Difference
ArithmeticThe 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.
Example
In $9 - 3 = 6$, the difference is $6$.
Key Insight
The difference tells you how far apart two numbers are. The difference between $9$ and $3$ is $6$ steps on the number line.
Definition
The difference of $a$ and $b$ is $a - b$. The difference 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.
Example
$15 - 8 = 7$. The difference between $15$ and $8$ is $7$. $|-3 - 5| = |-8| = 8$ (absolute difference).
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.
Definition
The difference operator $\Delta f(n) = f(n+1) - f(n)$ in the calculus of finite differences is the discrete analog of the derivative. It satisfies linearity and the product rule in the same form as differential calculus, enabling techniques like summation by parts (the discrete analog of integration by parts).
Example
$\Delta(n^2) = (n+1)^2 - n^2 = 2n + 1$. Summing: $\sum_{k=1}^{n} \Delta(k^2) = (n+1)^2 - 1 = n^2 + 2n$. This gives $\sum_{k=1}^{n}(2k+1) = n^2 + 2n$, confirming sum of first $n$ odd numbers $= n^2$.
Key Insight
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.