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Why does a minus times a minus equal a plus???

2

Okay this question seriously bugs me and i cant believe that i havent thought to ask on here.
Ive asked at school but not got a v good answer, my teacher basically said because it is and i just have to accept it.
So anyone know???

Reply 1

~NC~

11

This is ithink because the two minuses cancel each other out therefore just leaving a positive number, i think thats the reason.

Reply 2

Icy Ghost

2

eg: 3x9 is getting 9, 3 times.... and -3x-9 is getting -9, -3 times...
so the result is the same.

> says my friend.

:smile:

Reply 3

~NC~

11

Hmm that also makes sense icy.

Reply 4

C4>O7

5

Can be proved using axioms:
a(b+c)=ab+ac (distributive law)
=> (-1)(1+(-1))=(-1).1+(-1).(-1)

But a+(-a)=0 so 1+(-1)=0, and 0.a=0 so (-1).0=0,
=> 0=(-1).1+(-1).(-1)
and 1.a=a so 1(-1)=(-1)
=> 0=(-1)+(-1).(-1)
=> (-1).(-1)=1

Reply 5

steve2005

17

rachelrainbow32

Okay this question seriously bugs me and i cant believe that i havent thought to ask on here.
Ive asked at school but not got a v good answer, my teacher basically said because it is and i just have to accept it.
So anyone know???

Study the pattern.

Picture 2.png25.5KB

Reply 6

AlphaNumeric

10

steve2005

Study the pattern.

That isn't rigorous though. How do you know the pattern doesn't go something like 8, 6, 4, 2, 0, 2, 4, 6 etc?

Similarly things like "They cancel out" don't have any rigour.

The proper way to show minus times a minus is a plus is the way C4>O7 has done it, by using the axiomatic approach. Once you've defined how 0 and 1 behave, the concept of additive inverses and the fact multiplication is distributive over addition you can show -1*-1 = 1.

Reply 7

asadtamimi

13

yea but its like saying u have -1 orange and u had -2 oranges to get an orange?

Reply 8

ledzep15

1

- X - = + because if you multiply -1 x -1 you get 1!

Reply 9

AlphaNumeric

10

ledzep15

- X - = + because if you multiply -1 x -1 you get 1!

Thats totally circular reasoning though. Its like me saying "2+4 = 7 because when you add 2 and 4 you get 7". The statement could be (and is) incorrect because I've proved nothing.

Reply 10

ledzep15

1

haha, well you have proven me wrong, im not very smart

Reply 11

Chewwy

17

asadtamimi

yea but its like saying u have -1 orange and u had -2 oranges to get an orange?

you can't have a negative number of oranges. similarly you can't have a fractional number of oranges. we can make use of these numners however to help us solve other problems.

people in my classes are having this problem now with complex numbers - they can't comprehend them since they can't immediately see their uses. it's good the thread starter is actually asking questions about the nature of numbers here, instead of just blindly accepting them.

Reply 12

spy007

1

2 x 2 is 2 "lots" of 2 = 2+2 = 4

2 x -2 is 2 "lots" of -2 = -2-2 =-4

-2 x -2 is -2 "lots" of -2 = -(-2-2) = 4

Reply 13

AlphaNumeric

10

spy007

-2 x -2 is -2 "lots" of -2 = -(-2-2) = 4

That is still using the assumption to prove the assumption. You've assumed that multiplying (-4) by the - (which is really a -1 infront) gives 4, so you've used -1*-1=1 to prove -1*-1=1.

As I said, with these handwavey arguments you almost always end up using your assumption to prove the assumption. C4>07 proved it properly, algebraically using only basic axioms without any handwaving.

Reply 14

ledzep15

1

you could have a negative number of oranges if you owe someone oranges..

Reply 15

cosmik_debris

Positive and negative are relative. If you have two oranges and give one to your friend, the orange which you give away becomes negative, but only from your frame of reference. To your friend, the orange becomes positive, from negative.

When dealing with real numbers only we're dealing with a 1-dimensional space. Think of the multiplication of a negative as a rotation of through 180 degrees. Ie. if you multiply something by two negatives it will rotate a whole revolution through 360 degrees and end up with the same polarity you started with (if it was originaly negative it will still be negative, and likewise for positive).

When dealing with complex numbers we're dealing with a 2-dimensional space. The same concept can be applied, whereby multiplying by j (sqrt-1) will make a rotation through 90 degrees. So how about if we multiply by -j... the negative causes a rotation of 180 and j causes a rotation of 90... making a total rotation of 270 degrees.

By no means a proof, but its a lovely concept.

Reply 16

Dekota

I hate it when people say "2 times 2" or "8 times 6" etc. it doesn't sound right, even my teachers says it.
It should be "2 lots of 2" or "2 multiplied by 2" / "8 lots of 6" or "8 multiplied by 6"

Reply 17

KAISER_MOLE

7

Who said that two wrongs don't make a right, they're having a laugh! :p:

Reply 18

wippa_dippa

2

The only way I could remember that, is that if the signs are different, the answer will be negative; if the signs are the same, the answer will be positive.

Wippa :redface:

)

Why does a minus times a minus equal a plus???

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