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# Mathematics

## Natural numbers and integers

The positive whole numbers are the numbers that we use for counting. They are called the natural numbers.

$1$ and $10$ are natural numbers. $3.2$, $1/4$ and $-1$ are not.

The positive and negative whole numbers are collectively called integers.

$-1$, $-10$ and $5$ are integers. $-1/3$ and $2.4$ are not

The set of the natural numbers is written as $\mathbb{N}$. The set of the integers is written as $\mathbb{Z}$.

There are infinitely many integers.

This Babylonian clay tablet shows arithmetic using natural numbers. Babylonians also used fractions and square roots, but no negative numbers or zero.

## Laws of indices

There are three laws of indices. They relate operations on exponents with the same base to operations on the indices.

• Addition of indices: When the exponents are multiplied, the powers are added. $$\Tblue{3}^\Tred{2} \times \Tblue{3}^\Torange{5} = (\Tblue{3} \times \Tblue{3}) \times (\Tblue{3} \times \Tblue{3} \times \Tblue{3} \times \Tblue{3} \times \Tblue{3}) = \Tblue{3}^7 = \Tblue{3}^{\Tred{2}+\Torange{5}}$$
• Subtraction of indices: When the exponents are divided, the powers are subtracted. \begin{align*} \Tblue{2}^\Tred{4} \div \Tblue{2}^\Torange{2} &= (\Tblue{2} \times \Tblue{2} \times \Tblue{2} \times \Tblue{2}) \div (\Tblue{2} \times \Tblue{2}) \\&= \frac{\Tblue{2} \times \Tblue{2} \times \Tblue{2} \times \Tblue{2}}{\Tblue{2} \times \Tblue{2}} = \Tblue{2}^2 = \Tblue{2}^{\Tred{4}-\Torange{2}} \end{align*}
• Multiplication of indices: When an exponent is raised to a power, the powers are multiplied. $$(\Tblue{3}^\Tred{2})^\Torange{3} = (\Tblue{3} \times \Tblue{3}) \times (\Tblue{3} \times \Tblue{3}) \times (\Tblue{3} \times \Tblue{3}) = \Tblue{3}^6 = \Tblue{3}^{\Tred{2} \times\Torange{3}}$$
Index Exponent Formula Example
$+$ $\times$ $\Tblue{a}^{\Tred{x}+\Torange{y}}=\Tblue{a}^\Tred{x}\Tblue{a}^\Torange{y}$ $\Tblue{2}^{\Tred{2}+\Torange{1}} = \Tblue{2}^\Tred{2}\cdot \Tblue{2}^\Torange{1} = \Tblue{2}^3$
$-$ $\div$ $\displaystyle \Tblue{a}^{\Tred{x}-\Torange{y}}=\frac{\Tblue{a}^\Tred{x}}{\Tblue{a}^\Torange{y}}$ $\displaystyle \Tblue{2}^{\Tred{5}-\Torange{2}} = \frac{\Tblue{2}^\Tred{5}}{\Tblue{2}^\Torange{2}} = \Tblue{2}^3$
$\times$ Power $(\Tblue{a}^{\Tred{x}})^{\Torange{y}}=\Tblue{a}^{\Tred{x}\Torange{y}}$ $(\Tblue{2}^\Tred{2})^\Torange{2}= \Tblue{2}^{\Tred{2}\times \Torange{2}} = \Tblue{2}^4$

## Introduction to standard form

Small or big numbers are generally written in standard form or scientific notation. This makes the numbers easier to write, shows the scale of the number, and makes it easy to compare quantities.

A positive number is in standard form if it is written as $$\Tblue{A} \times 10^\Tred{n}.$$ The number $\Tblue{A}$ must be between 1 (included) and 10 (excluded) in absolute value. It is sometimes called the mantissa. The number $\Tred{n}$ is an integer, called the exponent.

