Denominator
The denominator is the bottom number in a fraction, showing how many equal parts the whole is divided into.
Definition
The denominator is the bottom number of a fraction: it tells you how many equal parts the whole has been cut into. In a fraction $a/b$, the denominator $b$ (which must not equal zero) specifies the size of each equal part relative to the whole and is also the divisor when the fraction is read as a division; it determines what kind of fractional unit is being used. For algebraic fractions, the denominator is a nonzero polynomial, and the zeros of the denominator are poles of the resulting rational function.
Example
In the fraction $2/6$, the denominator is $6$: think of an egg carton with $6$ slots, the $6$ tells you there are $6$ equal spots total. Fractions with the same denominator share the same fractional unit, so $3/8$ and $5/8$ both use eighths and can be added directly: $3/8 + 5/8 = 8/8 = 1$. The rational function $1/(x^2-4)$ has denominator $x^2-4 = (x-2)(x+2)$; at $x = 2$ and $x = -2$ the denominator is zero, creating vertical asymptotes (poles of order $1$) in the graph.
Key Insight
A bigger denominator means smaller pieces: $1/10$ of a pie is a much smaller slice than $1/2$ of the same pie, even though the numerator is the same. The denominator can never be zero because division by zero is undefined, there is no number of equal parts that zero represents, a simple rule with deep consequences throughout mathematics. In $p$-adic number theory, the denominator of a rational number controls its $p$-adic valuation: a rational with $p$ in its denominator has negative $p$-adic valuation, while integers have non-negative valuation, a complete inversion of the familiar size intuition.