Difference
The difference is the result of subtracting one number from another.
Formula
a - b = \text{difference}
Definition
The difference is the answer you get when you subtract one number from another, $a - b$, and it depends on order: $a - b$ is generally not equal to $b - a$. The absolute difference $|a - b|$ measures distance regardless of order and is always non-negative. In the calculus of finite differences, the difference operator $\Delta f(n) = f(n+1) - f(n)$ is the discrete analog of the derivative, satisfying linearity and the product rule in the same form as differential calculus, which enables techniques like summation by parts (the discrete analog of integration by parts).
Example
In $9 - 3 = 6$, the difference is $6$; the difference tells you how far apart two numbers are, so $9$ and $3$ are $6$ steps apart on the number line. $15 - 8 = 7$, while $|-3 - 5| = |-8| = 8$ shows the absolute difference in action. $\Delta(n^2) = (n+1)^2 - n^2 = 2n + 1$, and summing gives $\sum_{k=1}^{n} \Delta(k^2) = (n+1)^2 - 1 = n^2 + 2n$, confirming that the sum of the first $n$ odd numbers equals $n^2$.
Key Insight
When solving "how much more" or "how much less" problems, the absolute difference removes the need to worry about which number is larger. Finite differences are used in numerical analysis for approximating derivatives and in combinatorics for computing sums of polynomial sequences: the discrete derivative mirrors continuous calculus in nearly every fundamental theorem.