Why does (-1)(-1)=1

I think you guys will know that (-1)(-1)=1 as me too, however a lot of students still being confused of this! Therefore, I want someone to explain this simple thing with different ways that the students will know instantly.

Example

(-1)(-1)

=(-1)(-1)-1+1

=(-1)(-1+1)+1

=0+1

#Algebra

Easy Math Editor

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Comments

The proof I have isn’t very simple, but it shows that $(-a)·(-b) = ab$ is a direct consequence of the properties of numbers (or what we defined them to be). These properties are, of course, the following:

Associative law - addition

$a+(b+c)=(a+b)+c$
Additive identity

$a+0=0+a=a$
Additive inverse

$a+(-a)=(-a)+a=0$
Communtative law - addition

$a+b=b+a$
Associative law - multiplication

$a· (b· c)=(a· b)· c$
Multiplicative identity

$a· 1=1· a=a$
Multiplicative inverse

$a· a^{-1}=a^{-1}· a=1$
Communtative law - multiplication

$a·b=b·a$
Distributive law

$a·(b+c)=a·b+a·c$

Ok, now the rules are set and we can prove that $(-a)·(-b) = ab$ .

Note that $(-a)· (b)=-(a· b)$ , so $(-a)· (-b)+[-(a· b)]=(-a)· (-b)+(-a)· b$ $=(-a)· [(-b)+b]$ $=(-a)· 0$ $=0.$

Now adding $(a· b)$ to both sides gives us $(-a)· (-b)=a· b.$

Alex Greist - 1 year, 10 months ago

Thanks for your help XD

Isaac YIU Math Studio - 1 year, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$