1.01 Calculations with whole numbers

Order of operations

An operation is a mathematical process, such as addition, subtraction, multiplication and division. Other operations include raising a number to a power and taking a root of a number. An operator is a symbol that indicates the type of operation. For addition, subtraction, multiplication and division, the corresponding operators are $+$+, $-$−, $\times$× and $\div$÷​.

There are a number of conventions (or rules) which we follow in order to correctly interpret and work with expressions that include operations. In particular, there is a priority order for evaluating these operations, which is as follows:

 

Worked examples

Example 1

Evaluate the expression $5\times\left(6+6\right)$5×(6+6).

Think: According to the order of operations we should evaluate the sum $6+6$6+6 first, since it is inside grouping symbols. We can then multiply by $5$5 afterwards.

Do:

$5\times\left(6+6\right)$5×(6+6)	$=$=	$5\times12$5×12	Evaluating the addition inside the parentheses first
	$=$=	$60$60	Evaluating the multiplication next

 

example 2

Evaluate the expression $100-9\times6+18\div6$100−9×6+18÷​6.

Think: This time there are no grouping symbols involved. Using the order of operations we should evaluate the multiplication and division first, from left to right. We can then evaluate the subtraction and the addition, again from left to right.

Do:

$100-9\times6+18\div6$100−9×6+18÷​6	$=$=	$100-54+18\div6$100−54+18÷​6	Evaluating the multiplication first, since it is further left
	$=$=	$100-54+3$100−54+3	Evaluating the division next
	$=$=	$46+3$46+3	Evaluating the subtraction, since it is further left
	$=$=	$49$49	Evaluating the addition last

 

Example 3

Evaluate the expression $\left(48\div12+35\div5\right)\times3^2$(48÷​12+35÷​5)×32

Think: Using the order of operations, we start by working inside the grouping symbols. Inside, we should perform the divisions (from left to right) before the addition. After that, we should evaluate the power, and then finally the multiplication.

Do:

$\left(48\div12+35\div5\right)\times3^2$(48÷​12+35÷​5)×32	$=$=	$\left(4+35\div5\right)\times3^2$(4+35÷​5)×32	Evaluating division inside the grouping symbols first
	$=$=	$\left(4+7\right)\times3^2$(4+7)×32	Evaluating the other division inside the grouping symbols
	$=$=	$11\times3^2$11×32	Evaluating the last operation (addition) inside the grouping symbols
	$=$=	$11\times9$11×9	Evaluating the power next
	$=$=	$99$99	Evaluating the multiplication last

 

Reflect: In this expression it is possible to evaluate the power first, or at the same time as the division, since it doesn't affect the calculations going on inside the grouping symbols. In this way, we can often perform more than one operation at a time once we are comfortable with the order of operations:

$\left(48\div12+35\div5\right)\times3^2$(48÷​12+35÷​5)×32	$=$=	$\left(4+7\right)\times9$(4+7)×9
	$=$=	$11\times9$11×9
	$=$=	$99$99

 

Practice questions

Question 1

Question 2

Calculators

When using a calculator it is important to understand the calculation that is being performed and the order of the operations involved. In particular, it is a good idea to have a reasonable estimate about what the output of the calculation should be.

Try the following calculation on your calculator. Type it in exactly from left to right.

$6+3\times7$6+3×7

Some calculators will display $63$63, and others will display the correct answer of $27$27. This is because some calculators can correctly identify that, following the order of operations, the multiplication needs to be completed before the addition of $6$6. Other calculators will evaluate $6+3$6+3 once it is typed in, without waiting for the $\times7$×7.

When faced with a numeric calculation, you should first look at it and decide whether it is something that you can do in your head (mental computation), something that can be done quickly by hand, or if you need a calculator. This decision will be different for everyone, so you'll just have to try some questions out and see.

 

Using a calculator

Most high school students will have a scientific calculator that looks something like one of these:

 

Exploration

Suppose that in the process of answering a question we need to calculate the product $351\times996$351×996, and decide that it will be easier to use a calculator rather than writing it out by hand.

Before putting the values into the calculator, we can quickly estimate the answer mentally by rounding. $351$351 is close to $350$350, and $996$996 is close to $1000$1000, so we can expect the answer to be roughly $350\times1000=350000$350×1000=350000.

Now we can enter the product into the calculator by pressing each button individually in the exact order:

$\editable{3}$3

$\editable{5}$5

$\editable{1}$1

$\editable{\times}$×

$\editable{9}$9

$\editable{9}$9

$\editable{6}$6

$\editable{=}$=

The display screen on the calculator shows that the answer to this calculation is $349596$349596. This is close to our earlier estimate of $350000$350000, so we can be confident that we have pressed the correct buttons and gotten the right answer.

 

Practice questions

Question 3

Question 4

Question 5

 

Negative numbers on a calculator

Most calculators have a special button to make negative numbers, here are some examples on some common calculator types.

 

For example, to evaluate $-351\times996$−351×996 using a calculator we would press the same buttons as before, but we will also need to press a button to make the first number negative.

Some calculators require pressing the negative sign before putting in the rest of the number, like this:

$\editable{\left(-\right)}$(−)

$\editable{3}$3

$\editable{5}$5

$\editable{1}$1

$\editable{\times}$×

$\editable{9}$9

$\editable{9}$9

$\editable{6}$6

$\editable{=}$=

Some calculators require pressing the negative sign after putting in the number, like this:

$\editable{3}$3

$\editable{5}$5

$\editable{1}$1

$\editable{\left[+/-\right]}$[+/−]

$\editable{\times}$×

$\editable{9}$9

$\editable{9}$9

$\editable{6}$6

$\editable{=}$=

Some calculators will accept both ways.

Try this calculation on your calculator (note that the correct answer is $-349596$−349596). Which way do you need to enter the negative on your calculator to get the correct answer?

 

Practice question

Question 6

 

Consider the expression $9989-\left(-9885\right)$9989−(−9885). To enter this into a calculator, some calculators need the negative number inside parentheses, while other calculators will accept the subtraction of a negative number without parentheses. Try the two methods below, and see what your calculator does. Make sure you know which method gives the correct answer (which is $19874$19874 for this example).

 

Method 1:

$\editable{9}$9

$\editable{9}$9

$\editable{8}$8

$\editable{9}$9

$\editable{-}$−

$\editable{\left(-\right)}$(−)

$\editable{9}$9

$\editable{8}$8

$\editable{8}$8

$\editable{5}$5

$\editable{=}$=

 

Method 2: Some calculators (particularly older ones) require grouping symbols when negatives are involved in a calculation.

$\editable{9}$9

$\editable{9}$9

$\editable{8}$8

$\editable{9}$9

$\editable{-}$−

$\editable{(}$(

$\editable{9}$9

$\editable{8}$8

$\editable{8}$8

$\editable{5}$5

$\editable{\left(-\right)}$(−)

$\editable{)}$)

$\editable{=}$=

 

Finally, remember that when using a calculator to perform calculations, it is as much about learning how to read the mathematical questions as it is learning how to use your tool effectively!

Practice question

Question 7