Addition
ArithmeticAddition 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. The symbol for addition is the plus sign (+).
Example
$3 + 5 = 8$. You have $3$ red marbles and $5$ blue marbles. Put them together and you have $8$ marbles total.
Key Insight
Addition answers the question "how many altogether?" Counting up on a number line is the same as adding.
Definition
Addition is a binary operation on numbers that combines two quantities (addends) to produce their sum. Key properties: commutative ($a + b = b + a$), associative ($(a + b) + c = a + (b + c)$), and identity element ($a + 0 = a$).
Example
$(-3) + 7 = 4$ (adding a negative). $1/4 + 1/2 = 1/4 + 2/4 = 3/4$ (adding fractions by common denominator). $2.5 + 1.75 = 4.25$.
Key Insight
Because addition is commutative and associative, we can add numbers in any order and any grouping. This flexibility is what makes mental math tricks (like making tens) work.
Definition
Addition is the fundamental binary operation in any abelian group. In $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$, addition gives an abelian group structure with $0$ as identity and $-a$ as the inverse of $a$. In abstract algebra, a group $(G, +)$ need not have multiplication; addition alone defines the entire structure.
Example
In $\mathbb{Z}_5$ (integers mod $5$): $3 + 4 = 2 \pmod{5}$. This modular addition is still commutative, associative, and has identity $0$, so $\mathbb{Z}_5$ is an abelian group under addition.
Key Insight
Fourier analysis decomposes functions using addition of sinusoids. Vectors, matrices, polynomials, and functions all admit an "addition" operation that satisfies abelian group axioms, unifying vast areas of mathematics under a single framework.