Started a degree, well a course in 1st year uni maths with the OU (www.open.ac.uk), MST121 to be precise. I'm suprised on two levels first how much I've forgotten and how much I still remember. Fractions I completely forgot, but algebra I remembered quite well. I also worked through an OU Book "Countdown to Mathematics v.1".
In "CountDown to Maths v.1" there was a little margin note on negative numbers why -n x -n = +x ; you may get a proof later on in your maths study. This got me thinking a I could not sleep for until I came up with a proof.
Negative Number Proof.
This is my first orginal maths proof and no doubt its been done before.
Numbers including negative is usually displayed as below.
-3 -2 -1 0 1 2 3
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3 + -3 = = 0 ,
go right three then go left 3 to get the resultant value of ZERO.
So postive and negative numbers are all about direction to add numbers you must add the directions to get the correct anwser. Think of a tiled floor and select a tile as zero and stand on it. Pick a direction and walk two tile lengths, you have added two to zero and have a positive value of two. Take 4 from your tile value (2 - 4 = -2). You have to turn round , how far do you have to turn round, in the opposite direction so what is that in an angle ...euraka you have to turn round 180 degrees.
So positive and negative numbers are in effect 180 degrees out of phase or in easier english in the opposite direction ( think of going left or right, back and foward).
Multiplication
+5 x +5 = +25 ..... OK what about angles or direction.
I will use {n} as a direction or angle based on 360 degrees.
If you want a resultant directions from two powers add the angles. I remember this from my old electrical engineering days with Induction motors and phase angles.
5{0} x 5{0} = 5 x 5 {0 + 0 } = 25 {0)
Now we will try two negative numbers
-5 x -5 = +25
5{180} x 5{180} = 5 x 5 { 180 + 180 } = 25{360} = 25{0} = +25
360 degrees is also zero; even been told to do a 360 turn ?
Multipling Negative and a Positive number
-5 x +5 = -25
5{180} x 5{0} = 5 x 5 { 180 + 0 } = 25{180} = -25
Note 180 degrees is the exact opposite direction from 0 degrees. Stand up and try it.
So there you have it, maybe someoen some where will find this usefully.