$7600$ is written in standard form $\Tblue{7.6}\cdot 10^\Tred{3}$. The mantissa is $\Tblue{7.6}$ and the exponent is $\Tred{3}$. The scientific notation for $-0.02$ is $\Tblue{-2}\cdot 10^{\Tred{-2}}$

Some examples:

\begin{align*} 123.4 & = \Tblue{1.234}\times 10^\Tred{2},\quad-32,\!124 = \Tblue{-3.2124}\times 10^{\Tred{5}},\\ &0.32 = \Tblue{3.2}\times 10^{\Tred{-1}},\quad 0.032 = \Tblue{3.2}\times 10^{\Tred{-2}}. \end{align*}

The exponent gives the position of the first significant figure.

$100 = 10^\Tred{2}$ has $\Tred{2}+1 = 3$ digits left from the decimal point.

$0.01 = 10^{\Tred{-2}}$ has $\Tred{2}-1 = 1$ zero right from the decimal point.

Saturn is 116 000 000 m in diameter. E. coli cells are 0.000 002 m long. These numbers are easier to handle using standard form.

## Factors and multiples

A number is a multiple of another number if it is the product of this number with an integer.

An integer is a divisor of a number if the ratio of the number with the integer is an integer. It is also called a factor.

$\Tblue{6}$ is a multiple of $\Tred{2}$, $\Tred{2}$ is a divisor of $\Tblue{6}$, since $\displaystyle\frac{\Tblue{6}}{\Tred{2}} = 3$ is an integer.

To rephrase, take two integers $\Tblue{m}$ and $\Tred{d}$. $\Tblue{m}$ is a multiple of $\Tred{d}$ and $\Tred{d}$ is a divisor of $\Tblue{m}$ if the ratio $\displaystyle \frac{\Tblue{m}}{\Tred{d}}$ is an integer.

$\Tred{4}$ is a factor of $\Tblue{12}$, because $\displaystyle\frac{12}{\Tred{4}} = 3$ is an integer

$\Tred{8}$ is not a factor of $\Tblue{12}$ because $\displaystyle\frac{12}{\Tred{8}} = 1.5$ is not an integer.

60 has many factors. An hour can be divided into 12 times five minutes.

## Introduction to surds

The roots of most rational numbers cannot be written as fractions. They are irrational numbers. A surd is the root of a rational number that is irrational. A surd is also called a radical.

$\Tred{\sqrt{2}}$, $\Tred{\sqrt{3}}$, $\Tred{\sqrt{1/2}}$, $\Tred{\sqrt{8}}$ are surds.

$\Tblue{\sqrt{4}} = 2$, $\Tblue{\sqrt{1/9}} = 1/3$ and $\Tblue{\sqrt{9/16}} = 3/4$ are not surds because the roots are rational numbers.

In practice, we don't simplify surds and keep the roots $\sqrt{a}$ in the expressions.

We can simplify $\Tblue{\sqrt{4}}$ to $2$, but not $\Tred{\sqrt{2}}$.

$$1-\Tblue{\sqrt{4}} +\Tred{\sqrt{2}} = 1-\Tblue{2}+\Tred{\sqrt{2}} = -1 + \Tred{\sqrt{2}}.$$

To check if a number is a surd, write it in decimal form with a calculator. If you find recurring decimals, it is not a surd. Otherwise it is.

$\Tred{\sqrt{5}}$ is a surd because it has no recurring decimals.

$$\Tred{\sqrt{5}}\simeq 2.2360679774998$$

The sides of a square with area two have length equal to the square root of two. This is a surd.

## Addition and subtraction of fractions

If you want to add or substract fractions such as $$\displaystyle \frac{1}{\Tred{8}} + \frac{5}{\Tblue{6}},$$ you need to go through three steps.

