Why does -1*-1 = +1

breadmonkey

This might sound like a rediculous question and I'm sure the answer is embarressingly obvious, but I realised the other day that, half way through an engineering degree, I never questioned this, nor could I really explain it to anyone.

Tar.Aldarion

Think of it in terms of a simple number line. you have a point at -1. you then then go to the point of the line that is -1 times that point when you multiply.


if you were at -2 and multiplied x3 it would go three times further out on the number line to -6 etc.

Look at these patterns:
3 times 2 = 6 3 times (negative2) = negative6
2 times 2 = 4 2 times (negative2) = negative4
1 times 2 = 2 1 times (negative2) = negative2
0 times 2 = 0 0 times (negative2) = 0

Number line:
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

breadmonkey

I'm still not convinced. I want something CONCRETE!

BMH

Two wrongs make a right.

JoeB-

because multiplication works on the magnitude of the digits themselves, the signs are dealt with afterwards according to simple conventions... most people perform both steps in one go leading to this confusion...

consider -1 * 1 = -1
could be written as (-)(1*1) = -1

so now -1 * -1 = 1
written as (-)(-)(1 * 1) = (-)(-1) = 1

so now the question why does (-)(-) = a positive value????

A minus could be considered as reversing the value of an existing number (reflecting the current value through the point 0 on the number line and out the following side an equal distance, so minuses become pluses and pluses become minuses)

so (-)(-)(1) means reflect 1 twice through the 0 point, this has no effect so the convention is that two negative numbers multiply to produce a positive......

Cheers

Tar.Aldarion

breadmonkey wrote:

I'm still not convinced. I want something CONCRETE!

How is it not concrete.
Don't think of numbers on the line as being positive or negtive, you are simply moving a point by use of some operator and reading out what is written under it. Just think of it as a minus moving a point through zero on the number line and moving it out a distance on the other side.
Come on now breadmonkey, this is junior infants stuff!

Son Goku

Multiplication by -1 maps a number to its negative.

Or if you want a semi-proof:
-1 is the negative of 1

The definition of multiplication is:
a * b = a+a+a.........(b many a's)...........+a

So 2*3=2+2+2.

-1*1 = -1 (1 many -1's)

-1*-1 = ? (-1 many -1's)

What is "-1 many -1's"?
Whatever it is, it will be the negative of "1 many -1's" as -1 is the negative of 1.

"1 many -1's" was -1, so its negative is 1.
Therefore "-1 many -1's" is 1, so

-1*-1 = 1.

breadmonkey

Thanks guys, but I found JoeBallantine's the clearest.

Come on now breadmonkey, this is junior infants stuff!

I know it sounds stupid but I don't think you would find too many 6th years who could give you a definitive, clear answer. That's just my opinion, I could be wrong.

Karoma

(-)(-) (<-teeheehee)

David19

Firstly you should note that -1 is an element of the set of real numbers. The real numbers form a field under addition and multiplication. This is important because of one of the axioms says minus numbers(or additive inverses) exist. It can also be shown they're unique to each number.

http://en.wikipedia.org/wiki/Field_%28mathematics%29

To show (-1)(-1) = 1:

(-1) + (1) = 0 ........... an axiom of real numbers
(-1)(-1) + (-1)(1) = 0 .......... multiply across by (-1)
(-1)(-1) + (-1)(1) + (1)(1) = (1)(1) ...... add (1)(1) to both sides
(-1)(-1) + (1)[((-1) + (1)] = (1)(1) ............... distributive law
(-1)(-1) + (1)[0] = (1)(1) ................. from first line
(-1)(-1) = (1)(1) ...........it can be shown (1)(0) = 0
(-1)(-1) = 1 ..............it can be shown (1)(1) = 1

---

This stuff comes from an area of maths called real analysis and forms the basics of calculus. There's notes here if you're interested:
http://www.maths.tcd.ie/~richardt/121/121-ch1.pdf

I think that's all right. Hopefully it's concrete enough!

I actually would not have been able to explain it!Thanks for the enlightenment!:)

breadmonkey

To show (-1)(-1) = 1:

(-1) + (1) = 0 ........... an axiom of real numbers
(-1)(-1) + (-1)(1) = 0 .......... multiply across by (-1)
(-1)(-1) + (-1)(1) + (1)(1) = (1)(1) ...... add (1)(1) to both sides
(-1)(-1) + (1)[((-1) + (1)] = (1)(1) ............... distributive law
(-1)(-1) + (1)[0] = (1)(1) ................. from first line
(-1)(-1) = (1)(1) ...........it can be shown (1)(0) = 0
(-1)(-1) = 1 ..............it can be shown (1)(1) = 1

Now that's concrete!

Steveire

I saw a proof of 1+1 = 2 before.

I can't remember how it went, but I have some vague recollections. It started off by defining zero as zero, then saying 1 comes straight after zero, and has a magnitude of 1. Therefore 1+1=2, 1+1+1=3 etc.

Any clarity on this?

Son Goku

Okay let us assume the Natural numbers:

0, 1, 2, 3...................................

Every natural number a has a successor s(a).

