Explore Arithmetic operations

# Dividing integers

After you’ve learned how to add, subtract, and multiply integers, division is next up!

You’ve been asked to calculate how many times one number fits into another. This is when division comes in handy. Dividing just means subtracting the same number a specific number of times until you get $$0$$, to see how many times a number you are subtracting with fits into the number you started with. So instead of calculating $$12-4=8, ~8-4=4,~ 4-4=0$$, you can think of it as if you were calculating $$12\div4$$.

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

## What does it mean to divide integers?

Dividing integers means splitting integers into an equal number of parts.

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:

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

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

And that’s pretty much it!

### Why is dividing integers so useful?

Besides the fact that multiplying integers is one of the basic concepts we learn in math, think of it this way: there are many real-life problems it can solve! For example, a person scored $$-24$$ points because of many penalties in a game. Each penalty is $$-8$$ points. How many penalties did the person get? Well, that problem can be solved by dividing integers. The number of points has to be divided by the penalty points. It all comes down to a simple math problem of dividing integers:

$$-24\div (-8)$$

## How to divide integers

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

If two positive or two negative numbers are divided, their quotient is a positive number.

$$(-)\div(-)=+$$
$$(+)\div(+)=+$$

If two numbers of different signs are divided, their quotient is a negative number.

$$(-)\div(+)=(-)$$
$$(+)\div(-)=(-)$$

### Example 1

Divide the integers:

$$-6\div(-3)$$

Dividing two negatives equals a positive $$(-)\div(-)=+$$:
$$6\div 3$$
Divide the numbers:

$$2$$

### Example 2

Divide the integers:

$$-20 \div4$$

Dividing a negative and a positive equals a negative $$(-)\div(+)=(-)$$:
$$-(20 \div4)$$
Divide the numbers:

$$-5$$

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 division.
2. Divide 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!

Divide the integers:

1. $$-35\div 7$$
2. $$42\div(-6)$$
3. $$-63\div (-9)$$
4. $$-120\div 20$$

Solutions:

1. $$-5$$
2. $$-7$$
3. $$7$$
4. $$-6$$

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:

/

## Got arithmetic homework?

Head over to the Photomath app for instant, step-by-step solutions to all of your arithmetic problems.