Operations with Signed Integers

Learning outcome

Calculate with positive and negative integers, interpret different uses of the minus sign and explain why the sign rules work.

Concepts and reasons

Integers include zero, positive whole numbers and their negatives. Their signs describe position relative to zero. A number’s magnitude is its distance from zero, ignoring direction.

A minus sign has different jobs. In -7, it is a unary negative sign: it applies to one number. In 12 - 7, it is subtraction, an operation between two numbers. Parentheses in (-7) group the negative number; they do not create another subtraction.

On a number line, adding a positive integer moves right and adding a negative integer moves left. For equal signs, add magnitudes and retain the sign. For opposite signs, subtract the smaller magnitude from the larger and retain the larger magnitude’s sign. Equal opposite magnitudes cancel to zero.

Subtracting means adding the opposite: a - b = a + (-b). Removing a negative change reverses its effect, so subtracting a negative increases the result. Do not treat two nearby minus signs as interchangeable without identifying their roles.

Multiplication by a positive integer repeats addition. Reversing one factor’s sign reverses the product: a × [b + (-b)] = 0, so a × b and a × (-b) must be opposites. Reversing both signs restores the original product. Thus, for nonzero factors, equal signs give a positive product and different signs give a negative product.

Division reverses multiplication, so the same sign rule applies to a nonzero quotient. The division exercises here have integer quotients; division of arbitrary integers need not.

Zero times any integer is zero, and zero divided by a nonzero integer is zero. Division by zero is undefined. Multiplication by zero cannot produce a nonzero dividend; for 0 ÷ 0, every possible quotient would satisfy the multiplication check, so there is no unique answer.

Worked examples

Example 1 — Opposite signs. Calculate (-8) + 13. The positive magnitude is larger: 13 - 8 = 5. Therefore (-8) + 13 = 5.

Example 2 — Subtracting a negative. Calculate 6 - (-11). Replace subtraction by addition of the opposite: 6 + 11 = 17. The parentheses identify -11 as the whole number being subtracted.

Example 3 — Product followed by division. Calculate [(-7) × (-6)] ÷ (-3). Equal signs give a product of 42. Then 42 ÷ (-3) = -14 because the signs differ. Check: (-14) × (-3) = 42.

Common mistakes

Do not use multiplication’s sign rule for addition. Do not discard parentheses before identifying the signed number. Zero is neither positive nor negative, and an undefined division is not an answer of zero.

Practice questions

1. Calculate (-17) + 9.
2. Calculate (-6) - (-14).
3. Calculate [(-8) × (-9)] ÷ (-6).
4. A lift starts at level -3, with ground level labelled 0. It rises 8 levels, then descends 6. At which level does it stop?

Answers and explanations

1. -8. The negative magnitude is larger. Subtract magnitudes: 17 - 9 = 8, then retain the negative sign.
2. 8. Subtracting -14 means adding 14: (-6) + 14 = 8.
3. -12. First, (-8) × (-9) = 72. Next, 72 ÷ (-6) = -12. Checking by multiplication gives (-12) × (-6) = 72.
4. Level -1. After rising, its level is (-3) + 8 = 5. Descending gives 5 - 6 = -1, one level below ground.