Integers

Integers are the whole numbers and their negatives: ..., −3, −2, −1, 0, 1, 2, 3, .... They include zero but no fractions or decimals, so 5 and −12 are integers but ½ and 2.7 are not. The set of integers goes on forever in both directions and is written ℤ.

What are integers?

• Positive integers: 1, 2, 3, ... (also the counting or natural numbers).
• Negative integers: −1, −2, −3, ....
• Zero: an integer that is neither positive nor negative.

The symbol ℤ comes from the German word Zahlen, meaning numbers. Every integer is also a rational number, because it can be written over 1, such as −4 = −4/1. See rational numbers for how the number sets fit together.

Integers on a number line

Numbers get bigger as you go right. So −1 is bigger than −4, even though 4 is bigger than 1: −1 is further right. Each integer has an opposite the same distance from zero on the other side, so the opposite of 3 is −3. The distance from zero is the number's absolute value, written |−3| = 3.

Adding integers

• Same signs: add the sizes and keep the sign. −7 + (−5) = −12.
• Different signs: subtract the smaller size from the larger, and take the sign of the one with the larger size. −9 + 4 = −5, and 6 + (−2) = 4.

On a number line, adding a positive moves you right and adding a negative moves you left. For −9 + 4, start at −9 and move 4 right to land on −5. That is exactly how the US Common Core describes it for grade 7: p + q is the point a distance |q| from p, in the direction q's sign says.

Subtracting integers

Subtracting a number is the same as adding its opposite: p − q = p + (−q). So change the subtraction to addition and flip the sign of the second number, then follow the adding rules.

• 3 − 8 = 3 + (−8) = −5
• −4 − (−6) = −4 + 6 = 2
• 5 − (−3) = 5 + 3 = 8

"Minus a minus makes a plus" is the short version: taking away a debt leaves you better off.

Multiplying and dividing integers

Multiply or divide the sizes as usual, then use the signs to decide whether the answer is positive or negative:

In short: same signs give a positive, different signs give a negative. With several numbers, count the negatives. An even number of negatives gives a positive answer and an odd number gives a negative one. That is why (−2)⁴ = 16 but (−2)³ = −8.

Why is a negative times a negative positive? The rules have to keep the distributive law working. Since (−1) × (1 + (−1)) = (−1) × 0 = 0, the two parts (−1) × 1 and (−1) × (−1) must cancel out. The first is −1, so the second has to be +1. The Common Core uses this same argument in grade 7. Dividing two integers doesn't always give an integer, though: −7 ÷ 2 = −3.5.

Integers in real life

• Temperature: it is −3 °C at dawn and rises 8 degrees by noon, so it is −3 + 8 = 5 °C.
• Money: a balance of −$20 is $20 owed. Paying in $50 gives −20 + 50 = $30.
• Height: sea level is 0, a mountain is positive and the sea floor is negative.
• Buildings: floor −2 is two floors below the ground floor.

Consecutive integers

Consecutive integers follow each other with no gaps, such as 23, 24, 25. Call the first one n, and the next ones are n + 1 and n + 2. Then a question like "three consecutive integers add up to 72" becomes an equation:

The integers are 23, 24 and 25. Consecutive even or odd integers go up in 2s, so use n, n + 2, n + 4. Two consecutive even integers that add up to 94 are 46 and 48.

Common mistakes

• Thinking −8 is bigger than −2. On the number line, −2 is further right, so it is bigger.
• Forgetting to flip the sign when subtracting a negative. 5 − (−3) is 8, not 2.
• Mixing up the rules. "Two negatives make a positive" is for multiplying and dividing. −3 + (−4) is −7, not 7.
• Missing brackets on a calculator. Type (−2)⁴, not −2⁴, which most calculators read as −(2⁴) = −16.
• Leaving out zero. Zero is an integer.

To practise negative numbers among other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 paper. In the UK, the same practice opens as GCSE Maths Foundation.

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