Product
The product is the result of multiplying two or more numbers together.
Formula
a \times b = \text{product}
Definition
The product is the answer you get when you multiply two or more numbers, written $a \times b$ or $a \cdot b$ or (in algebra) simply $ab$; the product of any number and $0$ is $0$, and the product of any number and $1$ is that number. In a ring $(R, +, *)$, the product is defined by the multiplication operation $*$, and products generalize far beyond numbers: the Cartesian product of sets, the tensor product of vector spaces, and the direct product of groups all share structural properties with numerical products.
Example
In $6 \times 7 = 42$, the product is $42$, just as the sum is always the result of addition. The product of the first $5$ natural numbers, $1 \times 2 \times 3 \times 4 \times 5 = 120$, is written $5!$ in factorial notation, and pi notation generalizes this: $\prod_{k=1}^{n} k = n!$. The product $0 \times n = 0$ for any $n$ is the absorption property of zero, meaning that if even one factor is zero, the entire product is zero no matter how many other factors there are.
Key Insight
The product is always the result of multiplication. Multiplicativity, the property that $f(mn) = f(m)f(n)$ for coprime $m,n$, is a central theme in number theory: the Euler product formula expresses the Riemann zeta function as a product over primes, linking multiplication directly to prime distribution, and the Euler phi function ($\phi(mn) = \phi(m)\phi(n)$ for $\gcd(m,n)=1$) is a key example of this idea.