The idea

Negatives and the order of operations

Why the order of operations exists, why −3² and (−3)² are different numbers, and how the sign rules for negatives follow from what multiplication means.

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Number · Integers, powers & roots

Negatives and the order of operations

Why the order of operations exists, why −3² and (−3)² are different numbers, and how the sign rules for negatives follow from what multiplication means.

Why it works

A convention with a reason

The order of operations is a convention with a reason.

3 + 4 \times 2

could mean two things; maths picks one so every reader gets the same answer. Powers bind tightest, then multiplication/division, then addition/subtraction — because

4 \times 2

is shorthand for repeated addition and

2^2

for repeated multiplication, the shorthands get unpacked first:

3 + 4 \times 2^2 = 3 + 4 \times 4 = 3 + 16 = 19.

Left-to-right (

3 + 4

first) gives 28 — a different, wrong number. Brackets exist to override the convention:

(3 + 4) \times 2 - 1 = 13

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