• Re-write the fractions so that they have the same denominator. This can be done by cross multiplying by the denominators. $$\frac{1}{\Tred{8}} + \frac{5}{\Tblue{6}} = \frac{1\times\Tblue{6}}{\Tred{8}\times\Tblue{6}} + \frac{5\times\Tred{8}}{\Tblue{6}\times \Tred{8}} = \frac{6}{\Tviolet{48}} + \frac{40}{\Tviolet{48}}$$
• Add or subtract the numerators $\displaystyle \frac{\Tgreen{6}}{\Tviolet{48}} + \frac{\Tgreen{40}}{\Tviolet{48}} = \frac{\Tgreen{6} + \Tgreen{40}}{\Tviolet{48}}= \frac{\Tgreen{46}}{\Tviolet{48}}$
• Simplify the resulting fraction $\displaystyle \frac{46}{48} = \frac{23\times\Torange{2}}{24\times\Torange{2}} = \frac{23}{24}$

Here are some other examples

\begin{align*} &\frac{5}{6} + \frac{2}{3} = \frac{15 + 12}{18} = \frac{27}{18} = \frac{3}{2},\; &\frac{3}{5} + \frac{1}{3} = \frac{9 + 5}{15} = \frac{14}{15},\\ &\frac{5}{6} - \frac{2}{3} = \frac{15 - 12}{18} = \frac{3}{18} = \frac{1}{6}, \; &\frac{3}{5} - \frac{1}{3} = \frac{9 - 5}{15} = \frac{4}{15} \end{align*}
Fractions can be added when the denominators are the same.

## Decimal representation of rational numbers

Every number has a unique decimal representation.

• If the prime factorisation of the denominator of a rational number only has $2$s and $5$s, the decimal has finite length. It is called a terminating decimal. $$\frac{1}{2} = 0.\Tblue{5},\quad \frac{1}{5} = 0.\Tblue{2},\quad \frac{1}{8} = 0.\Tblue{125}, \quad\frac{231}{125} = 1.\Tblue{848}.$$
• All other rational numbers have recurring decimals. This means that they have an infinite decimal pattern that repeats itself. The recurring part is often written with a dot or a bar on top. \begin{align*} \frac{1}{3} &= 0.\Tgreen{3}\Tblue{3}\Tviolet{3}\dots = 0.\dot{\Tgreen{3}},\quad \frac{1}{7} = 0.\overline{\Tgreen{142857}},\\ &\frac{231}{162} = 1.4\Tgreen{259}\Tblue{259}\Tviolet{259}\ldots = 1.4\overline{\Tgreen{259}} \end{align*}
• Irrational numbers have infinite non-recurring decimals. $$\sqrt{2}= 1.\Torange{414\;213\;562\dots}, \quad \pi = 3.\Torange{141\;592\;653\dots}$$
Timings in races are given as decimals.

## What is a ratio?

A ratio is a relationship between two or more quantities. It expresses the relative proportions of various quantities. A ratio is written with a colon ($:$), as in $3:4$.

A café makes biscuits using $\Tblue{2}$ kilos of flour and $\Tred{1}$ kilo of sugar. The ratio of flour to sugar is $\Tblue{2}:\Tred{1}$.

I make biscuits with $\Tblue{8}$ ounces of flour and $\Tred{4}$ ounces of sugar. The ratio flour to sugar is the same $\Tblue{8}:\Tred{4} = \Tblue{2}:\Tred{1}$, because I need twice as much flour as I need sugar.

Ratios can compare more than two quantities.

My recipe uses as much butter as it does sugar and twice as much flour. The ratio of flour to sugar to butter is $\Tblue{2}:\Tred{1}:\Tviolet{1}$.

Recipes use ingredients in particular ratios.

## Direct proportion

Two variables $\Tblue{a}$ and $\Tgreen{b}$ are proportional, or in direct proportion, if the ratio $\Tblue{a}/\Tgreen{b}$ is constant for all values of the variables.

At constant speed, distance covered is proportional to time.

We write $\Tblue{a}\,\Torange{\propto}\,\Tgreen{b}$ using the proportionality symbol $\Torange{\propto}$.

