Multiplication

Multiplication is one of the four basic arithmetic operations. Here we focus on what multiplication means, how to read and write it, and practical methods to carry it out.

The idea of multiplication

At the beginner level, multiplication is most often understood as repeated addition of the same number.

If you have $4$ groups of $3$ apples each:

• One way to describe this is: $3 + 3 + 3 + 3$
• Another way is: $4 \times 3$

So:
$$4 \times 3 = 3 + 3 + 3 + 3 = 12$$

In words, $4 \times 3$ can be read as:

• “4 times 3”
• “4 multiplied by 3”
• “4 groups of 3”

This repeated-addition meaning works well when multiplying whole numbers (natural numbers and integers) and is a good starting point.

Notation and ways to say it

Multiplication appears in several notations:

• With a $\times$ sign:
• $4 \times 3$
• With a dot:
• $4 \cdot 3$
• With parentheses (often used in algebra):
• $(4)(3)$ or $4(3)$

All of these mean “multiply these numbers.”

When you say them:

• $4 \times 3$ is “4 times 3”
• $4 \cdot 3$ is also “4 times 3”
• $4(3)$ is “4 times 3” or “4 multiplied by 3”

In everyday arithmetic with numbers, the $\times$ sign is most common. In algebra (with variables) the dot and parentheses are more common to avoid confusion with the letter “x.”

Basic multiplication facts: times tables

For working with whole numbers, it is very useful to memorize multiplication tables (also called “times tables”) for small numbers, usually from $1$ to $10$ or $12$. These are just the results of multiplying pairs of small whole numbers.

Some basic examples:

• For $2$:
• $2 \times 1 = 2$
• $2 \times 2 = 4$
• $2 \times 3 = 6$
• $2 \times 4 = 8$
• $2 \times 5 = 10$
• For $5$:
• $5 \times 1 = 5$
• $5 \times 2 = 10$
• $5 \times 3 = 15$
• $5 \times 4 = 20$
• $5 \times 5 = 25$
• For $10$:
• $10 \times 1 = 10$
• $10 \times 2 = 20$
• $10 \times 3 = 30$

Knowing these by heart makes all later arithmetic much faster.

You don’t need to memorize everything at once. You can:

• Practice one number at a time (for example, all the $2$’s, then all the $3$’s)
• Use patterns (for example, $5$-times table always ends in $0$ or $5$)
• Notice that $3 \times 4$ and $4 \times 3$ are the same result, so you really remember fewer separate facts.

Using arrays and area to picture multiplication

Multiplication of whole numbers can be pictured as a rectangular array of objects.

For example, to represent $3 \times 4$:

• Draw $3$ rows
• Each row has $4$ objects

You might picture:

• Row 1: ● ● ● ●
• Row 2: ● ● ● ●
• Row 3: ● ● ● ●

You can count that there are $12$ dots in total, so $3 \times 4 = 12$.

This same idea leads to viewing multiplication as an area problem:

• A rectangle that is $3$ units tall and $4$ units wide has area $3 \times 4$ square units.

Arrays and area pictures can help you:

• Understand why certain multiplication properties are true
• Check your work or estimate answers

Properties specific to multiplication

These properties are rules that multiplication of numbers follows. They are helpful when computing and when simplifying expressions.

Commutative property

The commutative property of multiplication says that the order of the two numbers does not change the product:

$$a \times b = b \times a$$

Examples:

• $3 \times 4 = 4 \times 3 = 12$
• $7 \times 2 = 2 \times 7 = 14$

If you think of arrays, turning a $3$-by-$4$ rectangle sideways gives a $4$-by-$3$ rectangle, but it still has the same number of dots.

Associative property

The associative property says that when you multiply three numbers, it doesn’t matter which pair you multiply first:

$$(a \times b) \times c = a \times (b \times c)$$

Example:

• $(2 \times 3) \times 4 = 6 \times 4 = 24$
• $2 \times (3 \times 4) = 2 \times 12 = 24$

You get the same result either way, so you can group the numbers in the way that makes calculation easier for you.

Distributive link with addition

Multiplication connects to addition through the distributive property. In words, multiplying a sum is the same as multiplying each part and then adding:

$$a(b + c) = ab + ac$$

Example:

• $3(4 + 2) = 3 \times 6 = 18$
• $3 \times 4 + 3 \times 2 = 12 + 6 = 18$

This shows a way to break large products into simpler ones.

Multiplicative identity

There is a special number for multiplication called the multiplicative identity: it is $1$.

Multiplying any number by $1$ leaves it unchanged:

$$a \times 1 = 1 \times a = a$$

Examples:

• $5 \times 1 = 5$
• $1 \times 100 = 100$
• $(-7) \times 1 = -7$

Multiplicative zero

Multiplying any number by $0$ always gives $0$:

$$a \times 0 = 0 \times a = 0$$

Examples:

• $7 \times 0 = 0$
• $0 \times 12345 = 0$

This is because “zero groups of something” means “you have none.”

