Can Negative Numbers Exist Beyond Relative Terms?

[gonegahgah]

May I just check something please.
You can't actually have -3 apples can you?
Negative numbers are only relative numbers aren't they?

eg.
a. You can be 10m west of a particular point
b. You can be -20m west of a particular point (which makes you 20m east)

Same with temperate. 0 is just a point that is set and any temp below is negative.

Is it correct that negative numbers are only relative numbers?
Or can negative numbers exist in another form(s)?

 

[Gear300]

It depends on how you attach a relative meaning to it. Let's say that to remain as a buyer, you need 3n coins (n>=0)...and after that you need n coins for every apple. If you have 0 coins on you, you're not even qualified as a buyer...then that would mean that it is somewhat the same as saying you having -3 apples.

Negative numbers are additive inverses to positive numbers, and could also be referred to as symmetric to positive numbers; if you can find the appropriate symmetry, then you could involve negative numbers for all or a certain range of positive numbers.

 

[HallsofIvy]

If you have no apples and you owe someone 3 apples, you could say you have "-3" apples. As Gear300 says, it all depends upon what meaning you attach to the numbers.

 

[gonegahgah]

Cool. Thanks.
Then it becomes like a ledger item rather than actual physical apples but which will be met by future production of apples or comparative compensation (unless filing for bankruptcy). Cool.

I also have another question about negatives.
When we talk about √-1 or any other negative number (these are called imaginary numbers aren't they?) can this be shown using area diagrams on an x-y axis where the axis have marked 0 and negative numbers, and it has 4 quadrants? ie 2x2 quadrant gives area 4, -2x-2 quadrant gives area 4, -2x2 quadrant gives area -4, 2x-2 quadrant gives area -4.

For area, in these respects, the negative number is only a relative to origin number and negative area results only denote position relative to origin being top-left or bottom-right where right and top are considered positive and for positive areas top-right or bottom-left.

For a physical area we could say it consists of 2 lengths to the west x 2 lengths to the north giving an area of 4 in the nw quadrant.
We could do a different one where an area consists of 2 lengths to west x -2 lengths to the north giving an area of 4 in the sw quadrant.

Mathematically would the above be said to be a negative area when we treat both n-s and e-w orientations as the same unit - eg metres - and the n distance as -2 distance while the e distance as 2 distance?

 

[arildno]

The area interpretation of the operation of multiplication is of little use, even for negative numbers.

What IS crucial, is how the operation is DEFINED; only subsequently is it of interest to find out what it may model.

Take multiplication of negatives as an example:

CAN we find some neat illustration of this in "the real world"?

Sure we can!

Let's say we are placed at a road (at what we call the "spatial origin"), and at a particular moment (called the "temporal origin"), a car whizzes by at a speed of 50 m/s.

Now, can we make sense of the different multiplications:

a) 50*1, b) (-50*1), c) 50*(-1), d) (-50)*(-1) ?

Sure we can!
a) describes the position of the car one second AFTER t=0, and specifies that the car was moving towards the right

d) (-50)*(-1) describes the position of the car one secong BEFORE t=0, its direction of movement leftwards.

And so on.

None of these examples can be said (easily) to be simplistic "areas", and similarly, nor is multiplication of complex/imaginary numbers to be so interpreted.

 

[uart]

***Just for comic relief***

I once saw a stand up comic making a joke about his lack of understanding of negative numbers and it really gave me a good laugh. I can't remember the exact dialog but it went something like this :

If I had 2 apples and I gave away 4 of them then I'd have negative 2 apples.

Now if I was walking down the steet with those 2 apples which I don't have and I meet up with another fellow who also didn't have any apples, only he didn't have 6 of them, then if i took the six apples he didn't have away from the 2 apples that I don't have then we'd have positve 4 apples! What I want to know is why the poor countries with food shortages aren't already doing this!

Does any else find that fairly amusing?

 

[gonegahgah]

@uart, yes it is funny.

@arildno, I agree that those examples don't represent simple areas as one of the units is time. You could also say that someone covered two lots of 25 metres making a total distance of 50 metres and still not be representing simple areas though the 2 then has no unit at all.

Slightly off topic, could it even be put forward that area calculations can be considered to have different units on each axis? For example could we say that a rectangle of 4m long by 2m wide gives an area of 8(mlong)(mwides) instead of 8m². This is just to say that a metre in one axis is not identical to a metre in another axis; even though by rotation you can achieve transposition. If they were identical you could simply add them together instead of multiplying them. Obviously this is implicit to our understanding of area; but the vital distinction remains between the two axis as a critical differentiation between one measure and the other in dictating how we calculate. Is that so?

Also, to help me explore this further, can you arildno, or someone, provide me with real world examples where we use imaginary numbers that aren't area based? I want to explore this idea further so that would be a great help. Thanks.

 

[arildno]

Sure, gonegahgah:

Caspar Wessel, a norwegian surveyor made in the late 18th century a note on how we might interpret multiplication of complex numbers in a geometric manner (the French mathematician Argand made a similar note, independently, a decade later).

A simple reflection on multiplication of real numbers (both positive and negative) can give the gist of the idea:

Consider "1" as the arrow from the origin to one unit along the positive (right) side of number scale, "-1" as the arrow from the origin to one unit on the negative (left) side of the number scale.

Both arrows has TWO properties:
a) A LENGTH, i.e the positive number "1"
b) An angle to the positive number scale (0 degrees for the 1-arrow, 180 degrees for the (-1)-arrow)

Now, "define" multiplication as two intertwined operations:
Multiply the lengths together (as defined for positive numbers), ADD the angles two such arrows make with the positive half-axis.

1*1 will therefore have LENGTH 1; the rsultant arrow making 0+0=0 degrees to the positive half-axis, i.e, be the 1-arrow itself.

1*(-1) will have LENGTH 1, the resultant arrow will make 0+180=180 degrees to the positive half-axis, i.e, be the (-1)-arrow

(-1)*(-1) will have LENGTH 1, the resultant arrow will make 180+180=360 degrees to the positive half-axis, be the 1-arrow.


In an entirely similar manner, construct the COMPLEX NUMBER PLANE, where an imaginary axis is drawn perpendicularly to the real number line through the origin.

Every point in the plane can be represented as an "arrow" that has a length, and makes a certain angle to the positive half-axis of the real number line.

Multiplication of two such arrows produce a new arrow whose length is the product of the factors' lengths, and whose angle to the positive half-axis will be the SUM of the factors' respective angles.

That's the geometric interpretation of multiplication of complex numbers.

 

[gonegahgah]

Thanks arildno.

I will have to think on this.
My initial feel is to be worried that a multiplication is being transformed into a paired (multiply,addition).
I would have thought that 1 arrow left * 1 arrow right would answer: 1(arrow left)(arrow right) or 1(0deg)(180deg)? Like: 4a * 4b = 16ab?
I'm not sure how the coupled addition gets in there?
Where am I going wrong?

Example wise, I think I heard that physics or engineering uses √-1 or other √negatives in experiments/engineering feats. I wondered if you could recall any of these that I might explore to see how they work the √negatives physically?

 

[Bohrok]

This page might be of interest to you about \sqrt{-1}
http://mathforum.org/library/drmath/view/53879.html

 

[KevinItIs]

Nice Question..

 

[gonegahgah]

Thanks Bohrok. That will give me something to explore.