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Lesson 7: Comparing absolute values

# Negative numbers FAQ

## What are negative numbers?

Negative numbers are numbers that are less than zero. They have a negative sign (minus) in front of them, like minus, 3 or minus, 17. Negative numbers can represent things that are opposite, missing, or below something else.
For example, if we own 5 apples and we eat 2 of them, we can say that we have 5, minus, 2, equals, 3 apples left. But if we own 2 apples and we eat 5 apples, we can say that we have 2, minus, 5, equals, minus, 3 apples left. That means we ate 3 more apples than we had, or we owe 3 apples to someone.
Negative numbers can also represent temperatures that are below freezing, heights that are below sea level, or debts that we owe.

## How do we compare negative numbers?

Comparing negative numbers means finding out which one is smaller or larger than another one. It works the same as comparing positive numbers. Numbers farther to the right (or farther up, on a vertical number line) are greater than numbers that are farther left (or down) on the same line.
To compare negative numbers, we need to remember that the farther left a number is relative to 0 on the number line, the smaller it is. For example, minus, 3 is smaller than minus1, because minus, 3 is farther left of 0 than minus, 1 is. We can write this as minus, 3, is less than, minus, 1 or as minus, 1, is greater than, minus, 3.
We can also use the opposite numbers to compare two negative numbers. For example, the opposite of minus, 3 is 3, and the opposite of minus, 5 is 5. Since positive 5 is farther right of 0 than positive 3 is, 5, is greater than, 3. The negative numbers will have the opposite relationships, since they move left of 0 instead. So minus, 5 is less than minus, 3, which we write as minus, 5, is less than, minus, 3.
A positive number is always greater than a negative number.

## What is absolute value?

Absolute value is the distance of a number from zero on the number line. It is always a positive number or zero. We write it with two vertical bars around the number. For example, the absolute value of minus, 3 is 3, and the absolute value of 3 is also 3. We can write this as vertical bar, minus, 3, vertical bar, equals, 3 and vertical bar, 3, vertical bar, equals, 3. The absolute value of 0 is 0, since there is no distance between 0 and itself.
The absolute value of a rational number is the same as the value of the number without the sign. For example, the absolute value of minus, 1, point, 25 is 1, point, 25, and the absolute value of 0, point, 75 is 0, point, 75. We can write this as vertical bar, minus, 1, point, 25, vertical bar, equals, 1, point, 25 and vertical bar, 0, point, 75, vertical bar, equals, 0, point, 75.

## Where do we use negative numbers in the real world?

Negative numbers and absolute value can help us describe and measure many things that we encounter every day. Here are some examples of where we use negative numbers and absolute value in the real world:
• We use negative numbers to show temperatures that are below freezing, like minus, 10, degree Celsius or minus, 15, degree Fahrenheit. We can also use negative numbers to show how much colder or warmer a place is than another place, like minus, 5 degrees difference or minus, 10 degrees difference.
• We use negative numbers to show heights that are below sea level, like minus, 100 meters or minus, 300 feet. We can also use negative numbers to show how much deeper or higher a place is than another place, like minus, 50 meters difference or minus, 150 feet difference.
• We use negative numbers to show debts that we owe, like minus, dollar sign, 500 or minus, dollar sign, 1000.
• We use negative and positive numbers to show movement in specific directions, like saying that we moved minus, 10 seconds in a video to mean that we went 10 seconds backwards in the video. If moved 10 seconds forward in the video, we could use positive 10.
The absolute value tells us the amount of the change in each case, but removes the information about the direction of the change. For example, the absolute value of a height could tell us its distance from sea level, but would not tell us whether the object was above or below sea level.