Dividend
ArithmeticThe dividend is the number being divided in a division problem.
Formula
\text{dividend} / \text{divisor} = \text{quotient}
Definition
The dividend is the number being divided up in a division problem.
Example
In $30 / 6 = 5$, the dividend is $30$. We are splitting $30$ into equal groups.
Key Insight
The dividend is the "total" you start with before sharing. It is always the number that gets divided.
Definition
In the expression $a / b = q$, $a$ is the dividend, $b$ is the divisor, and $q$ is the quotient. The division algorithm guarantees: for integers $a, b$ ($b > 0$), unique $q$ and $r$ exist such that $a = bq + r$ with $0 \le r < b$. The dividend $a$ equals $bq + r$.
Example
In $47 / 8$: dividend $= 47$, divisor $= 8$, quotient $= 5$, remainder $= 7$. Check: $8 \times 5 + 7 = 40 + 7 = 47$.
Key Insight
In long division, you work with parts of the dividend one digit at a time, using partial dividends at each step. Understanding the dividend's role is key to tracking what you have "used up."
Definition
The term "dividend" extends to any element being divided in a divisibility relation. In a Euclidean domain, every pair $(a, b)$ with $b \neq 0$ yields unique $q$ and $r$ such that $a = bq + r$ where the Euclidean norm $N(r) < N(b)$. The dividend $a$ is the element whose divisibility by $b$ is being analyzed.
Example
In the Gaussian integers $\mathbb{Z}[i]$, dividing $7+4i$ by $3+2i$: $7+4i = (3+2i)(2) + (1+0i)$, so quotient $= 2$ and remainder $= 1$ (since $|1| < |3+2i| = \sqrt{13}$).
Key Insight
The dividend / divisor relationship generalizes to polynomials: in polynomial long division, the dividend polynomial is divided by the divisor polynomial, yielding a quotient polynomial and remainder of lower degree.