Apply Properties to Add and Subtract Rational Numbers

Addends may be grouped in any order.

Opposites cancel each other.

A smarter grouping can make a friendlier sum.

Read it: Adding positive and negative numbers

−5 + 7 starts at −5 on the number line and hops 7 to the right. The hop passes 0 and lands on 2, so −5 + 7 = 2.

1. Turn any subtraction into adding the opposite: 3 − 5 is 3 + (−5).
2. Pair any opposites first: a number and its opposite make 0.
3. Add what is left on the line: hop right for a positive number, left for a negative.

When the two numbers have the same sign, the hop keeps going the same way: add the distances and keep the sign.

When the signs are opposite, the hop turns back: take the shorter distance from the longer, and the longer one's sign wins. Here 7 − 5 = 2.

Subtracting a number is the same as adding its opposite, and a number plus its opposite is 0.

Same signs: add the distances and keep the sign. Opposite signs: subtract the distances and keep the sign of the longer one.

7/8 + 6 + (−7/8) = ___ — 6. 7/8 and −7/8 are opposites and make 0, so the sum is 6.