Quotient
The quotient is the result of dividing one number by another.
Formula
\text{dividend} / \text{divisor} = \text{quotient}
Definition
The quotient is the answer you get when you divide one number by another; for $a$ divided by $b$ ($b \neq 0$), the quotient is $a / b$, and for integers the quotient $q$ and remainder $r$ satisfy $a = bq + r$ with $0 \le r < |b|$. In abstract algebra, a quotient structure is formed by partitioning a set by an equivalence relation: a quotient ring $R/I$ is formed by dividing a ring $R$ by an ideal $I$, and a quotient group $G/N$ is formed by dividing a group $G$ by a normal subgroup $N$, generalizing numerical division to entire structures.
Example
In $20 / 5 = 4$, the quotient is $4$. For $17 / 5$: the integer quotient is $3$ (with remainder $2$), while the real quotient is $3.4$, and checking, $5 \times 3.4 = 17$; knowing which one you need matters in programming, where integer division and floating-point division behave differently. $\mathbb{Z}/5\mathbb{Z}$ (integers mod $5$) is the quotient ring of $\mathbb{Z}$ by the ideal $5\mathbb{Z}$, with elements the equivalence classes $\{0,1,2,3,4\}$ and arithmetic performed mod $5$.
Key Insight
The quotient tells you the size of each equal share or how many times the divisor fits into the dividend. The quotient construction is one of the most powerful tools in algebra: it takes a complicated structure and collapses an equivalence to reveal a simpler one, and Galois theory classifies field extensions via quotient groups, ultimately proving which polynomial equations can be solved by radicals.