“How (-)×(-) = + ”?
As we all know that (plus × plus) is plus and it’s easy to
understand but have you ever wondered that how (minus × minus) is plus.
Many of us
are using this condition in our daily mathematical life knowingly and
unknowingly but you may have never thought that how this product of two
negative become a positive. Well, I’m going to show you logically how this
“-x-‘’ become “+” in a easy way .There might be many other proofs regarding
this awesome thing but I’m going to
explain in a simple ways.
Firstly,
you must have the knowledge of simple things like this (+ x-= -) and (-x+ =-)
First, let
us start with easiest way, (besides mathematics is all about the learning of
patterns).
(-1) x (+3) =-3
(-1) x (+2) =-2
(-1) x (+1) =-1
(-1) x (0) = 0
(-1) x (-1) =+1
(-1) x (-2) =+2
(-1) x (-3) =+3
From the above pattern it’s easy
to understand how negative times negative is positive. It sounds total
mathematical? How do you feel?
Here we can
go with proof which is kind of language type.
Let’s
suppose a town, consider the good guys as positive (+) or bad guys as negative (-)
in the town. If the guys enter the town take that as positive (+) and if the
guys leave the town take that as negative (-):
Now again
we go through the following pattern by considering the above assumption:
•
If the good guys
(+) enter (+) the town, that is good (+) for the town. (+) x (+) =+
•
If the good guys
(+) leave (-) the town, that is bad (-) for the town. (+) x (-) =-
•
If the bad guys (-)
enter (+) the town, that is bad (-) for the town. (-) x (+) =-
•
If the bad guys (-)
leave (-) the town, that is good (+) for the town. (-) x (-) =+
So, these
were the basic two proofs!
Here are
some other which will help you in different way to understand the problem!
We all know
that
-1+1=0.
Multiplying this equation by -1, we get:
(-1)*[(-1)+1]=(-1)*(0).
Using the
distributive property, we get,
(-1)(-1)+(-1)1=(-1)0
[ a(b+c)=ab+ac (distributive property)]
As we know,
(-1)1=-1 and (-1)0=0
So, by
applying this and adding 1 on both sides,
(-1)(-1)+(-1)+1=0+1
Or, (-1)(-1)+0=1
Or, (-1)(-1)=1
This proves
the relationship.
At last but
not the least:
From the
index rule.
We know that
(ax)y
= ax y,
Which holds
true for a=2 and x=y= -1.
Therefore,
(2-1)-1
= (2)(-1)(-1).
By the
definition of negative exponent, we get
(2-1)-1
=(1/2)-1 =2.
We can then
conclude that 21=2(-1)(-1)
•
1=(-1)(-1)
As we know
if the bases are equal, so must the exponents be equal.
These were
the illustration that proves how the product of two negative becomes positive.
If you have
got any other ways you can comment down.
Thank you!
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