Opposite Number
ArithmeticThe opposite of a number is the number the same distance from zero on the number line but on the other side; it is the additive inverse.
Formula
\text{opposite of } a = -a; \ a + (-a) = 0
Definition
The opposite of a number is the number on the other side of zero on the number line, the same distance away. Adding a number and its opposite always gives zero.
Example
The opposite of $5$ is $-5$. The opposite of $-3$ is $3$. The opposite of $0$ is $0$. $5 + (-5) = 0$.
Key Insight
Opposites always add up to zero. Think of them as perfectly balanced weights on either side of a scale.
Definition
The opposite (additive inverse) of $a$ is $-a$, defined by the property $a + (-a) = 0$. On the number line, $a$ and $-a$ are symmetric about zero and have the same absolute value. The double negative law: $-(-a) = a$.
Example
Opposite of $7/2$ is $-7/2$. Opposite of $-4.5$ is $4.5$. Simplify $-(-(-3))$: $-(-(-3)) = -(3) = -3$.
Key Insight
The notation "$-a$" does NOT mean $a$ is negative. If $a = -5$, then $-a = -(-5) = 5$. The negative sign means "opposite," and the opposite of a negative is positive.
Definition
In any abelian group $(G, +)$, the additive inverse of $a$ is the unique element $-a$ such that $a + (-a) = 0$ (the identity). Existence of additive inverses is one of the group axioms. In a ring, the additive inverse satisfies $(-a)(-b) = ab$ (product of negatives), derivable from the distributive law.
Example
In $\mathbb{Z}/7\mathbb{Z}$: the opposite of $3$ is $4$ (since $3 + 4 = 7 \equiv 0 \pmod 7$). In the group of $2\times2$ invertible matrices, the additive inverse of matrix $A$ is $-A$ (negate all entries).
Key Insight
The existence of additive inverses distinguishes groups from monoids. Adding inverses to the natural numbers gives the integers; adding multiplicative inverses (for non-zero elements) gives the rationals. This chain of extensions is the construction of the number line from scratch.