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 are very useful because they help us talk about things that are missing or going down, like owing money or temperatures below freezing.

We usually write negative numbers with a minus sign in front of them, like −3, which we say as "minus three" or "negative three". Numbers bigger than zero are called positive numbers, and 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 almost two thousand years ago in a book called the Nine Chapters on the Mathematical Art. Over time, people all over the world learned how to add, subtract, and multiply with negative numbers, making math even more powerful and 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 are used in many everyday situations to show a difference, loss, or measurement below zero. In sports, for example, a team’s goal difference can be negative if they concede more goals than they score. In science, temperatures below zero degrees Celsius or Fahrenheit use negative numbers, and map locations south of the equator or west of the prime meridian are also shown with negative values.

In finance, negative numbers show debts or losses, such as a bank account overdraft or a business with negative earnings. They can also show a country’s economic decline, like negative growth in its GDP. Negative numbers appear in many other areas, from counting down time on music players to showing a drop in a politician’s approval rating.

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, because both are debts that combine into a bigger debt.

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 negation of 3. When you add a number and its negation, the result is always zero: 3 + (-3) = 0. This idea works for all numbers, including zero and other negative numbers. The absolute value of a number is how big it is, without the sign. So, the absolute value of both -3 and 3 is 3.

Formal construction of negative integers

See also: Integer § Construction

We can expand the idea of natural numbers, like 1, 2, 3, and so on, to include negative numbers by using pairs of these natural numbers. For example, we might pair (3, 5) to represent a number. We can add and multiply these pairs using special rules.

This way of building numbers helps us understand how negative numbers work and how they relate to positive ones. It also shows that each positive number has a unique "opposite" negative number.

History

For a long time, people found it hard to understand negative numbers because you can't have something like "minus-three apples". Early mathematicians thought equations with negative answers were silly or wrong.

The first known use of negative numbers comes from an ancient Chinese book called the Nine Chapters on the Mathematical Art. Chinese mathematicians used red and black counting rods to show positive and negative numbers. In India, negative numbers were used to mean debts. Later, Islamic and European mathematicians began to use and understand negative numbers better, helping them solve more complex problems.