negative number

A negative number is a number that is less than zero. It is the “opposite” of a positive number. It can be the result of a subtractionPlanetmathPlanetmath in which the subtrahend is greater than the minuend. In everyday life, negative numbers are most often used to indicate debts.

For example, Bob’s checking account has $47.20 and Alice cashes the check he gave her for $80.00, and the bank gives the requested amount. Bob’s account is then down to -32.80, and could be further indebted if the bank collects some sort of fee for giving more money than was available.

In a number line, negative numbers are usually to the left of zero:


In a 2-dimensional coordinate plane, negative numbers on the vertical axis are usually below zero. (In the complex planeMathworldPlanetmath, technically, they may be above if desired).

Whereas giving a plus sign for positive numbers is optional, giving the minus sign for negative numbers is a must.

Addition and subtraction of negative numbers is fairly straightforward and intuitive. Suppose Bob also gave Carol and Dick a check for $80.00 each and they both cash their checks after Alice cashed hers. This is -32.80-80.00-80.00=-192.80. Multiplication of a negative number by a positive number is also straightforward. Suppose Bob’s bank charges a fee of $25.00 for each instance of overdraft. This translatesMathworldPlanetmath to -25.00×3=-75.00.

However, multiplication of a negative number by another negative number gives a positive number. In all honesty, I can’t think of a situation in everyday life in which it would be necessary to multiply two negative numbers. At any rate, the rule of sign changes has the consequence that (-x)a>0 if a is even and (-x)a<0 if a is odd. This comes in very handy in number theoryMathworldPlanetmathPlanetmath for creating alternating sums or checking the relative density of one kind of number to another. For example, for squarefreeMathworldPlanetmath n, the Möbius functionMathworldPlanetmath μ(n)=(-1)ω(n) (where ω(n) is the number of distinct prime factors function).

As a consequence of these sign changes, a positive real number x2 technically has two square rootsMathworldPlanetmath, x and -x. The specific case of x2=25 was used in The Simpsons episode “Girls Just Want to Have Sums,” (first aired April 30, 2006) in which Lisa Simpson dressed up as a boy to sneak into a math class. Asked for the solution, Lisa answers 5, but the teacher says this is wrong. Martin then gives the two correct answers, 5 and -5. Even computer algebra systems, and certainly most scientific calculators, will only give the positive answer.

What numbers are the square roots of a negative real number? No such numbers exist. More precisely, they are imaginary numbersMathworldPlanetmath, multiplesMathworldPlanetmath of the imaginary unitMathworldPlanetmath. For example, -25=5i or -5i.

A question that comes up much less often is: What is a negative number raised to a fractional power? As you may know, raising a positive number to the reciprocal of n has the same effect as taking the nth root of n. We can plot the integer powers of, say, 2, with dots and then connect the dots with straight lines; the result, though not technically correct, would not be too far off the mark (which would be a smooth curve connecting the dots). From such a graphic we might draw the incorrect conclusion that 21.5=3, while the correct answer (the base 2 logarithm of 3) is more like 1.5849625007211561815.

Whether we connect the dots in a plot of (-2)n with either straight lines or curves, we might be fooling ourselves. The incorrect conclusion of (-2)1.5=0 just doesn’t make sense.

On most scientific calculators, trying to raise a negative number to a fractional power will result in an error exception (unless of course you enter it in such a way that the calculator interprets as merely a positive number raised to a fractional power and then multiplied by -1).

Once again, imaginary numbers come to the rescue. Since the square root of a negative number is an imaginary number, it only makes sense to extend this to negative numbers raised to fractional powers. For example, (-2)32=-2i2 (and of course the complex conjugateDlmfMathworldPlanetmath of that).

In Mathematica, I made the following plot of (-2)n in steps of 110 and then separated the resulting complex numbersMathworldPlanetmathPlanetmath into their real and imaginary partsDlmfMathworld.

Not entirely convinced this is right, I tried a 3-dimensional plot (in the plot, x runs from 1 to 2, y runs from 1 to 40 and z runs from -20 to 20):

As a final sanity check, I compared this to a similar plot of 2n (with the axes oriented in the same way as in the previous plot):

In the end, though the 3-dimensional plots may look “cooler,” the 2-dimensional plot is actually more enlightening, showing the powers of a negative number fall on a logarithmic spiralMathworldPlanetmath.


  • 1 Screen name “Cromulent Kwyjibo”. Personal communication, June 8, 2007.
  • 2 Alberto A. Martínez, Negative Math: How Mathematical Rules Can Be Positively Bent. Princeton and Oxford: Princeton University Press (2006)
Title negative number
Canonical name NegativeNumber
Date of creation 2013-03-22 17:13:34
Last modified on 2013-03-22 17:13:34
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 12
Author PrimeFan (13766)
Entry type Definition
Classification msc 00A05
Related topic ClassificationOfComplexNumbers