Product
ArithmeticThe 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.
Example
In $6 \times 7 = 42$, the product is $42$.
Key Insight
The product is always the result of multiplication, just like the sum is always the result of addition.
Definition
The product of $a$ and $b$ is written $a \times b$ or $a \cdot b$ or (in algebra) $ab$. For a set of numbers, the product of all elements uses pi notation (capital Greek Pi). The product of any number and $0$ is $0$. The product of any number and $1$ is that number.
Example
Product of the first $5$ natural numbers: $1 \times 2 \times 3 \times 4 \times 5 = 120$ (written as $5!$ in factorial notation).
Key Insight
The product $0 \times n = 0$ for any $n$ is the absorption property of zero. It means if even one factor is zero, the entire product is zero, no matter how many other factors there are.
Definition
In a ring $(R, +, *)$, the product is defined by the multiplication operation $*$. Products generalize 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. The product formula for the Euler phi function ($\phi(mn) = \phi(m)\phi(n)$ for $\gcd(m,n)=1$) is a key example of multiplicativity in number theory.
Example
Pi notation: $\prod_{k=1}^{n} k = n!$ ($n$ factorial). $\prod_{p \text{ prime}, p \le n}(1/(1-1/p))$ diverges and equals $\zeta(1)$, encoding information about prime density.
Key Insight
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 to prime distribution.