Now let us define an operation called addition which is labeled with the symbol + with the properties:

(1) a + 0 = a.
(2) s(a) + b = a + s(b).

Now set b = 1 and s(a) = 1.

So statement (2) becomes:
(3) 1 + 1 = a + s(b).

If s(a) = 1, then a = 0.
If b = 1, is s(b) = 2.

So statement (3) becomes:
(4) 1 + 1 = 0 + 2.

From statement (1) a + 0 = a, so statement (4) becomes:

1 + 1 = 2.

Tar.Aldarion

Didn't think he was serious...
Oh well.

breadmonkey

Tar.Aldarion wrote:

Didn't think he was serious...
Oh well.

I think you underestimate the absolute fundamentals Tar.

Tar.Aldarion

I know it and obviously they are important, thought Steve was making a joke tbh.

Karoma

God said it should be so! It's all part of God's design.

David19

Here's a pretty clear and easy proof of 1 + 1 = 2:

http://www.idt.mdh.se/~icc/1+1=2.htm

Pretty obvious stuff really.

breadmonkey

David19 wrote:

Here's a pretty clear and easy proof of 1 + 1 = 2:

http://www.idt.mdh.se/~icc/1+1=2.htm

Pretty obvious stuff really.

The only recognisable symbols for me are beta, colon, and square brackets. That's it.

Pft.
You have one apple. You get another one apple. You have two apples.

Gurgle

You have -1 apples.
How many of -1 apples do you have?

1

QED

breadmonkey

bluewolf wrote:

Pft.
You have one apple. You get another one apple. You have two apples.

Oh, now I get it!:D

breadmonkey

Gurgle wrote:

You have -1 apples.
How many of -1 apples do you have?

1

QED

WHat the hell is -1 apples?:p

breadmonkey

breadmonkey wrote:

Oh, now I get it!:D

Simpler than all the alphas and betas!!

WHat the hell is -1 apples?

It's when that annoying mathematician next door steals one of your apples!

Gurgle

bluewolf wrote:

It's when that annoying mathematician next door steals one of your apples!

and you didn't have one in the first place.
Its maths, not to be confused with science or reality.

(of means multiply, multiply means of)

Gurgle

Gurgle wrote:

and you didn't have one in the first place.
Its maths, not to be confused with science or reality.

Yep.
Maths is fun.
And this is far better than studying for my complex analysis exam...

we could start talking about i apples

Gurgle

bluewolf wrote:

we could start talking about i apples

:rolleyes: Mathematicians!
thats j to an engineer.

The square root of -1 is j? o.O

breadmonkey

bluewolf wrote:

The square root of -1 is j? o.O

Not to turn this thread into a spam fest or anything, but I must mention that I prefer:

o_Ó

breadmonkey

breadmonkey wrote:

Not to turn this thread into a spam fest or anything, but I must mention that I prefer:

o_Ó

My eyebrow doesn't grow into my eye though.

Steveire

ò_Ó

Yep, that was it.

Tar.Aldarion

Lmao at the mathematician stealing your apple, that you did not have in the first place.

Tar.Aldarion

Tar.Aldarion wrote:

Lmao at the mathematician stealing your apple, that you did not have in the first place.

We had to write him an IOU, so now when we get one apple we will have to take it away

singingstranger

I think I like Gurgle's answer to that one best...

...anyway, you can't look for a "concrete" answer, numbers are only mental playthings after all. Numbers have no context without being attached to a particular good ("I have -1 apples. How many bundles of -1 apples have I got?"), they can't exist as a self-entity.

[/rant]

mikhail

breadmonkey wrote:

Now that's concrete!

Very nice proof indeed.

bluewolf wrote:

The square root of -1 is j? o.O

Yeah, engineers use j for sqrt(-1). Mainly because i already stands for current in circuit analysis. Why they used i for current, I'll never know.

Gurgle wrote:

You have -1 apples.
How many of -1 apples do you have?

1

QED

To paraphrase:
You have 2 apples.
How many of 2 apples do you have?

1

=> You've no idea what you're talking about.
QED

WhiteWashMan

Son Goku wrote:

.
Therefore "-1 many -1's" is 1, so

.

actually, "-1 many -1's" = + or - (1)

/smartalec

Son Goku

WhiteWashMan wrote:

actually, "-1 many -1's" = + or - (1)

/smartalec

How so?
The statement "many" is equivalent to multiplication.

Gurgle

mikhail wrote:

You have 2 apples.
How many of 2 apples do you have?

1

You seem to be trying to imply theres something wrong with this maths?

WhiteMashMan wrote:

actually, "-1 many -1's" = + or - (1)

no actually its not.
its +1 every time.
with or without the smartalec tag

Dr. Octagon

I was puzzled by the question but Tar... answered it simply and accurately.

Tar.Aldarion

Dr. Octagon wrote:

I was puzzled by the question but Tar... answered it simply and accurately.

I please to aim.

spacetweek

breadmonkey wrote:

I know it sounds stupid but I don't think you would find too many 6th years who could give you a definitive, clear answer. That's just my opinion, I could be wrong.

It's not stupid, there's nothing wrong with asking yourself the philosophical implications of maths. I often mull over the deeper meaning of e^i.pi = -1.