Adding and Subtracting Integers
Theory
To add and subtract integers, change every double sign first, then work from left to right. Adding a negative is subtracting: \(4 + (-3) = 1\). Subtracting a negative is adding: \(4 - (-3) = 7\). This Year 8 maths page shows chains of integers, changes across zero and missing integers.
The size of an integer is how far it is from zero: the size of \(-31\) is \(31\). The sign says which way; the size says how far.
Adding a negative is the same as subtracting: \(+(-8) = -8\). Subtracting a negative is the same as adding its opposite: \(-(-18) = +18\), so \(-9 - (-18) = -9 + 18 = 9\).
Different signs. Subtract the sizes and keep the sign of the larger size: \(-31 + 19 = -12\). One minus is not a double sign: \(-14 - 9 = -23\) just moves further left.
Chains. Change each double sign, then work left to right. A quicker way: add the positives, add the negatives, then combine the two totals. Keep each sign with its own number.
Change or difference? A change is new \(-\) old and can be negative: from \(12^\circ\)C to \(-5^\circ\)C the change is \(-5 - 12 = -17\), a fall of \(17^\circ\)C. A difference is higher \(-\) lower, straight across zero: from \(-13\) up to \(21\) is \(21 - (-13) = 34\).
Missing integer. Undo the operation, then check. In \({?} - (-15) = -4\), the \({?}\) has \(15\) added to it, so \({?} = -4 + (-15) = -19\). Check: \(-19 + 15 = -4\).
Time zones are written as offsets from UTC. The time difference between two places is one offset minus the other. Going back past midnight gives the day before.
The four sign patterns (a double sign changes first: \(+(-3)\) is \(-3\) and \(-(-3)\) is \(+3\)):
| Adding | Subtracting |
|---|---|
| \(4 + 3 = 7\) | \(4 - 3 = 1\) |
| \(4 + (-3) = 1\) | \(4 - (-3) = 7\) |
| \(-4 + 3 = -1\) | \(-4 - 3 = -7\) |
| \(-4 + (-3) = -7\) | \(-4 - (-3) = -1\) |
How to add and subtract integers
- Change each double sign: \(+(-) \to -\) and \(-(-) \to +\).
- Work left to right, or group the positives and the negatives.
- Combine the totals: subtract the sizes and keep the sign of the larger size.
How to find a missing integer
- Undo the operation: undo adding by subtracting, and subtracting by adding.
- Check by putting your answer back in.
(i) \(3 + (-4)\)
(ii) \(3 - (-4)\)
| \(3 + (-4)\) | \(=\) | \(3 - 4 = -1\) |
| \(3 - (-4)\) | \(=\) | \(3 + 4 = 7\) |
(i) \(-1\): start at \(3\) and go back \(4\).
(ii) \(7\): start at \(3\) and go forward \(4\).
(ii) A stone falls from a height of \(100\) m. How far has it fallen when it is \(65\) m above the ground?
| \(12 + 6\) | \(=\) | \(18\) |
| \(100 - 65\) | \(=\) | \(35\) |
(i) \(18^\circ\)C
(ii) It has fallen \(35\) m.
(i) \(-31 + 19\)
(ii) \(-26 - (-42)\)
| \(-31 + 19\) | \(=\) | \(-(31 - 19) = -12\) |
| \(-26 - (-42)\) | \(=\) | \(-26 + 42 = 16\) |
(i) \(-12\) (different signs: subtract the sizes, keep the sign of \(31\))
(ii) \(16\)
| \(\text{positives}\) | \(=\) | \(48 + 27 = 75\) |
| \(\text{negatives}\) | \(=\) | \(-63 - 19 = -82\) |
| \(75 + (-82)\) | \(=\) | \(-7\) |
\(-7\). Check left to right: \(48 - 63 = -15\), \(-15 + 27 = 12\), \(12 - 19 = -7\).
(i) How many hours ahead of \(Q\) is \(P\)?
(ii) It is 09:00 on Wednesday in \(P\). What is the time in \(Q\)?
| \(10 - (-7)\) | \(=\) | \(10 + 7 = 17\) |
| \(9 - 17\) | \(=\) | \(-8\) |
| \(24 - 8\) | \(=\) | \(16\) |
(i) \(17\) hours
(ii) \(-8\) means \(8\) hours before midnight: 16:00 on Tuesday.
(i) Find the total.
(ii) How much higher is the highest round than the lowest?
(iii) What fifth score gives a total of \(-10\)?
| \(\text{total}\) | \(=\) | \(-8 + 4 = -4\) |
| \(3 - (-6)\) | \(=\) | \(9\) |
| \({?}\) | \(=\) | \(-10 + 4 = -6\) |
(i) \(-4\)
(ii) \(9\)
(iii) \(-6\). Check: \(-4 + (-6) = -10\).
(ii) Two integers add to \(-5\). The larger minus the smaller is \(13\). Find them.
| \(-5 + 3\) | \(=\) | \(-2\) |
| \(3 + (-8)\) | \(=\) | \(-5\) |
| \(-2 + (-5)\) | \(=\) | \(-7\) |
(i) \(-7\)
(ii) Try pairs that add to \(-5\): \(4\) and \(-9\) work, because \(4 + (-9) = -5\) and \(4 - (-9) = 13\). Or: larger \(= (\text{sum} + \text{difference}) \div 2\)
\(= (-5 + 13) \div 2 = 4\); smaller \(= -5 - 4 = -9\).
Common pitfalls
Frequently asked questions
What is minus a negative?
Subtracting a negative is the same as adding: \(5 - (-2) = 7\).
What is plus a negative?
Adding a negative is the same as subtracting: \(5 + (-2) = 3\).
How do you add integers with different signs?
Subtract the sizes and keep the sign of the larger size: \(-9 + 4 = -5\).
How do you find the difference between a negative and a positive number?
Higher minus lower: \(21 - (-13) = 34\).
Is there a quick way to add a long chain of integers?
Add all the positives, add all the negatives, then combine the two totals.