Subtraction and Difference¶

Definition¶

The operation of subtraction $x-y$ of two integers $x$ and $y$ is defined as

The result is called the difference between $x$ and $y\text{.}$

Interpretation¶

The difference $x-y$ measures how much larger $x$ is than $y$: it answers "what must be added to $y$ to reach $x$?", since $y+(x-y)=x$$x$. Its sign tells which number is larger, and its absolute value $|x-y|$ is the distance between $x$ and $y$ on the number line.

Subtraction is defined using addition and negation, $x-y=x+(-y)$$(-y)$, to ensure algebraic consistency: every property of subtraction follows from the rules of addition. This also explains why subtraction is less well behaved. It is not commutative, since $y-x=-(x-y)$$y)$, and not associative: $(5-3)-1=1$$1$, but $5-(3-1)=3$$3$. A minus sign before brackets therefore changes every sign inside, $x-(y-z)=x-y+z$$z$. The difference set $A\setminus B$ uses the same word for removing the elements of $B$ from $A$.

Key Results¶

• Subtraction is not commutative.
• The difference $x-y$ equals $x+(-y)\text{.}$
• Subtraction extends addition within the integers.

Examples¶

• $5-3=2\text{.}$
• $3-5=-2\text{.}$

Prerequisites¶

• Negation

Related Concepts¶

• Difference Set
• Cauchy Sequence
• Integer
• Addition and Sum

Sources¶

• Math I: Calculus · Lecture 2

Last updated: 20 October 2026 03:02