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Letters and arithmetic

The simplest thing that can be done with numbers is adding, subtracting, multiplying or dividing them. When two numbers are added, subtracted, multiplied or divided, you are performing arithmetic Arithmetic is derived from the Greek word arithmos meaning number . . These four basic operations can be performed on any two real numbers.

Mathematics as a language uses special notation to write things down. So instead of:

mathematicians write

In earlier grades, place holders were used to indicate missing numbers in an equation.

However, place holders only work well for simple equations. For more advanced mathematical workings, letters are usually used to represent numbers.

These letters are referred to as variables , since they can take on any value depending on what is required. For example, $x=1$ in [link] , but $x=26$ in $2+x=28$ .

A constant has a fixed value. The number 1 is a constant. The speed of light in a vacuum is also a constant which has been defined to be exactly 299 792 458 m $·$ s ${}^{-1}$ (read metres per second). The speed of light is a big number and it takes up space to always write down the entire number. Therefore, letters are also used to represent some constants. In the case of the speed of light, it is accepted that the letter $c$ represents the speed of light. Such constants represented by letters occur most often in physics and chemistry.

Additionally, letters can be used to describe a situation mathematically. For example, the following equation

can be used to describe the situation of finding how much change can be expected for buying an item. In this equation, $y$ represents the price of the item you are buying, $x$ represents the amount of change you should get back and $z$ is the amount of money given to the cashier. So, if the price is R10 and you gave the cashier R15, then write R15 instead of $z$ and R10 instead of $y$ and the change is then $x$ .

We will learn how to “solve” this equation towards the end of this chapter.

Addition and subtraction

Addition ( $+$ ) and subtraction ( $-$ ) are the most basic operations between numbers but they are very closely related to each other. You can think of subtracting as being the opposite of adding since adding a number and then subtracting the same number will not change what you started with. For example, if we start with $a$ and add $b$ , then subtract $b$ , we will just get back to $a$ again:

If we look at a number line, then addition means that we move to the right and subtraction means that we move to the left.

The order in which numbers are added does not matter, but the order in which numbers are subtracted does matter. This means that:

The sign $\ne$ means “is not equal to”. For example, $2+3=5$ and $3+2=5$ , but $5-3=2$ and $3-5=-2$ . $-2$ is a negative number, which is explained in detail in "Negative Numbers" .

Commutativity for addition

The fact that $a+b=b+a$ , is known as the commutative property for addition.

Multiplication and division

Just like addition and subtraction, multiplication ( $\times$ , $·$ ) and division ( $÷$ , /) are opposites of each other. Multiplying by a number and then dividing by the same number gets us back to the start again: