2 Calculation with numbers

To perform calculations with numbers we use the operations , $+$ , $-$ , $\times$ and $÷$ .

2.1 Addition ( $+$ )

We say that $4+5$ is the sum of 4 and 5. Note that $4+5$ is equal to $5+4$ so that the order in which we write down the numbers does not matter when we are adding them. Because the order does not matter, addition is said to be commutative . This first property is called commutativity .

When more than two numbers are to be added, as in $4+8+9$ , it makes no difference whether we add the 4 and 8 first to get $12+9$ , or whether we add the 8 and 9 first to get $4+17$ . Whichever way we work we will obtain the same result, 21. Addition is said to be associative . This second property is called associativity .

2.2 Subtraction ( $-$ )

We say that $8-3$ is the difference of $8$ and $3$ . Note that $8-3$ is not the same as $3-8$ and so the order in which we write down the numbers is important when we are subtracting them i.e. subtraction is not commutative. Subtracting a negative number is equivalent to adding a positive number, thus $7-\left(-3\right)=7+3=10$ .

2.3 The plus or minus sign ( $\pm$ )

In engineering calculations we often use the notation plus or minus , $\pm$ . For example, we write $12\pm 8$ as shorthand for the two numbers $12+8$ and $12-8$ , that is 20 and 4. If we say a number lies in the range $12\pm 8$ we mean that the number can lie between 4 and 20 inclusive.

2.4 Multiplication ( $\times$ )

The instruction to multiply, or obtain the product of, the numbers 6 and 7 is written $6\times 7$ . Sometimes the multiplication sign is missed out altogether and we write $\left(6\right)\left(7\right)$ .

Note that $\left(6\right)\left(7\right)$ is the same as $\left(7\right)\left(6\right)$ so multiplication of numbers is commutative. If we are multiplying three numbers, as in $2\times 3\times 4$ , we obtain the same result whether we multiply the 2 and 3 first to obtain $6\times 4$ , or whether we multiply the 3 and 4 first to obtain $2\times 12$ . Either way the result is 24. Multiplication of numbers is associative.

Recall that when multiplying positive and negative numbers the sign of the result is given by the rules given in Key Point 1.

For example, $\left(-4\right)\times 5=-20$ , and $\left(-3\right)\times \left(-6\right)=18$ .

When dealing with fractions we sometimes use the word ‘of’ as in ‘find $\frac{1}{2}$ of 36’. In this context ‘of’ is equivalent to multiply, that is

$\frac{1}{2}\text{of}36\text{is equivalent to}\frac{1}{2}\times 36=18$

2.5 Division ( $÷$ ) or ( $∕$ )

The quantity $8÷4$ means $8$ divided by $4$ . This is also written as $8∕4$ or $\frac{8}{4}$ and is known as the quotient of $8$ and $4$ . In the fraction $\frac{8}{4}$ the top line is called the numerator and the bottom line is called the denominator . Note that $8∕4$ is not the same as $4∕8$ and so the order in which we write down the numbers is important. Division is not commutative.

When dividing positive and negative numbers, recall the following rules in Key Point 2 for determining the sign of the result:

2.6 The reciprocal of a number

The reciprocal of a number is found by inverting it. If the number $\frac{2}{3}$ is inverted we get $\frac{3}{2}$ . So the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ . Because we can write 4 as $\frac{4}{1}$ , the reciprocal of 4 is $\frac{1}{4}$ .

Task!

State the reciprocal of

1. $\frac{6}{11}$ ,
2. $\frac{1}{5}$ ,
3. $-7$ .

Answer

2.7 The modulus notation ( $\left|\right|$ )

We shall make frequent use of the modulus notation $\left|\right|$ . The modulus of a number is the size of that number regardless of its sign. For example $\left|4\right|$ is equal to 4, and $|-3|$ is equal to 3. The modulus of a number is thus never negative.

Task!

State the modulus of

1. $-17$ ,
2. $\frac{1}{5}$ ,
3. $-\frac{1}{7}$
4. $0$ .

Answer

2.8 The factorial symbol (!)

Another commonly used notation is the factorial , denoted by the exclamation mark ‘!’. The number $5!$ , read ‘five factorial’, or ‘factorial five’, is a shorthand notation for the expression $5\times 4\times 3\times 2\times 1$ , and the number $7!$ is shorthand for $7\times 6\times 5\times 4\times 3\times 2\times 1$ . Note that $1!$ equals 1, and by convention $0!$ is defined as 1 also. Your scientific calculator is probably able to evaluate factorials of small integers. It is important to note that factorials only apply to positive integers.

Example 1

1. Evaluate 4! and 5! without using a calculator.
2. Use your calculator to find $10$ !.

Solution

1. 4! = $4\times 3\times 2\times 1=24$ . Similarly, 5! = $5\times 4\times 3\times 2\times 1=120.$ Note that $5!=5\times 4!=5\times 24=120$ .
2. $10!=3,628,800$ .

Task!

Find the factorial button on your calculator and hence compute 11!.

(The button may be marked ! or $n!$ ). Check that $11!=11\times 10!$

Answer