The ratio $\Tred{k}$ between proportional variables is the proportionality constant. $$\Tblue{a}=\Tred{k}\,\Tgreen{b},\qquad \Tblue{a}:\Tgreen{b} = 1:\Tred{k}.$$

Several variables are collected in the table below. The variables $\Tblue{a}$ and $\Tgreen{b}$ are proportional with proportionality constant $\Tred{3}$. The variables $\Tblue{a}$ and $\Tgreen{c}$ are not proportional.

 Variable Variable Ratio Variable Ratio $\Tblue{a}$ $\Tblue{3}$ $\Tblue{6}$ $\Tblue{9}$ $\Tblue{12}$ $\Tblue{15}$ $\Tgreen{b}$ $\Tgreen{1}$ $\Tgreen{2}$ $\Tgreen{3}$ $\Tgreen{4}$ $\Tgreen{5}$ $\Tred{a/b}$ $\Tred{3}$ $\Tred{3}$ $\Tred{3}$ $\Tred{3}$ $\Tred{3}$ $\Tgreen{c}$ $\Tgreen{3}$ $\Tgreen{12}$ $\Tgreen{27}$ $\Tgreen{48}$ $\Tgreen{75}$ $\Tred{a/c}$ $\Tred{1}$ $\Tred{0.5}$ $\Tred{0.33}$ $\Tred{0.25}$ $\Tred{0.2}$
The price paid is directly proportional to the weight of the fruit. The proportionality constant depends on the units used to weigh the fruit.

## What is a percentage?

A percentage is one-hundredth of a number. It uses the symbol $\%$. A percentage is a fraction where the denominator is $\Tblue{100}$.

$$\Tred{50}\% = \frac{\Tred{50}}{\Tblue{100}} = \frac{1}{2} = 0.5,\quad \Tred{40}\% = \frac{\Tred{40}}{\Tblue{100}} = \frac{2}{5} = 0.4.$$

To express a number as a percentage, write it in decimal form and multiply it by $\Tblue{100}$.

$$0\Torange{.43}2 = \Tblue{100}\times 0\Torange{.43}2\% = \Torange{43.}2\%,\quad \frac{2}{7} = 0\Torange{.28}6 = \Torange{28.}6\%$$

Percentages are often used to represent a fraction of a quantity.

Half a cake is the same as $50\%$ of the cake.

$25\%$ of $247$ is $\Torange{25}\%\times 247 = 0\Torange{.25}\times 247 = 61.75$

$26$ is $14.6\%$ of $178$ because $\displaystyle \frac{26}{178} = 0\Torange{.14}6 = \Torange{14.}6\%.$

## Expressions and variables

An algebraic expression is a formula using letters.

$$\Tred{x}+3,\quad 2\times \Tred{n},\quad (\Tred{x}+\Tred{y})^2,\quad 3\Tred{x} - 2,\quad \Tred{t}^2.$$

The letters are called the variables, because they can take several values or inputs. The different values that a letter can take is the range.

The value of the expression is called the output.

To compute the expression, we substitute the letter with its value and we work out the answer.

The value of the expression $$\Tred{x}+3$$ for the input $\Tred{x}=\Tblue{2}$ is the number $\Tblue{2} + 3 = 5$.

The values of an expression can be represented in table.

Table for the expression $\Tred{x}+3$.

 Input Calculation Value $-1$ $0$ $2$ $3$ $5$ $\Tblue{-1}+3$ $\Tblue{0}+3$ $\Tblue{2}+3$ $\Tblue{3}+3$ $\Tblue{5}+3$ $2$ $3$ $5$ $6$ $8$

A quadratic expression is an expression involving a variable and its square.

$\Tred{t^2}-1$, $\Tred{x^2}+\Tred{x}-1$, $\Tred{y}-3$ are quadratic expressions.

$\Torange{t^3}-\Tred{t}$ and $\Torange{\sqrt{x}}+\Tred{x}-1$ are not quadratic expressions.

