number is $a$ and the scale factor is $b$, then the value after $n$ steps is $ab^n$ Suppose that a positive quantity changes in steps, where its value at each step is obtained from its value at the previous step by multiplying by the same scale factor, which is between 0 and 1 (exclusive). Then the quantity is said to decay exponentially. ##### Percentages and Increases & Decreases by Factors Suppose, for example, that you want to increase the price of an item costing £18 by 15%. Since 100% + 15% = 115%, this means that the new price is 115% of the old price. The new price is $£18 ×1.15 = £20.70$ If it's easier to think about with a formula, then the following can be used: To increase a number by $r\%$: $\Large \frac{100 + r}{100}$ Conversely: $\Large \frac{100 - r}{100}$ ##### Exponential Regression Just as linear regression exists, exponential regression also exists for plotting lines along exponentially growing or decaying data points. In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the outcome or response variable, or a label in machine learning parlance) and one or more error-free independent variables (often called regressors, predictors, covariates, explanatory variables or features) # Statistics ### Introduction & Basics ##### Scope of Statistics The scope of statistics was the topic of some debate in the early days of the discipline. In 1834, the Statistical Society of London was founded, with a wheatsheaf as its emblem. This was inscribed with the Latin motto ‘aliis exterendum’, translated as ‘to be threshed by others’, thou8gh the accuracy of this translation has since been disputed. This was interpreted to mean that while statisticians collect and present data (the wheat, in this analogy), they should leave the interpretation of data (the threshing) to others. This was hotly disputed at the time, the opposing view – generally accepted by modern statisticians – being that statistics also involves interpreting and drawing inferences from data. Eventually, the Latin motto was dropped. The Royal Statistical Society, which succeeded the Statistical Society of London, originally used the wheatsheaf as its emblem, as shown in Figure 2(a). It now uses the emblem shown in Figure 2(b). ##### False security: a costly rounding error During the first Gulf War, an American Patriot missile system in Saudi Arabia failed to intercept an incoming Iraqi missile. The missile destroyed an American Army barracks, killing 28 soldiers and leaving many wounded. The cause of the failure of the missile system was eventually tracked down to a rounding error in the system’s internal clock, which grew as time went on. ##### Stemplots, or Stem-and-leaf plots The three basic elements in the stemplot are the stem, the levels and the leaves. A stemplot is like a histogram — they are both tools to help you visualize a data set. Stemplots show a little more information than a histogram and have been a common tool for displaying data sets since the 1970s. They are typically used when there is a medium amount of quantitative variables to analyze; Stemplots of more than 50 observations are unusual. ![[stemplot3.jpg]] They were widely used before the advent of the personal computer, as they were a fast way to sketch data distributions by hand. They are used less frequently today, but you’ll still see some here and there. The numbers are arranged by place value. The largest place-value digits are placed in the stem. Stemplots in the real world aren’t usually labeled with the place-values that the stem represents — it’s usually up to you to figure it out based on the context and the data. However, in textbooks and other education materials it’s common for stemplots to be labeled. For example, in the above image, all the numbers in level of the 2 stem would be 21, 24, 25, 25, 26. It is shorthand for these values. ### Percent, Permille, & Permyriad | Word | Symbol | | --------- | ------ | | Percent | % | | Permille | ‰ | | Permyriad | ‱ | Most people are familiar with percent, meaning per 100; but permille and permyriad also existing, meaning per 1000 and per 10000, respectively. To put it into context, 1% is equivalent to 10‰ and 100‱. In day-to-day life, permille can be used to report blood alcohol content. Some countries use percent instead. For example, in the US, driving with a BAC 0.08% is illegal and this is the most common way to write it there. However, this can also be written as 0.8‰. In finance, the related term basis point means the same as permyriad, however the contexts they're used in are different. Changes of interest rates are often stated in basis points. For example, if an existing interest rate of 10 percent is increased by 1 basis point, the new interest rate would be 10.01 percent. Like percentage points, basis points avoid the ambiguity between relative and absolute discussions about interest rates by dealing only with the absolute change in numeric value of a rate. For example, if a report says there has been a "1% increase" from a 10% interest rate, this could refer to an increase either from 10% to 10.1% (relative, 1% of 10%), or from 10% to 11% (absolute, 1% plus 10%). However, if the report says there has been a "100 basis point increase" from a 10% interest rate, then the interest rate of 10% has increased by 1.00% (the absolute change) to an 11% rate ### Miscellaneous This section holds notes I have not yet sorted into appropriate sections #### Sigma Notation $\Large \sum_{i=1}^6 2i = 42$ In sigma notation, the number above sigma represents the number your variable will go up to, in this case 6, so we would substitute i for numbers until getting to 6. The number below sigma is the number you will start at. The equation to the right of sigma is the one you will fill in values for to get individual answers, and then finally, each of those results will be summed. Thus, the notation above could have been written as: $\Large (2\cdot 1)+(2\cdot 2)+(2\cdot 3)+(2\cdot 4)+(2\cdot 5)+(2\cdot 6)$ Conversely, if one wanted to represent the sequence of subtraction, one would just use the negative of a summation: $\Large \sum_{i=1}^6 (-2i) = -42$ #### Pi Notation $\Large \prod_{i=1}^{6} 2i = 46,080$ In pi notation, the number above pi represents the number your variable will go up to, in this case 6, so we would substitute i for numbers until getting to 6. The number below pi is the number you will start at. The equation to the right of pi is the one you will fill in values for to get individual answers, and then finally, each of those results will be multiplied. Thus, the notation above could have been written as: $\Large (2\cdot 1)(2\cdot 2)(2\cdot 3)(2\cdot 4)(2\cdot 5)(2\cdot 6)$ Conversely, if one wanted to represent the sequence of division, one would just use the product of the reciprocals: $\Large \prod_{i=1}^{6} \frac{1}{2i} = \frac{1}{46,080}$ autodacty # History of Math ### Algebraic Notation The history of algebraic notation reflects the evolution of mathematical thought and communication over millennia, progressing through three main stages: rhetorical, syncopated, and symbolic algebra. Each stage represents a significant shift in how mathematicians expressed and solved equations, moving from verbose descriptions to the concise symbolic language used today. **Rhetorical Algebra** was prevalent in ancient civilizations such as Babylonian (c. 2000–1600 BCE) and early Greek mathematics. During this period, mathematical problems and solutions were written entirely in words without any symbols. For example, a Babylonian scribe might express an equation as: "If I add seven to twice my number, I get twenty-one." This approach made complex calculations cumbersome and limited the ability to generalize solutions. The transition to **Syncopated Algebra** began around the 3rd century CE with mathematicians like Diophantus of Alexandria. Syncopated algebra introduced abbreviations and some symbolic representations, serving as an intermediary between rhetorical and symbolic algebra. Diophantus, often called the "Father of Algebra," used symbols for unknowns and operations, such as using the Greek letter sigma (ς) to represent the unknown and abbreviations for powers and operations. In medieval Islamic mathematics (c. 9th–12th centuries), scholars like Al-Khwarizmi also employed syncopated notation, blending words with abbreviations to simplify expressions. An example of syncopated algebra in Arithmetica is as follows: $K^{\upsilon} \bar{\alpha}\ \zeta \bar{\iota}\ \pitchfork\ \Delta^{\upsilon} \bar{\beta}\ M \bar{\alpha}\ ἴσ\ M \bar{\epsilon}$ Where: $\bar{\alpha}$ is 1, as it is the 1st letter of the Greek alphabet $\bar{\beta}$ is 2 $\bar{\epsilon}$ is 5 $\bar{\iota}$ is 1 $ἴσ$ is equals, short for ἴσος $\pitchfork$ is subtraction of everything that follows up to ἴσ $M$ is the zeroth power, thus making something 1 $\zeta$ is equivalent to modern $x$ variable $\Delta^{\upsilon}$ is the second power, from δύναμις meaning "power" $K^{\upsilon}$ is the second power, from κύβος meaning "cube" $\Delta^{\upsilon} \Delta$ is the fourth power $\Delta^{\upsilon} K$ is the fifth power $K^{\upsilon} K$ is the sixth power A literal symbol-for-symbol translation of Diophantus's syncopated equation into a modern symbolic equation would be the following: ${x^{3}}1{x}10-{x^{2}}2{x^{0}}1={x^{0}}5$ if the modern parentheses and plus are used then the above equation can be rewritten as: $(x^3 1 + x10) - (x^2 2 + x^0 1) = x^0 5$ The development of **Symbolic Algebra** emerged during the Renaissance, revolutionizing mathematical notation. Pioneers like François Viète (1540–1603) introduced the use of letters to represent both known and unknown quantities systematically. René Descartes (1596–1650) further refined this notation by popularizing the use of $x, y, z$ for unknowns and $a, b, c$ for known quantities, as well as introducing the exponential notation $x^n$. An example of symbolic algebra is the modern equation $x^2 + 5x = 14$, which concisely represents relationships and allows for generalization and manipulation of equations. This evolution from rhetorical to symbolic algebra significantly enhanced the efficiency and clarity of mathematical communication. The adoption of symbols allowed mathematicians to develop more complex theories, share ideas more effectively, and laid the groundwork for advances in various fields of science and engineering.