Explore Arithmetic operations

# Multiplying integers

After you’ve learned how to add and subtract, multiplication is next in line. If you’ve been asked to add the same number multiple times, this is when multiplication comes in handy. Multiplying just means adding the same number a specific number of times. So instead of calculating $$2+2+2$$, you can think of it as calculating $$3× 2$$.

But what if one or both of the numbers was negative? Well, that’s when we don’t multiply whole numbers anymore – that’s when integers come into the picture!

## What does it mean to multiply integers?

Multiplying integers means adding the same integer multiple times. An integer is any number from a set of whole numbers and their additive inverses – the numbers with the same absolute value but with an opposite sign:

$$\{\ldots-3,-2,-1,0,1,2,3,\ldots\}$$

‘’Wait… what is the absolute value of a number?!’’

Good question! The absolute value of a number is that same number, but only without the sign in front of it. Why? Because the absolute value actually shows what the distance is from that number to $$0$$ on the number line. For example, the absolute value of $$|-4|=4$$. That’s the distance from $$-4$$ to $$0$$ on the number line; $$4$$ units:

Multiplying integers comes down to multiplying their absolute values and determining the sign of the result by applying the following rules:

1. The product of two positives equals a positive
2. The product of two negatives equals a positive
3. The product of a negative and a positive equals a negative.

And that’s pretty much it!

### Why is multiplying integers so useful?

Besides the fact that multiplying integers is one of the most basic things we learn in math, there are many real-life problems it can solve! For example, a submarine is $$20$$ feet below the sea level and it need to dive $$5$$ times deeper, on which depth will the submarine be? Well, that problem can be solved by multiplying integers. You have to calculate $$5$$ times the number $$-20$$. It all comes down to a simple math problem of multiplying integers:

$$-20\times 5$$

## How to multiply integers

Before we learn how to multiply integers, let’s check out some rules for multiplication.

If two positive or two negative numbers are multiplied, their product is a positive number.

$$(-)\times(-)=+$$
$$(+)\times(+)=+$$

If two numbers of different signs are multiplied, their product is a negative number.

$$(-)\times(+)=(-)$$
$$(+)\times(-)=(-)$$

Now, let’s walk through a problem together.

### Example 1

Multiply the integers:

$$-5\times 8$$

Remember that multiplication is just a shorter way of writing repeated addition.

Therefore, if we have a product $$a\times b$$ where $$a$$ is positive and $$b$$ is negative, we are adding $$b$$ to itself, $$a$$ number of times.

Since adding negative numbers will always result in a negative number, each multiplication problem with a negative and a positive term will result in a negative term.

So, factor out the negative sign:
$$-({5\times 8})$$
Multiply the numbers:

$$-40$$

### Example 2

Multiply the integers:

$$-3\times(-7)$$

Remember that multiplication is just a shorter way of writing repeated addition.

The first factor, $$-3$$, represents the repeated subtraction of a number, $$\text{three times in a row}$$:
$$(-3) \times (-7) = -~ ? -~?-~?$$
The second factor, $$-7$$, means that the number that is subtracted three times is $$-7$$:
$$(-3) \times ({-7}) = -({-7})-({-7})-({-7})$$
Use the fact that the opposite of a negative is a positive:
$$(-3) \times (-7) = {+} 7 {+} 7 {+} 7$$
When we add the numbers, we get a positive result, $$21$$:

$$(-3) \times (-7) = 21$$

That wasn’t so bad, right? Now that we’ve walked through detailed examples, let’s review the overall process so you can learn how to use it with any problem:

## Study summary

1. Use the rules for multiplication.
2. Multiply the numbers.

## Try it yourself!

Practicing math concepts like this one is a great way to prepare yourself for the math journey to come! So, when you’re ready, we’ve got some practice problems for you!

Multiply the integers:

1. $$-5\times4$$
2. $$7\times(-3)$$
3. $$-9\times (-4)$$
4. $$-8\times2$$

Solutions:

1. $$-20$$
2. $$-21$$
3. $$36$$
4. $$-16$$

If you’re struggling through the solving process, that’s totally okay! Stumbling a few times is good for the learning process. If you get stuck or lost, scan the problem using your Photomath app and we’ll walk you through it!

Here’s a sneak peek of what you’ll see:

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