Subtraction
ArithmeticSubtraction 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. The symbol for subtraction is the minus sign (-).
Example
$9 - 4 = 5$. You have $9$ grapes and eat $4$. You have $5$ left.
Key Insight
Subtraction answers the question "how many are left?" or "how many more does one have than the other?" It is the opposite operation of addition.
Definition
Subtraction of $b$ from $a$ gives the difference $a - b$. 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).
Example
$5 - (-3) = 5 + 3 = 8$ (subtracting a negative is adding). $12 - 7 = 5$, but $7 - 12 = -5$ (not commutative). $(10 - 4) - 3 = 3$, but $10 - (4 - 3) = 9$ (not associative).
Key Insight
Thinking of subtraction 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.
Definition
Subtraction is not a primary operation in group theory; it is defined via the additive inverse: $a - b = a + (-b)$. In any abelian group, this is well-defined. The set of integers under subtraction is NOT a group (not associative, no identity relative to subtraction from the right), confirming that addition, not subtraction, is the fundamental operation.
Example
In a ring $R$, $a - b = a + (-b)$. The operation $a \mapsto a - b$ (subtract a fixed $b$) is a bijection on $R$, which is used to shift sets and prove results in combinatorics and number theory.
Key Insight
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.