Multiplication with sign (positive and negative)

When multiplying integers, the sign of the answer follows a simple pattern:

• Positive $\times$ positive = positive
• $3 \times 4 = 12$
• Negative $\times$ positive = negative
• $(-3) \times 4 = -12$
• Positive $\times$ negative = negative
• $3 \times (-4) = -12$
• Negative $\times$ negative = positive
• $(-3) \times (-4) = 12$

A helpful way to remember:

• If the two numbers have the same sign, the product is positive.
• If they have different signs, the product is negative.

Mental strategies for multiplication

You don’t always need a written method; you can often use mental tricks.

Some common strategies:

• Using known facts:
• If you know $6 \times 7 = 42$, then you also know $7 \times 6 = 42$.
• Breaking numbers apart (using distributive property):
• To compute $7 \times 8$:
  $ \times 8 = 7 \times (5 + 3) = 7 \times 5 + 7 \times 3 = 35 + 21 = 56$$
• Using nearby easy numbers:
• To compute $9 \times 7$, think of $10 \times 7 = 70$ and subtract one group of $7$:
  $ \times 7 = 10 \times 7 - 1 \times 7 = 70 - 7 = 63$$
• Doubling and halving:
• For $4 \times 25$:
  $ \times 25 = 2 \times (2 \times 25) = 2 \times 50 = 100$$

These ideas prepare you for more advanced arithmetic and algebra.

Written multiplication methods (multi-digit numbers)

When numbers have more than one digit, mental methods might be difficult, so we use written procedures.

Example: multiplying a one-digit by a multi-digit number

Compute $7 \times 326$:

• Multiply $7$ by each digit of $326$, starting from the right, and carry when needed.

1. $7 \times 6 = 42$
• Write down $2$, carry $4$
2. $7 \times 2 = 14$
• Add the carried $4$: $14 + 4 = 18$
• Write down $8$, carry $1$
3. $7 \times 3 = 21$
• Add the carried $1$: $21 + 1 = 22$

So $7 \times 326 = 2282$.

Example: multiplying two multi-digit numbers

Compute $23 \times 47$:

1. Multiply $23$ by $7$ (the ones digit of $47$):
• $7 \times 3 = 21$, write $1$, carry $2$
• $7 \times 2 = 14$, plus $2$ is $16$
• So $23 \times 7 = 161$
2. Multiply $23$ by $4$ (the tens digit of $47$), but this $4$ actually means $40$:
• $4 \times 3 = 12$, write $2$, carry $1$
• $4 \times 2 = 8$, plus $1$ is $9$
• So $23 \times 4 = 92$, which represents $23 \times 40 = 920$
• In the usual layout, we write $92$ shifted one place to the left (with a zero at the end) to show it’s “tens”.
3. Add the two partial results:
• $161 + 920 = 1081$

So $23 \times 47 = 1081$.

The key ideas are:

• Multiply each digit of one number by the entire other number.
• Keep track of the place value (ones, tens, hundreds, …).
• Add all the partial products at the end.

Multiplication with zero, one, and powers of ten

Certain numbers make multiplication especially simple.

• Multiplying by $0$:
• Always gives $0$: $45 \times 0 = 0$
• Multiplying by $1$:
• Leaves the number unchanged: $1 \times 356 = 356$
• Multiplying by $10, 100, 1000$, etc.:
• For whole numbers, this shifts digits to the left and adds zeros:
• $34 \times 10 = 340$
• $34 \times 100 = 3400$
• $7 \times 1000 = 7000$

These facts are very helpful when estimating products and checking if your answer is reasonable.

Checking multiplication results

It is easy to make small mistakes when multiplying, especially with large numbers. Here are simple checks:

• Estimate using rounded numbers:
• For $198 \times 51$, think of $200 \times 50 = 10000$.
  If your exact answer is close to 000$, it is probably reasonable.
• Use the reverse operation (division):
• If you think $37 \times 24 = 888$, divide: $888 \div 24$.
  If you do not get $, then a mistake has been made.
• Check with a calculator (when allowed):
• Use a calculator to confirm your final result, especially in real-life calculations.

Multiplication in everyday life

Multiplication appears naturally in many situations:

• Repeated payments:
• $5$ bus trips costing $2$ dollars each:
  $ \times 2 = 10 \text{ dollars}$$
• Counting items in groups:
• $8$ boxes each containing $12$ cookies:
  $ \times 12 = 96 \text{ cookies}$$
• Area of rectangles:
• A $4$ m by $6$ m rectangular room:
  $$\text{area} = 4 \times 6 = 24 \text{ square meters}$$

Becoming comfortable with multiplication makes it easier to handle money, measurements, and many other everyday tasks, as well as preparing you for more advanced topics in mathematics.