Table of Contents
Division is one of the four basic arithmetic operations, alongside addition, subtraction, and multiplication. In this chapter, we focus only on what is specific to division: what it means, how to write it, how to perform it in simple cases, and some typical patterns and difficulties beginners encounter.
The meaning of division
Division answers a “sharing” or “grouping” question.
- Sharing (fair share) meaning
“If I have $12$ objects and $3$ people, and I share them equally, how many does each person get?”
This is written as:
$$
12 \div 3
$$
and read as “12 divided by 3”.
Here, each person gets $4$, so:
$$
12 \div 3 = 4
$$
- Grouping meaning
“If I have $12$ objects and I want to put them into groups of $3$, how many groups can I make?”
This is the same division:
$$
12 \div 3 = 4
$$
but now the $4$ means “4 groups”.
These two interpretations—sharing and grouping—are different stories for the same operation.
Division notation
You will see division written in several ways:
- With a division sign:
$$
12 \div 3 = 4
$$ - As a fraction:
$$
\frac{12}{3} = 4
$$ - With a slash:
$$
12 / 3 = 4
$$
All three notations mean exactly the same operation in this context.
The number being divided (here $12$) is called the dividend.
The number you divide by (here $3$) is the divisor.
The result (here $4$) is the quotient.
So, in $12 \div 3 = 4$:
- dividend: $12$
- divisor: $3$
- quotient: $4$
Division and multiplication
Division is closely related to multiplication. In fact, you can think of division as “undoing” multiplication.
If you know that:
$$
3 \times 4 = 12,
$$
then you can form two related division facts:
$$
12 \div 3 = 4, \quad 12 \div 4 = 3.
$$
So, to answer $12 \div 3$, you can ask:
“What number multiplied by $3$ gives $12$?”
Since $3 \times 4 = 12$, the answer is $4$.
This relationship is very useful for basic division with small whole numbers, especially when you know some multiplication facts.
Simple whole-number division
We focus here on whole-number division where the answer is also a whole number (no remainder yet).
Examples:
- $15 \div 5$
We ask: “$5$ times what equals $15$?”
Since $5 \times 3 = 15$, we have
$$
15 \div 5 = 3.
$$
- $21 \div 7$
Since $7 \times 3 = 21$,
$$
21 \div 7 = 3.
$$
- $8 \div 2$
Since $2 \times 4 = 8$,
$$
8 \div 2 = 4.
$$
Often, division like this can be done by recalling or figuring out a multiplication fact.
Division with 1 and with the same number
Certain division patterns occur so often that it is helpful to know them as rules.
- Dividing by 1
When you divide any number by $1$, the result is the original number:
$$
a \div 1 = a \quad \text{(for any number } a\text{)}.
$$
Examples:
- $7 \div 1 = 7$
- $100 \div 1 = 100$
This fits the sharing idea: if you share $100$ things among $1$ person, that person gets all $100$.
- Dividing a nonzero number by itself
When you divide a nonzero number by itself, the result is $1$:
$$
a \div a = 1 \quad \text{for any nonzero number } a.
$$
Examples:
- $5 \div 5 = 1$
- $23 \div 23 = 1$
This matches the idea that “$a$ times $1$ equals $a$,” so division “backwards” from $a$ to $a$ gives $1$.
Division by zero is not allowed
A very important special case is division by zero. Expressions like
$$
12 \div 0 \quad \text{or} \quad \frac{7}{0}
$$
are not allowed. We say division by zero is undefined.
To see the problem using the multiplication relationship, suppose $12 \div 0$ had some answer, call it $x$. Then we would have:
$$
0 \times x = 12.
$$
But $0$ times any number is $0$, never $12$. There is no such $x$.
So:
- You can divide $0$ by a nonzero number: $0 \div 5 = 0$.
- You cannot divide by $0$: expressions like $a \div 0$ make no sense in ordinary arithmetic.
Whenever you see a problem with $0$ as a divisor, you should recognize that the expression is undefined.
Division with remainders (informal introduction)
Sometimes, when we divide whole numbers, they do not divide evenly into a whole number. In these cases, we may have a remainder.
For example, consider $14 \div 3$.
We ask: “How many groups of $3$ fit into $14$?”
- $3 \times 4 = 12$ (fits into $14$),
- $3 \times 5 = 15$ (too large).
So, we can make $4$ full groups of $3$, which use $12$ objects, and there are $2$ objects left over.
We can write:
$$
14 \div 3 = 4 \text{ remainder } 2,
$$
often shortened to:
$$
14 \div 3 = 4\text{ R }2.
$$
This means:
- quotient: $4$
- remainder: $2$
When working only with whole numbers, this is a common way to express division that does not come out evenly.
Checking a division answer
You can usually check a division involving whole numbers by using multiplication (and possibly adding a remainder).
- For division with no remainder:
If
$$
a \div b = q,
$$
then you should have
$$
b \times q = a.
$$
Example: $18 \div 6 = 3$ can be checked by $6 \times 3 = 18$.
- For division with a remainder:
If
$$
a \div b = q \text{ R } r,
$$
then
$$
b \times q + r = a.
$$
For $14 \div 3 = 4\text{ R }2$, we check:
$$
3 \times 4 + 2 = 12 + 2 = 14.
$$
This gives a quick way to see whether a division answer is reasonable.
Division facts and mental strategies
For small whole numbers, you can often divide in your head by relying on:
- Your knowledge of multiplication facts.
- Breaking numbers into easier parts.
Examples:
- $24 \div 4$
Since $4 \times 6 = 24$,
$$
24 \div 4 = 6.
$$
- $30 \div 5$
Since $5 \times 6 = 30$,
$$
30 \div 5 = 6.
$$
- $40 \div 10$
Since $10 \times 4 = 40$,
$$
40 \div 10 = 4.
$$
For slightly larger numbers, you can think in steps. For example, to estimate $52 \div 4$:
- $4 \times 10 = 40$
- $4 \times 3 = 12$
- $40 + 12 = 52$
So $4$ goes into $52$ exactly $13$ times:
$$
52 \div 4 = 13.
$$
These are simple beginnings to mental division, useful before more systematic written methods are introduced elsewhere.
Division and order: not commutative
Addition and multiplication are commutative: for them, changing the order of the numbers does not change the result (for example, $3 + 5 = 5 + 3$ and $4 \times 6 = 6 \times 4$).
Division is not commutative. In general:
$$
a \div b \neq b \div a.
$$
Example:
- $12 \div 3 = 4$,
- $3 \div 12$ is a completely different calculation (and does not give $4$).
So, with division, the order of the numbers matters a lot. The first number is the dividend, and the second is the divisor; swapping them changes the meaning.
Summary
- Division is the operation of sharing or grouping a quantity into equal parts.
- It is written with $\div$, as a fraction, or with
/. - Division is closely tied to multiplication: $a \div b$ asks for the number that, when multiplied by $b$, gives $a$.
- Dividing by $1$ leaves a number unchanged; dividing a nonzero number by itself gives $1$.
- Division by $0$ is never allowed in ordinary arithmetic.
- Whole-number division may have a remainder; you can always check your work using multiplication (and adding the remainder).
- Division is not commutative: swapping the numbers changes the result.