Factorisation and simplification of quadratic expressions use the following formulae.

\begin{align*} &(a+b)^2 = a^2 + 2ab + b^2,\\ &(a-b)^2 = a^2 - 2ab + b^2,\\ &(a+b)(a-b) = a^2 - b^2. \end{align*}

Here are applications of the rules:

$$x^2 - 1 = (x-1)(x+1),\quad (2x-3)^2 = 4x^2 -12x +9$$

The first rule can be used to compute squares of numbers $\ge 10$

\begin{align*} 12^2 = (\Tblue{10}+\Tgreen{2})^2 &= \Tblue{10}^2 + 2 \times \Tblue{10}\times \Tgreen{2} + \Tgreen{2}^2\\& = 100 + 40 + 4 = 144 \end{align*}

## Solving a quadratic equation (overview)

Summary of the solutions for the quadratic equation $$\Tblue{a}\Tred{x}^2 + \Tblue{b}\Tred{x}+\Tblue{c} = 0$$ for numbers $\Tblue{a}\ne0$, $\Tblue{b}$ and $\Tblue{c}$.

We use the discriminant $\Tviolet{\Delta} = \Tblue{b}^2 - 4 \Tblue{ac}$.

Case Formula of solution Number of solutions $\Tviolet{\Delta}\lt 0$ $\Tviolet{\Delta}= 0$ $\Tviolet{\Delta}\gt 0$ No solution $\displaystyle\frac{-\Tblue{b}}{2\Tblue{a}}$ $\displaystyle\frac{-\Tblue{b}\pm\sqrt{\Tviolet{\Delta}}}{2\Tblue{a}}$ $0$ $1$ $2$ $\Tred{x}^2 + 1 = 0$ $\Tred{x}^2 - 2\Tred{x} + 1 = 0$ $\Tred{x}^2 +2\Tred{x} = 0$ $-1$ $0$ $4$ None $1$ $-2$, $0$

## Set membership

We say that an element belongs to a set if it is part of the set. We use the symbol $\in$.

$$10\in\Z,\;O\in\{T,\Tred{O},K,L\}$$

When an element does not belong to a set we use $\notin$.

$$A\notin\{T,O,K,L\},\;-1\notin\N,\;\pi\notin\Q$$
The set of the primary and secondary colours contains the element 'green'.

## Union of two sets

The union of two sets $A$ and $B$ is a set that consists of the elements that are either in $A$ or $B$ or in both. The set is written $A\cup B$.

The union of the odd and even numbers is the set of all the integers.

$$\{\Tblue{1},\Tblue{2},\Tblue{3}\}\cup \{\Tgreen{0},\Tgreen{2},\Tgreen{4}\} = \{\Tgreen{0},\Tblue{1},\Tred{2},\Tblue{3},\Tgreen{4}\}$$

$A$ and $B$ are a subset of $A\cup B$.

Union of two sets

## Definition of a matrix

A matrix is a table or array of numbers or letters arranged in rows and columns. The table is delimited by brackets [ ] or parentheses ( ). The plural of matrix is matrices.

These are four examples of matrices of different sizes: $$\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix},\quad \begin{bmatrix}1&x\\1&y\\1&z \end{bmatrix},\quad \begin{pmatrix}1&1\\0&1\end{pmatrix},\quad \begin{bmatrix}1\\0\\-1\end{bmatrix}$$

The numbers contained within a matrix are called elements or entries.

To add two matrices, we add each entry of one matrix to the corresponding entry of the matrix.

$$\begin{pmatrix}\Tred{1}&\Tblue{2}\\\Tgreen{-1}&\Torange{0}\end{pmatrix} + \begin{pmatrix}\Tred{-3}&\Tblue{1}\\\Tgreen{1}&\Torange{2}\end{pmatrix} = \begin{pmatrix}\Tred{1-3}&\Tblue{2+1}\\\Tgreen{-1+1}&\Torange{0+2}\end{pmatrix} = \begin{pmatrix}\Tred{-2}&\Tblue{3}\\\Tgreen{0}&\Torange{2}\end{pmatrix}$$

The two matrices must have the same size. The resulting matrix has the same size.

