Division

Division is the inverse of multiplication: it tells you how many times one number fits into another, or into how many equal parts something can be split. The notation $a : b = c$ reads "a divided by b equals c". The number $a$ is the dividend, $b$ is the divisor (it must be nonzero), and $c$ is the quotient.

The same division sign is written two ways: $a : b$ and $a \div b$ mean exactly the same thing.

The names in division

• dividend — the number being divided ($a$),
• divisor — the number you divide by ($b$),
• quotient — the result ($c$).

Division and multiplication

Division and multiplication are inverse operations, so you can check every division with multiplication:

$a : b = c \iff c \cdot b = a$

Since $56 : 8 = 7$, we have $7 \cdot 8 = 56$. Knowing the times tables is therefore the key to dividing fluently.

Division with a remainder

Not every number divides evenly. Then we write division with a remainder:

$a = b \cdot q + r, \qquad 0 \le r < b$

where $q$ is the quotient (the whole part) and $r$ is the remainder — always smaller than the divisor. For example $17 : 5$ gives quotient $3$ and remainder $2$, because $17 = 5 \cdot 3 + 2$.

Properties and division by zero

• dividing by one leaves the number unchanged: $a : 1 = a$,
• a number divided by itself is one: $a : a = 1$ (for $a \neq 0$),
• you must never divide by zero — if $a : 0 = c$, then $c \cdot 0 = a$ would have to hold, but a product with zero is always $0$, so for $a \neq 0$ no such $c$ exists. Division by zero stays undefined.

Division is neither commutative nor associative: $a : b$ is generally not equal to $b : a$.

Common mistakes

• Dividing by zero — an undefined operation; there is no result.
• Swapping the dividend and divisor — $12 : 3 = 4$, but $3 : 12$ is something else entirely (less than one).
• Dropping the remainder — if the numbers don't divide evenly, the answer has a quotient and a remainder; check it with $a = b \cdot q + r$.