Addition
Addition is the arithmetic operation of combining two or more numbers to find their total, or sum.
Formula
a + b = \text{sum}
Definition
Addition is the operation of putting amounts together to find a total, written with the plus sign (+); as a binary operation on numbers, it combines two quantities (addends) to produce their sum. Its key properties are commutative ($a + b = b + a$), associative ($(a + b) + c = a + (b + c)$), and identity ($a + 0 = a$). More abstractly, addition is the fundamental binary operation in any abelian group: in $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$, it gives an abelian group structure with $0$ as identity and $-a$ as the inverse of $a$.
Example
$3 + 5 = 8$: you have $3$ red marbles and $5$ blue marbles, and putting them together gives $8$ marbles total. $(-3) + 7 = 4$ (adding a negative), $1/4 + 1/2 = 3/4$ (common denominators), and $2.5 + 1.75 = 4.25$. In $\mathbb{Z}_5$ (integers mod $5$): $3 + 4 = 2 \pmod{5}$, still commutative, associative, and with identity $0$, confirming $\mathbb{Z}_5$ is an abelian group under addition.
Key Insight
Addition answers the question "how many altogether?" and counting up on a number line is the same as adding. Because it is commutative and associative, numbers can be added in any order and grouping, which is exactly what makes mental math tricks like "making tens" work. This flexibility extends far beyond ordinary numbers: vectors, matrices, polynomials, and functions (as in Fourier analysis, which decomposes functions using addition of sinusoids) all admit an "addition" satisfying the same abelian group axioms, unifying vast areas of mathematics under a single framework.