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Introduction to Negative Numbers (page 1 of 4)

Sections: Introduction, Adding and subtracting, Multiplying and dividing, Negatives and exponents

When you first learned your numbers, way back in elementary school, you started with the counting numbers: 1, 2, 3, 4, 5, 6, and so on. Your number line looked something like this:

    Your first number line

Later on, you learned about zero, fractions, decimals, square roots, and other types of numbers, so your number line started looking something like this:

    A more advanced number line

Addition, multiplication, and division always made sense — as long as you didn't try to divide by zero — but sometimes subtraction didn't work.
If you had "9 – 5", you got 4:

    9 - 5 = 4

what if you had "5 – 9"? You just couldn't do this subtraction, because there wasn't enough "space" on the number line to go back nine units:


You can solve this "space" problem by using negative numbers. The "whole" numbers start at zero and count off to the right; these are the positive integers. The negative integers start at zero and count off to the left:

    advanced number line

Note the arrowhead on the far right end of the number line above. That arrow tells you the direction in which the numbers are getting bigger. In particular, that arrow also tells you that the negatives are getting smaller as they move off to the left. That is,
–5 is smaller than –4.

This might seem a bit weird at first, but that's okay; negatives take some getting used to. Let's look at a few inequalities, to practice your understanding. Refer to the number line above, as necessary.

  • Complete the following inequality:  3 _____ 6
  • Look at the number line: Since 6 is to the right of 3, then 6 is larger, so the correct inequality is:

      3    <    6

  • Complete the following inequality: –3 _____ 6
  • Look at the number line: Every positive number is to the right of every negative number, so the correct inequality is:   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

      –3    <    6

  • Complete the following inequality: –3 _____ –6
  • Look at the number line: Since 6 is to the left of 3, then 3, being further to the right, is actually the larger number. So the correct inequality is:

      –3    >    –6

  • Complete the following inequality: 0 _____ 1
  • Zero is less than any positive number, so:

      0    <    1

  • Complete the following inequality: 0 _____ –1
  • Zero is greater than any negative number, so:

      0    >    –1

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Cite this article as:

Stapel, Elizabeth. "Introduction to Negative Numbers." Purplemath. Available from Accessed



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