Subtraction
Subtraction is the arithmetic operation of finding the difference between two numbers by taking one away from the other.
Formula
a - b = \text{difference}
Definition
Subtraction is taking one number away from another to find the difference, written with the minus sign (-); it is defined as adding the additive inverse, $a - b = a + (-b)$. Unlike addition, subtraction is neither commutative ($a - b \neq b - a$ in general) nor associative ($(a - b) - c \neq a - (b - c)$ in general), and it is not a primary operation in group theory: the integers under subtraction alone do not form a group, confirming that addition, not subtraction, is the fundamental operation.
Example
$9 - 4 = 5$: you have $9$ grapes and eat $4$, leaving $5$. $5 - (-3) = 5 + 3 = 8$ (subtracting a negative is adding), and $12 - 7 = 5$ while $7 - 12 = -5$, showing subtraction is not commutative; likewise $(10 - 4) - 3 = 3$ but $10 - (4 - 3) = 9$, showing it is not associative either. Two's complement arithmetic in computers represents subtraction as addition of the bit-flipped-plus-one value, exploiting the group structure of integers modulo $2^n$ to avoid a separate subtraction circuit.
Key Insight
Subtraction answers "how many are left?" or "how many more does one have than the other?"; it is the opposite operation of addition. Thinking of it as "adding the opposite" is the key insight that makes it work on negative numbers and unifies it with addition in the structure of any group, since in any ring $a - b = a + (-b)$ is well-defined and the map $a \mapsto a - b$ (subtract a fixed $b$) is a bijection used to shift sets in combinatorics and number theory.