mirrored maths?

[michel123456]

The line of numbers goes like this, from negatives to positives through zero:

 

.....-3 -2 -1 0 +1 +2 +3 ......

 

The symbols are mirrored around zero. They are "opposite"

The negative numbers are less than zero.

And minus 3 is evidently a lesser value than +3

 

Because the arrow of small to big goes from left to right

 

small...................0......................BIG >

 

from-small-to-big does not change direction.

 

 

but the symbols of the line are mirrored. Minus 3 is the "opposite" of +3, but that is only a symbolic feature.

.....-3 -2 -1 0 +1 +2 +3 ......

 

Though mathematical operations keep the small/big arrow going from left to right, no matter you are in the left or right part of the line. It is like an elevator going from level -3 to level +3. All levels are levels: a flat platform on which you walk.

Under this concept, the position of number zero is simply conventional. One can change the position of zero and use a simple transformation to go from one system to the other.

.....-3 -2 -1 0 +1 +2 +3 ......

.....-4 -3 -2 -1 0 +1 +2 ......

 

 

Now, what if one made a new concept, where numbers are not like levels, but are like "warriors".

 

Say on the positive part would be the warriors of Alexander The Great, and on the negative part the warriors of Darius: they would be really "opposite"

 

minus 3 then should mean 3 units opposite to +3 units.

And the small-to-big arrow would be like this

 

< BIG................. small0small......................BIG >

 

That would be in agreement with the symbols.

 

There would be nothing smaller than zero.

Minus 3 would be bigger than minus 2.

And minus 3 would not be a less value than +3

 

Is it possible to build mathematics upon this last concept?

Edited December 31, 2013 by michel123456

[ajb]

This depends on how you define size and value. It looks to me that all you need is to define a norm on the real line thought of as a vector space as

 

[math]||x|| = |x-0| [/math],

 

which just agrees with out notion of absolute value. It defines a distance of x from 0. This will satisfy your needs, unless I misunderstand what you are trying to do.

[michel123456]

What I am trying to do?

Yes it could be about distance but not only.

Basically the concept is that a negative, instead of being simply a loss of something like a debt, can also be considered as a real opposite, something that is of opposite nature.There will be a notation issue.

 

In my thought, mathematical operations entirely on the positive part do not change.

Operations entirely on the negative part are absolutely the same as in the positive part: you can forget the minus sign everywhere and simply put back the minus on the result. It is an exact mirror.

 

Where it becomes a mess is when you make operations accross the mirror, mixing opposites.

 

Let's go back at the "warriors" analogy.

Say you are in Alexander's camp (the positive part).

There you have 1,2,3 warriors. You can make "operations" upon your army (the set of positive numbers), like adding warriors or substracting warriors (either paying for more mercenaries, or sending you warriors back home). That's all conventional using pos. and neg. numbers as usual.

 

But you cannot make operations simply upon opposite Darius warriors (the set of negative numbers-here comes the notation issue). You cannot pay Darius warriors nor you can send them home. They have another "nature" of opposite value. Operations accross zero are not that simple.

 

For example, if you want to reduce Darius warriors, you must engage a fight. (sorry for the analogy, I haven't found anything better so far)

 

Say you put together +1 and -1.

 

You have a fight of 2 warriors who have both the same value.

At the end, you get 2 warriors dead and thus no warrior at all, that is equal to zero.

Mathematics are saved.

 

But a fight has occured, war has happened. It is not the same thing as having no soldier in Alexander & Darius army. Alexander army has an infinte set of warriors ans so has Darius army. The total makes 2 (opposite) armies, not zero. Of course if they engage fight (making a mathematical operation) the sum might be zero.

 

When +1 and -1 meet, there is war and operation has a cost: 2 men will die.

So I wondered how one could show this cost in a mathematical way.

 

For example

on a graph, zero would be a vertical axis, and the +1-1 operation could be represented as a rotation of the numbers upon the vertical axis, meeting at the vertical of zero at height 1.

One could then say that the operation +1-1 gives a result of zero at level 1.

 

Am I getting into trouble?

Edited January 2, 2014 by michel123456

[studiot]

You seem to be feeling your way towards the Yin and Yang of Mathematics.

 

Mathematicians love to generalise, to extend an idea to larger and larger classes of (mathematical) objects.

But not all objects can be classed as positive or negative, for example complex numbers, matrices, operators etc.

 

This idea appears many times in many guises, you should investigate the following

 

Adjoint,

Conjugate,

Dual

Transpose

 

Maybe ajb will add to my list.

 

Edited January 2, 2014 by studiot

[michel123456]

Not sure.

 

Thinking more about this

 

 

 

[ajb]

Thinking more about this

 

mirroredmaths.jpg

But is this really just notation for the negative numbers? Does this really add anything?

 

I think you need to be very clear what you mean by opposite. As studiot suggests, it is something to do with a natural pairing of numbers in this case. The standard notion I guess would be

 

[math] (x,y) = x+y[/math]

 

and if this is zero, then the numbers are "opposite". In standard notation for the real line with addition we have

 

[math](x,y) = 0 \longrightarrow y = -x[/math].

