Integers
Integers are the positive and negative whole numbers and zero. This NSW Year 9 Core maths subtopic revises directed numbers from Stage 4 (NESA outcome MA4‑INT‑C‑01), but with longer chains of operations, grouping symbols and squared brackets instead of single sums.
You will rewrite double signs such as \(3 - (-2) = 3 + 2\), use the sign rules to multiply and divide, work a chain like \(98 \div (-7) \times (-3)\) from left to right, and use the order of operations on expressions such as \(4 - 2 \times (-4 + 1) - 7\) and \((3 - 4)^2 - (7 - 1)^2\). You will also turn a falling temperature, a bank balance and a diver's depth into number sentences. It is part of Year 9 Core Working with Numbers.
Every question with a fully worked solution.
- Integers - Video - Adding Integers | How to Add Positive and Negative Integers Watch
- Integers - Vidoe - Subtracting Integers | How to Subtract Positive and Negative Integers Watch
- Integers - Video - Multiplying Integers | How to Multiply Positive and Negative Integers Watch
- Integers - Video - Dividing Integers | How to Divide Positive and Negative Integers Watch
Theory
Integers are whole numbers that can be positive, negative or zero. This Year 9 maths page revises directed numbers: adding and subtracting negatives, the sign rules for multiplying and dividing, the order of operations, squaring a negative number, and writing number sentences for temperatures, money and depths.
An integer is a whole number that may be positive, negative or zero:
On a number line, negative numbers sit to the left of zero.
When two signs sit together, rewrite them as one sign before doing anything else. Adding a negative is the same as subtracting: \(5 + (-2) = 5 - 2\). Subtracting a negative is the same as adding: \(5 - (-2) = 5 + 2\).
To multiply or divide, work out the size first, then the sign. If the two signs are the same, the answer is positive. If they are different, the answer is negative.
The order of operations is the same as always: brackets, then powers, then \(\times\) and \(\div\) from left to right, then \(+\) and \(-\) from left to right.
Real situations become directed number sentences. Below zero, down, owing and colder are negative. Above zero, up, a deposit and warmer are positive.
Two signs together become one:
The sign rules for multiplying and dividing:
| Signs | Multiply | Divide | Answer |
|---|---|---|---|
| same | \((+) \times (+)\), \((-) \times (-)\) | \((+) \div (+)\), \((-) \div (-)\) | positive |
| different | \((+) \times (-)\), \((-) \times (+)\) | \((+) \div (-)\), \((-) \div (+)\) | negative |
How to evaluate an integer expression
- Rewrite any pair of signs as a single sign.
- Work inside the brackets first, then any powers.
- Multiply and divide from left to right. For each step, find the size, then use the sign rule.
- Add and subtract from left to right. A number line helps: adding moves right, subtracting moves left.
- Check the sign: count the negatives in a multiplying chain, or estimate on a number line.
(i) \(-7 + (-5)\)
(ii) \(-4 - (-9)\)
(iii) At 6 am the temperature is \(-4^\circ\text{C}\). By 1 pm it has risen \(11\) degrees. What is the temperature at 1 pm?
(i) Rewrite the two signs as one: \(+(-5) = -5\).
| \(-7 + (-5)\) | \(=\) | \(-7 - 5\) |
| \(\) | \(=\) | \(\mathbf{-12}\) |
(ii) Subtracting a negative is adding: \(-(-9) = +9\).
| \(-4 - (-9)\) | \(=\) | \(-4 + 9\) |
| \(\) | \(=\) | \(\mathbf{5}\) |
(iii) Write a directed number sentence. Below zero is negative and a rise is a plus:
| \(-4 + 11\) | \(=\) | \(\mathbf{7}\) |
The temperature at 1 pm is \(7^\circ\text{C}\).
(i) \(-8 \times 7\)
(ii) \(-2 \times (-5) \times (-6)\)
(iii) \(60 \div (-5) \div (-3)\)
(i) Different signs give a negative:
| \(-8 \times 7\) | \(=\) | \(\mathbf{-56}\) |
(ii) Work one pair at a time, left to right, deciding the sign each time:
| \(-2 \times (-5) \times (-6)\) | \(=\) | \(10 \times (-6)\) |
| \(\) | \(=\) | \(\mathbf{-60}\) |
Check: there are three negatives, an odd number, so the answer is negative.
(iii) The same rules work for division:
| \(60 \div (-5) \div (-3)\) | \(=\) | \(-12 \div (-3)\) |
| \(\) | \(=\) | \(\mathbf{4}\) |
Multiplication comes before addition, so do it first. Different signs give a negative:
| \(6 \times (-3)\) | \(=\) | \(-18\) |
Now the addition:
| \(-20 + (-18)\) | \(=\) | \(-20 - 18\) |
| \(\) | \(=\) | \(\mathbf{-38}\) |
Working left to right instead would give \(-14 \times (-3) = 42\), which is wrong. The order of operations does not change because numbers are negative.
(i) \((-5)^2\)
(ii) \(-5^2\)
(iii) \((2 - 7)^2 - 3^2\)
(i) The bracket says the whole of \(-5\) is squared:
| \((-5)^2\) | \(=\) | \((-5) \times (-5)\) |
| \(\) | \(=\) | \(\mathbf{25}\) |
(ii) With no bracket, only the \(5\) is squared and the minus stays in front:
| \(-5^2\) | \(=\) | \(-(5 \times 5)\) |
| \(\) | \(=\) | \(\mathbf{-25}\) |
(iii) Brackets first, then powers, then subtract:
| \((2 - 7)^2 - 3^2\) | \(=\) | \((-5)^2 - 3^2\) |
| \(\) | \(=\) | \(25 - 9\) |
| \(\) | \(=\) | \(\mathbf{16}\) |
Common pitfalls
Frequently asked questions
What is a negative times a negative?
A positive. When the two signs are the same, the answer is positive. For example, \(-4 \times (-6) = 24\).
What happens when you subtract a negative number?
Subtracting a negative is the same as adding the positive. For example, \(2 - (-7) = 2 + 7 = 9\).
Is negative 3 squared positive or negative?
It depends on the brackets. \((-3)^2 = 9\), but \(-3^2 = -9\), because without brackets only the \(3\) is squared.
What is an integer?
An integer is a whole number. It can be positive, negative or zero, such as \(-8\), \(0\) or \(15\). Fractions and decimals such as \(2.5\) are not integers.
How do you add a negative and a positive number?
Think of a number line. Start at the first number and move right for a positive or left for a negative. For example, \(-9 + 4 = -5\).