Adding, subtracting, multiplying and dividing integers and rational numbers all come down to handling the signs.
On a number line, adding a positive number moves you right and adding a negative number moves you left, and subtracting a number is the same as adding it with its sign changed.
When multiplying or dividing, like signs give a positive result and unlike signs a negative one; with several negative factors, the sign depends on whether there is an even or odd number of them.
On this page you change two numbers and follow additions with arrows on a number line, then watch how the sign of a product changes.
Addition on the number line
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Addition on the number line: move a from 0 (blue), then b more (red); where you stop is the result (gold)
3
-5
👀 Addition on the number line
①Start at 0, move a, then move b from there; where you stop is a + b
②Adding a positive number moves right, and adding a negative number moves left
③The starting example (+3) + (−5) goes 3 steps right, then 5 steps back left, and stops at −2
④If a = 0, the blue arrow has length 0, so you move only by b
Sign rules for addition, and subtraction
Adding numbers with the same sign
Add the absolute values and keep the shared sign
(+3) + (+5) = +8, (−3) + (−5) = −8
Adding numbers with different signs
Subtract the absolute values and take the sign of the one with the larger absolute value
(+3) + (−5) = −2, (−3) + (+5) = +2
💡 Turn subtraction into addition
①Change the sign of the number being subtracted and add it: a − b = a + (−b)
②3 − (−5) = 3 + (+5) = 8. Subtracting a negative makes the result larger
③On the number line, subtracting −5 is the opposite of moving 5 steps left, which means moving 5 steps right
④Once everything is addition, you can use the commutative and associative laws to calculate in a convenient order
Signs in multiplication and division
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The sign of a product: positive in blue, negative in red, zero in grey
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Signs in multiplication and division
(+) × (+) = (+), (−) × (−) = (+), (+) × (−) = (−)
Like signs give a positive, unlike signs a negative. Division follows the same rule
Check with examples
Example 1
Calculate (−7) + (+3) − (−5).
1
Turn the subtraction into addition: −(−5) becomes +(+5).
(−7) + (+3) + (+5)
2
Add from left to right.
= (−4) + (+5) = +1
▸ 1
On a number line, you go 7 steps left from 0, 3 steps right, then 5 more steps right, and stop at +1.
Example 2
Calculate (−2)³ × (−3).
1
Work out the power first. It multiplies three negatives, an odd number, so it is negative.
(−2)³ = (−2) × (−2) × (−2) = −8
2
A negative times a negative is positive.
(−8) × (−3) = +24
▸ 24
There are four negative factors in all: three −2s and one −3. Four is even, so the result is positive.
Exam checklist
Addition essentials
Same signs: add the absolute values; different signs: subtract them
With different signs, the result takes the sign of the larger absolute value
Multiplication and division essentials
An even number of negative factors → +, an odd number → −
(−1)² = 1, (−1)³ = −1
Grade 7 school test style
What is the value of −2² + (−3)²?
①−13
②−5
③5
④13
⑤1
▸ ③ 5
1
−2² means the negative of 2², which is −4, and (−3)² = (−3) × (−3) = 9.
−2² = −4, (−3)² = 9
2
Add the two values.
−4 + 9 = 5
🎯 Exam checklist
①Any number times 0 is 0
②You cannot divide by 0
③The distributive law a × (b + c) = a × b + a × c sometimes makes a calculation easier
④(−2)² = 4, but −2² = −(2²) = −4. Always check for brackets
⑤Division can be rewritten as multiplying by the reciprocal: a ÷ b = a × (1/b) (with b ≠ 0)
⑥Calculations with rational numbers follow the same sign rules as integers