An approach to the formal definition of negative numbers, using the ideas of abstract algebra.
Section 1 – is background. Skip it if you like.
What is a negative number?
1: It’s a number with a “-” in front.
2: It’s the opposite of a positive number.
Well, 1 is very poor, and 2 is no good, as there are no positive numbers until we have negative numbers, they are just numbers (referred to later as the original numbers).
There is a need to compare numbers, and one way is to ask “What is the difference between this number and that number. This is easy – the difference between 3 and 7 is 4, and we all learn to write 7 – 3 = 4.
Everything is fine for a while until someone says “But what about 3 – 7 ?”.
“Cannot be calculated. Has no meaning. You cannot take 7 things away from 3 things. You cannot cut a 7 inch piece of wood off a 3 inch piece.” are the answers.
These original numbers are usable for counting and measurement of quantity, but numbers can also be used to measure position, leading to questions of the form “How far is it from this number to that number?”. Temperature is the most obvious situation. “How much warmer is it today, compared to yesterday?”. With numbers we can ask “How far is it from 3 to 7 ?” and get a response ” 4 “, but we can also ask the question “How far is it from 7 to 3 ?”. The response is the same, with the extra “but in the opposite direction”.
Thus there arises a need for numbers capable of dealing fully with this new situation , the measurement of changes in position. So negative numbers are born (or created), and we hope they obey the same rules as the original numbers. Playing around seems to support this position, with a few mysteries, such as (-1) times (-1) equals 1, and two negatives make a positive.
However, in math we should not be satisfied by “Well, it seems to work OK”.
What follows is a formal definition of an extended number system, in which every number has an “opposite”, or an “additive inverse”, and in which every number not equal to zero has a multiplicative inverse, and in which the “properties of operations” are still valid.
The definition only uses brackets (parentheses), commas, and pairs of original numbers. It does NOT use the negative sign, and subtraction b – a, is only applied where it makes sense, that is when b>a.
An extended number (or “thing”), written ( a , b ), is defined as the distance from a to b.
It is immediately obvious that ( a , b ) = ( a + 1 , b + 1 ) = ( a + 2 , b + 2 ) and so on.
So we can write ( a , b ) as ( 0 , b – a ) when b>a, and as ( b – a , 0 ) ) when b<a, using subtraction only with the original numbers.
Addition needs to match the original number addition, so
(definition) ( a , b ) + ( p , q ) = ( a +p , b + q ) check it
Zero is the “thing” which when added to anything has no effect, so the zero “thing” is ( p , p ) for any p.
Now we can have the additive inverse of a “thing”, the one which when added to the “thing” gives zero.
(definition) Additive inverse of ( a , b ) is ( b , a ) since ( a , b ) + ( b , a ) = ( a + b , a + b )
Subtraction for “things” can now be defined as addition of the additive inverse.
We can define multiplication of “things” by looking at the product of two differences.
(original number definition) ( b – a )( d – c ) = bd + ac – ( ad + bc ) , so we have
(definition) ( a , b ) X ( c , d ) = ( ad + bc , bd + ac )
For multiplication we need a unit or identity “thing”, and the obvious choice is ( 0 , 1 ), or anything where the second number is one bigger than the first, for example ( 12 , 13 ). Using the “multiply” definition we have ( 0 , 1 ) X ( 0 , 1 ) = ( 0 , 1 ),
and ( 0 , 1 ) X ( 12 , 13 ) = ( 12 , 13 ) = ( 0 , 1 ),
and ( 0 , 1 ) X (3, 7 ) = (3, 7 )
Division is defined as the inverse of multiplication, so to divide a “thing” by another “thing” we multiply the “thing” by the multiplicative inverse of the “other thing”, which we now define.
(original number definition) multiplicative inverse of ( b – a ) is 1/(b-a)
(definition) Multiplicative inverse of ( a , b ) is ( 0 , 1/(b-a)) if b>a and ( 1/(a-b) , 0 ) if b<a
If we multiply a “thing” by its inverse we should get the unit or identity ( 0 , 1 ), and so we do:
( a , b ) X ( 0 , 1/(b-a)) = ( a/(b-a) , b/(b-a) ) = ( 0 , b/(b-a) – a/(b-a) ) = ( 0 , 1 )
We have enough here to show that the new operations of + and X have the same properties as add and multiply in the original numbers. Go on, show it !!!!
Now what has this got to do with negative numbers ? Well, the first thing is that ( 0 , 1 ) has an additive inverse, namely ( 1 , 0 ), or any of its other representations, say ( 5 , 4 ) for example.
The second thing is that ( 1 , 0 ) X ( 1 , 0 ) = ( 0 , 1 ) .
The third, and most important thing is that we have an arithmetic for the “How far is it from A to B” quantities which incorporates direction. When A<B the direction is one way. When A>B the direction is the other way. These directions are coventionally called “the positive direction” and “the negative direction”.
So, finally, we identify distances in the positive direction with the original numbers, and distances in the negative direction with new numbers, each of which is the “opposite” or “additive inverse” of one of the original mumbers.
Using the minus sign for “the additive inverse of” makes it quicker to write, at a cost of some possible confusion. We see now that for example ( 3 , 7) is identified with the original number 4 and ( 7 , 3 ) is identified with the new number -4. Also ( 0 , 1 ) is identified with the original number 1 and ( 1 , 0 ) is identified with the new number -1.
So since we have ( 1 , 0 ) X ( 1 , 0 ) = ( 0 , 1 ) it follows that -1 x -1 = 1, and there is no mystery about it!