 

But you could define this pairing differently...

[michel123456]

What I mean by opposite? Right, I am happy that you asked the question.

 

Here we are talking about (real) numbers.

 

on the number line, you have the positive part and the negative part.

 

1, 2, 3 are positive.

You can make an operation like addition say

1+2=3

 

From a semantic point of vue, one should make a distinction between what is the mathematical operation (here addition) and what is the number (the operand here 1,2,3)

 

If (i say if) the operator, the + sign, is different from the number, then one has to conclude that numbers (1,2,3) are neutral. Numbers have intrisically no sign, they are neither positive neither negative, they are neutral.

 

If the above is correct, then when inspecting the negative part of the number line there is a kind of issue, in the sense that one cannot separate the minus sign from the number. The actual concept is that in the negative part, the minus sign is intrisically part of the number in order to make it different from the conventionnal positive number.

 

Thus, in the negative part, the numbers are -1,-2,-3

 

wich are different by nature of numbers 1,2,3 that should be written +1,+2,+3

As if the + and - sign was part of the nature of the number and not only an operator.

 

So, coming back to the question "what do I mean by opposite", the answer is that, to me, opposites are objects of intrisically opposite values relatively to an axis of symmetry. And in order to get this "intrisic" property of opposite, the object itself (the number) must be written down in such a manner that it shows its oppositeness (if such a word exists) without the use of any operator. Otherwise there will exist a confusion between the operation and the operand.

Edited January 4, 2014 by michel123456

[studiot]

 

wich are different by nature of numbers 1,2,3 that should be written +1,+2,+3

As if the + and - sign was part of the nature of the number and not only an operator.

 

 

 

Not 'as if' this is actually the case, we just do not normally write the + by convention. But the + or - are definitely part of the number and not just an 'operator'.

 

for instance what happens if you add -2 to -3?

 

(-2) + (-3) = (-5)

 

The + here is an operator but all the minus signs are part of the number.

 

Strictly we should perhaps write

 

(+2) + (+3) = (+5)

 

where are the plus signs in the brackets are part of the number as before and the connective plus sign is an operator.

[michel123456]

Isn't that confusing?

Is it correct to insert an operator as it was an intrinsic property of a number?

 

For example, if I am not abused, in the binary system there exist only "zero" and "one", like an on/off switch, in which case there is no positive or negative meaning. In this case the number 1 is stricly neutral.

I don't know if there are other examples where numbers are neutral.

 

-------------

Anyway, you agree that the + and - sign are part of the number. In this case the notation I proposed shouldn't bother you.

 

The change with the traditional notation consists in that the nature of the opposite is not related to the operator, it can be anything. Simply, 2 opposite numbers have the same value but in opposite way, wathever it means. As AJB noted, it could be a same distance in opposite direction. That would be a question of units.

Edited January 5, 2014 by michel123456

[studiot]

 

Isn't that confusing?

Is it correct to insert an operator as it was an intrinsic property of a number?

 

Most people find it helpful rather than confusing, that is why they do it.

 

No it is not inserting an operator. The + or - is part of the number so -9 is a different number from +9 and can stand alone in an operation that has nothing to do with subtraction or addition. They are not a 'neutral' 9 with an appended sign.

The fact that the distance between 0 and - 9 is the same as the distance ebtween 0 and +9 is coincidence.

The distance between -18 and -9 is also the same.

In the complex domain there are an infinite number of numbers att he same distance from any given (complex) number.

That is getting into metric space theory, by the way.

 

 

For example, if I am not abused, in the binary system there exist only "zero" and "one", like an on/off switch, in which case there is no positive or negative meaning. In this case the number 1 is stricly neutral.

I don't know if there are other examples where numbers are neutral.

 

There are many different number systems with many different purposes.

 

Some of these do not have positive and negative numbers,

For example the complex numbers, and the transfinite numbers.

 

Some do not have a zero

For example the natural or counting numbers (this was the first system invented)

 

Some have strange looking numbers that cannot be directly expressed in the decimal system for example

[math]\left( {5 - \sqrt 7 } \right)[/math] The expression in brackets is one single number.

These are known as surds.

 

I mention the last because in my post#4 I said that you are onto the trail of something much bigger and you should look at certain (?new) ideas.

 

one such is the conjugate, which has many variations.

 

One of these is called the algebraic conjugate. These are the roots of the same algebraic equation.

 

For example the equation [math]{x^2} - 6x + 1 = 0[/math]

has two conjugate roots

[math]3 - 2\sqrt 2 [/math] and [math]3 + 2\sqrt 2 [/math]

 

neither of which are negative.

 

 

Anyway, you agree that the + and - sign are part of the number. In this case the notation I proposed shouldn't bother you.

 

 

I didn't say your notation bothers me and it doesn't.

 

But I don't see the point of changing something to introduce a new notation that does not add anything new.

This has already been done too many times in maths in my opinion.

 

I seriously suggest you will find much rich food for thought if you follow up my references in post#4

Edited January 5, 2014 by studiot