History of Negative Numbers

This is only a brief history of negative numbers. For a more comprehensive coverage and a longer reading, read The History of Negative Numbers on NRICH. For a concise, but right to the point introduction, stay right here with me.

In this lesson, you will learn

  • how people worked with subtraction long before negative numbers were accepted
  • who first used negative numbers, from the Chinese and Indian mathematicians to Cardano
  • the different ways people have written negative numbers, from rods with a slash to the minus sign

A Timeline of Negative Numbers

  1. Ancient times

    Subtraction without negative numbers

    Early writings of the Babylonians, Hindus, Egyptians, or Chinese do not contain any trace of the recognition of negative numbers.

    Nonetheless, they had no problems correctly doing computations involving subtraction such as:

    Interactive: tap to work it out

    Every step stays at zero or above, so no negative number is needed.

  2. 200 B.C.E.China

    Red rods and black rods

    The first mention of negative numbers can be traced to the Chinese around 200 B.C.E.

    The Chinese used red rods to represent positive numbers, but black rods to represent negative numbers.

    Interactive: build a number with rods

  3. 3rd centuryAlexandria

    Diophantus calls an equation absurd

    In the third century, around A.D. 250, the Alexandrian mathematician Diophantus said in his text Arithmetica that the following equation is absurd.

    4x + 20 = 4

    He said this because x would have to be −4.

    Interactive: check it

  4. A.D. 620–630India

    Brahmagupta’s rules

    In India, negative numbers started appearing around A.D. 620–630 in the work of the Indian mathematician Brahmagupta.

    He called positive numbers affirmative quantities. For instance, he said in his work that the sum of two affirmative quantities is affirmative.

    This means that the sum of two positive numbers is positive.

    3+5= 8
    Remember

    Brahmagupta also described positive numbers as fortunes and negative numbers as debts. A debt is easy to picture as an amount below zero.

  5. Around 1300China

    The rule of signs

    Around 1300, the Chinese mathematician Zhu Shijie (also written Chu Shih-chieh) gave the “rule of signs” for multiplication in his book of 1299.

    + × + = +− × − = ++ × − = −− × + = −

  6. Around 1545Italy

    Cardano and negative roots

    In the 16th century, around 1545, the study of solutions of equations began in Italy.

    This led the Italian mathematician Cardano to recognize negative roots as he tried to understand the meaning of the square root of a negative number such as

    √−2

    During that time, he also clearly stated rules for negative numbers.

  7. 1600s–1800sEurope

    The number line

    In the 1600s, the English mathematician John Wallis drew negative numbers on a number line, to the left of zero. By the 1800s, mathematicians had finally agreed on clear rules, and negative numbers were accepted as real numbers.

Which value of x makes 2x + 10 = 4 true, and would Diophantus have accepted it?

Show answer

x = −3, and no. 2(−3) + 10 = −6 + 10 = 4. Diophantus would have called this equation absurd, because the answer is negative.

History of Negative Numbers and the Use of Notation

Chart titled History of negative numbers showing different ways people wrote −6, including a slashed rod numeral, m:6, a small circle before 6 and a 6 with a dot above it

There have been various notations to represent negative numbers.

Interactive: see how −6 was written

The Hindus

The Hindus put a small circle or dot over or next to a number to denote that it was negative.

The Chinese

The Chinese drew a slash through a portion of the number to indicate that the number was negative.

For example, −1020 was written as shown below. Here, the slash is drawn over two bars or the number 2.

1 0 2 0

Cardano and the Europeans

Cardano used the symbol m: for negative numbers. For example, −6 can be written as m:6.

The Europeans invented the use of the symbols + and − to represent positive and negative numbers.

Today

Nowadays, textbooks can use any color to represent negative numbers. A popular color that is used today to represent negative numbers is red.

Common mistake

Thinking red has always meant negative. In ancient China it was the other way around: red rods were positive and black rods were negative.

Now that negative numbers are widely accepted, I do not know what we would do without them.

Other popular topics related to integers are modeling integers with chips and integers and inductive reasoning.

Summary

  • Ancient peoples could subtract, but only when the answer stayed at zero or above.
  • Around 200 B.C.E., the Chinese used red rods for positive numbers and black rods for negative numbers.
  • Diophantus (around A.D. 250) called equations with negative answers, like 4x + 20 = 4, absurd.
  • Brahmagupta (around A.D. 628) gave rules for positive and negative numbers, and Zhu Shijie gave the rule of signs around 1300.
  • Negative numbers have been written as a dot over the number, rods with a slash, m:6, the minus sign, and red ink.

Test Yourself: Interactive 10-Question Quiz

Now check what you learned. Every answer is explained, and each time you play you get new questions.

Question 1 of 10Score: 0

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