Negative number

In mathematics, a negative number is the opposite of a positive real number. It is a number that is less than zero. Negative numbers help us talk about things that are missing or going down, like owing money or very cold temperatures.

We write negative numbers with a minus sign in front of them, like −3. We say this as "minus three" or "negative three". Numbers bigger than zero are called positive numbers. We can write them with a plus sign, like +3, to show they are positive. Zero is special because it is not positive or negative.

Negative numbers have been used for a very long time. Ancient Chinese mathematicians wrote about them a long time ago in a book called the Nine Chapters on the Mathematical Art. Over time, people learned how to add, subtract, and multiply with negative numbers, which made math more useful.

Introduction

The number line

Main article: Number line

Negative numbers are the opposite of positive numbers. They are numbers that are less than zero. We can imagine numbers on a number line, where zero is in the middle. Positive numbers are to the right of zero, and negative numbers are to the left.

A bigger negative number, like -8, is actually smaller than a smaller negative number, like -5, even though 8 is bigger than 5 when we talk about positive numbers. So, -8 is less than -5. Every number that isn’t zero is either positive or negative. We sometimes add a plus sign in front of positive numbers, like +3, to show they are positive.

As the result of subtraction

Negative numbers often come from subtracting a bigger number from a smaller one. For example, if we subtract 3 from 0, we get -3:

0 − 3  =  −3.

In general, when you subtract a larger number from a smaller one, the result is negative. The size of this result is just the difference between the two numbers. For example,

5 − 8  =  −3

because 8 minus 5 equals 3.

Everyday uses of negative numbers

Negative numbers help us understand differences, losses, or measurements below zero.

In sports, a team may have a negative goal difference if they allow more goals than they score. In science, temperatures below zero degrees Celsius or Fahrenheit are written with negative numbers. Map locations south of the equator or west of the prime meridian also use negative values.

In finance, negative numbers show debts or losses, such as an overdraft in a bank account or a business with negative earnings. They can also show when a country’s economy is not growing. Negative numbers are used in many other ways, like counting down time on music players or showing when a politician’s approval rating goes down.

Arithmetic involving negative numbers

The minus sign shows when we are talking about subtraction or turning a number into its opposite. For example, −5 means the opposite of 5.

Adding two negative numbers is like adding two debts together. For example, adding −3 and −5 gives −8.

When we mix positive and negative numbers in addition, think of the negative numbers as amounts being taken away. For example, 8 + (−3) is the same as 8 − 3, which equals 5. If the negative number is bigger, the result is negative, like (−8) + 3 equals −5.

Negation

Main article: Additive inverse

A negative number is the opposite of a positive number. For example, -3 is the opposite of 3. When you add a number and its opposite, the result is always zero: 3 + (-3) = 0. This works for all numbers, including zero and other negative numbers. The absolute value of a number tells us how big it is, without looking at the sign. So, the absolute value of both -3 and 3 is 3.

Formal construction of negative integers

See also: Integer § Construction

We can start with natural numbers, like 1, 2, 3, and so on. We can then make negative numbers by using pairs of these natural numbers. For example, we might pair (3, 5) to stand for a number.

We can add and multiply these pairs using special rules.

This method helps us see how negative numbers work and how they relate to positive numbers. It also shows that each positive number has a special "opposite" negative number.

History

For a long time, people found negative numbers hard to understand. They thought about things like "minus-three apples" and didn't know what that meant.

The first known use of negative numbers was in an ancient Chinese book called the Nine Chapters on the Mathematical Art. Chinese mathematicians used red and black counting rods to show numbers. In India, negative numbers meant debts. Later, mathematicians in Islamic and Europe started using negative numbers to solve harder problems.