Multiplying and Dividing Integers
Grade 7 · Math · Study Guide
Master the sign rules for multiplying and dividing integers, and learn why negative times negative equals positive.
Multiplying and dividing integers follows a set of sign rules that never change. The key idea is this: if the two numbers have the same sign, the answer is positive. If they have different signs, the answer is negative. That is it — the sign rule is the same for both multiplication and division.
Same signs → positive result. Positive × positive = positive (5 × 4 = 20) and negative × negative = positive (-5 × -4 = 20). Different signs → negative result. Positive × negative = negative (5 × -4 = -20) and negative × positive = negative (-5 × 4 = -20). These rules hold for division too: -20 ÷ -4 = 5 (same signs, positive) and 20 ÷ -4 = -5 (different signs, negative).
Why does negative times negative equal positive? Think of it this way: multiplying by a negative flips the sign once. Multiplying by another negative flips it back. Two flips return you to positive. It is the same idea as two negatives in language: "I do not have no homework" means you do have homework.
When you multiply or divide a chain of integers, count the negative signs. An even number of negatives gives a positive answer. An odd number of negatives gives a negative answer. For example, (-2)(-3)(-4) = -24 because there are 3 negatives — an odd count.
These rules connect directly to algebra. When you see expressions like -3(x - 5) or need to evaluate -48 ÷ (-6), the sign rules guide every step. Getting comfortable with them now makes everything from solving equations to graphing lines much smoother.
| First number | Second number | Result sign | Example |
|---|---|---|---|
| Positive | Positive | Positive (+) | 6 × 4 = 24 |
| Negative | Negative | Positive (+) | (-6) × (-4) = 24 |
| Positive | Negative | Negative (−) | 6 × (-4) = -24 |
| Negative | Positive | Negative (−) | (-6) × 4 = -24 |
| Expression | Negative count | Sign of answer | Answer |
|---|---|---|---|
| (-2)(3)(-1) | 2 (even) | Positive | 6 |
| (-2)(-3)(-4) | 3 (odd) | Negative | -24 |
| (-1)(-1)(-1)(-1) | 4 (even) | Positive | 1 |
Key terms
- Sign rules
- Same signs → positive product/quotient; different signs → negative product/quotient
- Product
- The result of multiplying two or more numbers
- Quotient
- The result of dividing one number by another
- Factor
- A number being multiplied in a multiplication problem
- Even/odd negative count
- Even number of negative factors → positive product; odd number → negative product
Practice questions
What is (-7) × (-8)?
- -56
- 56
- -1
- 15
Show answer
Answer: 56
The product of a negative number and a positive number is always negative.
Show answer
Answer: True
(-36) ÷ (-9) = ___.
Show answer
Answer: 4
What is the sign of the product (-3)(-2)(-5)?
- Positive, because all three are negative
- Negative, because there are 3 negative factors
- Positive, because the numbers are small
- Negative, because the first factor is negative
Show answer
Answer: Negative, because there are 3 negative factors
(-4)² equals -16.
Show answer
Answer: False
Fun facts
- The rule that two negatives make a positive was formally proven by mathematicians in the 1800s, but merchants had been using it intuitively for centuries when tracking debts and credits.
- Computer processors use integer multiplication billions of times per second using the same sign rules you are learning right now.
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