negative number

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

subtraction

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 plane

, 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

translates

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 theory

for creating

alternating sums

or checking the relative

density

of one kind of number to another. For example, for

squarefree

n

, the Möbius

function

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

x

2

technically has two

square roots

,

x

and

-

x

. The specific case of

x

2

=

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 numbers

,

multiples

of the

imaginary unit

. For example,

-

25

=

5

⁢

i

or

-

5

⁢

i

.

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

n

th 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

2

1.5

=

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

)

3

2

=

-

2

⁢

i

⁢

2

(and of course the

complex conjugate

of that).

In Mathematica, I made the following plot of

(

-

2

)

n

in steps of

1

10

and then

separated

the resulting

complex numbers

into their real and

imaginary parts

.

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

2

n

(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 spiral

.

References

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

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