product of negative numbers

product of negative numbers

Why is

⁢

(

-

a

)

⁢

(

-

b

)

=

a

⁢

b

⁢

?

Negative numbers

are less than zero,

positive

ones are more than zero. One can study the former, concrete question in fact from a more general of view, where the used letter

variables

in the latter question may be thought equally well positive as negative (but it may be simpler and better to think that

a

and

b

there mean positive numbers). We use the notation convention, usual in mathematics, that

+

a

and

+

b

mean the same as

a

and

b

(see

plus sign

), when it is easy to speak of changing the sign.

The following three

properties

of

multiplication

are well known:

•

The product of two numbers equals to zero always when one of the numbers equals to zero.

•

Commutative law:

a

⁢

b

=

b

⁢

a

•

Distributive law

:

a

⁢

(

b

+

c

)

=

a

⁢

b

+

a

⁢

c

It is natural to require these properties regardless of whether the numbers are positive or negative.

We need in the following calculation only the first and the last property:

0

=

(

+

a

)

⋅

0

=

(

+

a

)

⁢

[

(

+

b

)

+

(

-

b

)

]

=

(

+

a

)

⁢

(

+

b

)

+

(

+

a

)

⁢

(

-

b

)

=

a

⁢

b

+

(

+

a

)

⁢

(

-

b

)

Because the value of the sum in the end is zero, the latter summand

(

+

a

)

⁢

(

-

b

)

must be the opposite number of the former addend

a

⁢

b

. Accordingly we may write:

(

+

a

)

⁢

(

-

b

)

=

-

(

a

⁢

b

)

(1)

This result means that as the sign of the second factor (

http://planetmath.org/Product

) of the product

(

+

a

)

⁢

(

+

b

)

is changed, the sign of the whole product changes. The same concerns of course also the first factor of the product, since by the commutative law, the order of the factors can be changed.

But if one changes in the product

(

+

a

)

⁢

(

+

b

)

the signs of both factors, first one and then the other, the sign of the product changes twice, i.e. it remains unchanged. Thus we obtain the final result

(

-

a

)

⁢

(

-

b

)

=

a

⁢

b

.

(2)

If we think that

a

and

b

are positive numbers, the result (2) must be understood that

the

product of two negative numbers

is necessarily positive

, in order to keep the three properties of multiplication in for all numbers.

Remark.

The justification of (1) and (2) makes apparent, that these formulae are in in every ring (cf. the parent entry (

http://planetmath.org/XcdotYXcdotY2

) where one assumes the ring unity).

References

1

K. Väisälä:

Algebran oppi- ja esimerkkikirja I

. Fifth edition. Werner Söderström osakeyhtiö, Porvoo & Helsinki (1952).

Title

product of negative numbers

Canonical

name

ProductOfNegativeNumbers

Date of creation

2013-03-22 17:35:41

Last modified on

2013-03-22 17:35:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Derivation

Classification

msc 97D40

Classification

msc 13A99

Synonym

product of two negative numbers

Related topic

OppositeNumber

Related topic

PlusSign

Related topic

Multiplication

Related topic

KalleVaisala

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