Subtrahend
ArithmeticThe subtrahend is the number being subtracted from the minuend in a subtraction problem.
Formula
\text{minuend} - \text{subtrahend} = \text{difference}
Definition
The subtrahend is the number being taken away in a subtraction problem.
Example
In $20 - 8 = 12$, the subtrahend is $8$. It is the amount being removed from $20$.
Key Insight
The subtrahend is what you "subtract." The minuend is what you start with. The difference is what remains.
Definition
In $a - b = c$, $b$ is the subtrahend. Increasing the subtrahend decreases the difference (while the minuend stays fixed). The subtrahend can be any real number, including negative values.
Example
Subtracting a negative subtrahend: $10 - (-3) = 13$. Here $-3$ is the subtrahend, and subtracting it is the same as adding $3$.
Key Insight
Subtracting a negative subtrahend increases the result. This is the algebraic reason why "two negatives make a positive" in subtraction: $a - (-b) = a + b$.
Definition
Formally, the subtrahend $b$ in $a - b$ is replaced by its additive inverse: $a - b = a + (-b)$. In any abelian group, this construction identifies the "subtrahend role" as that of the element whose inverse is being added. The notation clarifies that subtraction is not a primitive operation but addition of the inverse.
Example
In modular arithmetic, $2 - 5 \bmod 7 = 2 + (-5) \bmod 7 = 2 + 2 \bmod 7 = 4$ (since $-5 \equiv 2 \pmod 7$). The subtrahend $5$ is replaced by its additive inverse $2$ in $\mathbb{Z}/7\mathbb{Z}$.
Key Insight
Recognizing the subtrahend as an additive inverse unifies subtraction with addition across all algebraic structures: groups, rings, fields, and vector spaces all handle subtraction this way.