Explore Arithmetic operations

# Subtracting integers

One of the first things in math that we learn is counting. We count fingers, toys, apples, and oranges. What if you have $$5$$ apples, but then you eat $$1$$ of them? How will you determine how many apples you have left? You’re actually subtracting $$1$$ from the number of apples you already have. Then we learn how to subtract more than one at a time – this is called subtracting whole numbers: $$5-1=4$$.

But what if the minuend (the amount you’re subtracting from) is less than the subtrahend (the amount you’re subtracting)? This is where we put whole numbers aside, and where integers come into the picture!

## What does it mean to subtract integers?

Subtracting integers means performing subtraction over a set of integer numbers. An integer is any number from a set of whole numbers and their additive inverses – the numbers with the same absolute value but 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 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:

The minuend tells us where to start on the number line. The sign on the subtrahend will tell us which way to move on the number line. The absolute value of the second summand is just the number without its sign.

### Why is subtracting integers so useful?

Besides the fact that subtracting integers is one of the first basic things we learn in math, think of it this way: there are many real-life problems it can solve! For example, a mountain’s highest point is $$161$$ meters, and the deepest point in the sea beneath it is $$80$$ meters. What is the height difference between those two points?

This problem can be solved by subtracting integers. When we need to find the height difference, we need to subtract the numbers. In this case, this means we need to subtract the deepest point from the highest point and since the deepest point is $$80$$ meters below the sea surface, we can represent it with an integer $$-80$$.

Now, it all comes down to a simple math problem of subtracting integers:

$$161-(-80)$$

## How to subtract integers

Now that we know what are integers and why they’re useful, it’s time to see it in action! Let’s walk through a problem together.

### Example 1

Subtract the integers:

$$3-5$$

Determine the absolute value of each number:

$$|3|=3, |-5|=5$$

Notice that the negative number $$-5$$ has a greater absolute value since $$5$$ is greater than $$3$$. Hence, keep the negative sign of $$-5$$, and subtract the lesser absolute value $$3$$ from the greater one, $$5$$.

$$-(5-{3})$$

Subtract the numbers inside the parentheses:

$$-2$$

### Example 2

Subtract the integers:

$$-5-8$$

Remember, to factor out a term from an expression means to extract that term from all the terms in the expression. So, factor out the negative sign:

$${-}({5+8})$$

Add the numbers $$5$$ and $$8$$:

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 - when one number is negative

1. Keep the sign of the number with the larger absolute value  and subtract the smaller absolute value from the larger.
2. Subtract the numbers.

## Study summary - when both numbers are negative

1. Factor out the negative sign from the expression.
2. Add the numbers.

## Do 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!

Subtract the integers:

1. $$8-15$$
2. $$7-5$$
3. $$-9-1$$
4. $$-3-4$$

Solutions:

1. $$-7$$
2. $$2$$
3. $$-10$$
4. $$-7$$

If you’re still struggling with 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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