The sum below is NOT defined because the two matrices don't have the same size. $$\begin{pmatrix}1&2\\-1&3\end{pmatrix} + \begin{pmatrix}1\\-2\end{pmatrix}$$

## Definition of a function

A function is an expression of the form $$\Tred{f}(\Tblue{x}) = 2\Tblue{x}+5.$$ It relates an input $\Tblue{x}$ to an ouput $f(x)$. The output is sometimes written as $y$. The function is $\Tred{f}$.

The value of the function $\Tred{f}(\Tblue{x}) = 2\Tblue{x}+5$ for the input $\Tgreen{4}$ is $$\Tred{f}(\Tgreen{4}) = 2\times \Tgreen{4} + 5 = 13.$$

The input can be any number. It is represented by a letter, called the variable. The letter is generally $\Tblue{x}$, but it can be anything.

The functions $\Tred{f}(\Tblue{t}) = \Tblue{t}^2$ and $\Tred{g}(\Tblue{x}) = \Tblue{x}^2$ are the same, because the letters for the variable and for the function don't matter.

The function $\Tred{f}(\Tblue{x}) = 2 \Tblue{x} -3$ can be represented by a table for several inputs.

 $$x$$ $$f(x)$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$-7$$ $$-5$$ $$-3$$ $$-1$$ $$1$$ $$3$$ $$5$$

## Introduction to graphs

A simple way to represent a function is by drawing its graph. For each input $\Tblue{x}$ on the horizontal axis, called the $x$-axis, we plot the corresponding output $\Torange{y}=f(x)$ are on the vertical axis, called the $y$-axis.

The graph of a function gives its value for each possible input. Here, $f(\Tblue{2}) = \Torange{4}$

The graph is a two-dimensional representation of the input / output table associated with a function.

 x f(x) -2 -1 0 1 2 3 4 -7 -5 -3 -1 1 3 5
The function $\Torange{y} = 2 \Tblue{x} -3$ and its graph

A quadratic function takes the form $$f(x) = \Tred{a}x^2 + \Tgreen{b}x + \Tblue{c}$$ for three numbers $\Tred{a}\ne0$, $\Tgreen{b}$ and $\Tblue{c}$.

$f(x) = \Tred{-2} x^2 + \Tblue{1}$ and $g(x) = \Tred{2}x^2 \Tgreen{-3} x - \Tblue{6}$ are quadratic.

Constant functions and linear functions are NOT quadratic because they correspond to $\Tred{a}=0$.

The functions $x^3$ and $1/x$ are not quadratic.

• If $\Tred{a}\gt 0$, the graph has a U shape. It is symmetrical across the vertical line through its minimum point.
• If $\Tred{a}\lt 0$, the graph has an inverted U shape. It is symmetrical across the vertical line through its maximum point.

The number $\Tblue{c}$ is $f(0)$. It is the intercept with the $y$-axis.

Graphs of the two quadratic functions $y = 2x^2-2x-2$ (left) and $y = - x^2 +4$ (right) and a linear function $y = 2x + 2$ (centre).

## Power function

A power function is a function of the form $$f(x) = \Tblue{a} x^{\Torange{n}}.$$ The number $n$ is the power. It is often an integer.

Power functions have special name depending on the value of the power.

 function power $\Torange{n}$ $2^{\Torange{n}}$ reciprocal constant linear quadratic cubic $-1$ $0$ $1$ $2$ $3$ $0.5$ $1$ $2$ $4$ $8$
Graph of the power functions $x^n$ for several values of $n$

A power function with a negative exponent is not defined for $x=0$.

A number raised to a negative power is the reciprocal of the number at the positive power. $$\Tblue{3}^{-\Torange{2}} = \frac{1}{\Tblue{3}^{\Torange{2}}} = \frac{1}{\Tgreen{9}}.$$

The graph of the sum of power functions is obtained by adding, for each $x$, the $y$-values of each individual function.

Graph of the sum of power function $y=x^{-2} - 2x.$ The graph of $y=x^{-2}$ is on the left, $y=-2x$ in the centre, and the sum on the right.