Subtracting Integers
Subtracting Integers covers taking away positive and negative numbers in NSW Year 7 maths (NESA outcome MA4‑INT‑C‑01). Taking away a positive moves left on the number line, while taking away a negative is the same as adding its opposite, so \(7 - (-3) = 7 + 3 = 10\).
You will see why this works by continuing a pattern, use number lines, simplify double signs and find missing numbers. Problems include the height of a seagull above a diver below the surface and the time difference between two cities from their UTC offsets. It is part of Year 7 Integers.
Every question with a fully worked solution.
- Subtracting Integers - Video - Subtracting Integers | How to Subtract Positive and Negative Integers Watch
Theory
Subtracting a positive jumps left on a number line. Subtracting a negative is the same as adding its opposite, so it jumps right. This Year 7 maths page shows why \(7 - (-3) = 10\), how to change double signs, how to find a missing number and how to find differences such as temperature and time zone gaps.
Subtracting a positive moves left, even past zero: \(2 - 8 = 2 - 2 - 6 = -6\).
Subtracting a negative is the same as adding its opposite: \(-(-3) = +3\), so \(7 - (-3) = 7 + 3 = 10\).
Why? Follow the pattern. Each time the number taken away goes down by \(1\), the answer goes up by \(1\): \(8 - 1 = 7\), \(8 - 0 = 8\), \(8 - (-1) = 9\).
The difference between two values is higher minus lower, even across zero. A town at \(21^\circ\text{C}\) is \(21 - (-6) = 27^\circ\text{C}\) warmer than a town at \(-6^\circ\text{C}\).
Change each double sign, then work from left to right:
| Double sign | Becomes | Example |
|---|---|---|
| \(+\,(+)\) | \(+\) | \(2 + (+5) = 2 + 5\) |
| \(+\,(-)\) | \(-\) | \(2 + (-5) = 2 - 5\) |
| \(-\,(+)\) | \(-\) | \(2 - (+5) = 2 - 5\) |
| \(-\,(-)\) | \(+\) | \(2 - (-5) = 2 + 5\) |
How to subtract integers
- Change every double sign: \(-(-)\) becomes \(+\) and \(-(+)\) becomes \(-\).
- Jump on a number line, or use the adding rules.
- Work left to right when there are three or more numbers.
How to find a difference
- Write higher minus lower.
- Change the double sign if the lower value is negative.
- Add the sizes when the values are on opposite sides of zero.
How to find a missing number
- Simplify the known part first.
- Missing number taken away: start minus answer.
- Check by putting the number back in.
| \(1 - 5\) | \(=\) | \(1 - 1 - 4\) |
| \(\) | \(=\) | \(-4\) |
\(-4\)
(i) \(9 - (-7)\)
(ii) \(-12 - (-5)\)
(iii) \(-6 - 3\)
| \(9 - (-7)\) | \(=\) | \(9 + 7 = 16\) |
| \(-12 - (-5)\) | \(=\) | \(-12 + 5 = -7\) |
| \(-6 - 3\) | \(=\) | \(-9\) |
(i) \(16\)
(ii) \(-7\)
(iii) \(-9\) (no double sign: just jump left)
| \(4 - (-1)\) | \(=\) | \(5\) |
| \(4 - (-3)\) | \(=\) | \(7\) |
Each step the answer goes up by \(1\), so \(4 - (-2) = 6\) comes between them.
| \(-7 - (-15) - 4\) | \(=\) | \(-7 + 15 - 4\) |
| \(\) | \(=\) | \(8 - 4\) |
| \(\) | \(=\) | \(4\) |
\(4\)
| \(21 - (-6)\) | \(=\) | \(21 + 6\) |
| \(\) | \(=\) | \(27\) |
Higher minus lower: the coast is \(27^\circ\text{C}\) warmer.
\(-2 - \square = 5\). She moved right, so the card is negative. The number taken away \(=\) start \(-\) answer.
| \(-2 - 5\) | \(=\) | \(-7\) |
Check: \(-2 - (-7) = -2 + 7 = 5\). The card shows \(-7\).
| \(9 - (-3)\) | \(=\) | \(9 + 3\) |
| \(\) | \(=\) | \(12\) |
City A is \(12\) hours ahead.
Common pitfalls
Frequently asked questions
Why is subtracting a negative the same as adding?
Follow the pattern \(8 - 1 = 7\), \(8 - 0 = 8\), \(8 - (-1) = 9\): taking away less makes the answer bigger.
What does minus a negative become?
A plus. \(7 - (-3) = 7 + 3 = 10\).
Is -3 - 2 equal to 5?
No. There is no double sign. Start at \(-3\) and jump \(2\) left: \(-5\).
How do you find the difference between a positive and a negative?
Higher minus lower. \(21 - (-6) = 27\).
How do you find a missing number taken away?
Start minus answer. In \(-2 - \square = 5\), the missing number is \(-2 - 5 = -7\).