Math: Are negative numbers real things?

[joehilton]

It seems to me that negative numbers are a convenient way to measure movement between finite systems, but may not be real universally.

For example, if money leaves my account (negative number) it doesn't disappear or go into a universal money vacuum somewhere, it just goes (positive number) to another account. Same for body temperature, photon emission, or anything else in the universe - don't the laws of conservation mean that negative numbers are just measurements for movement between systems but there's never an real "subtraction" of anything anywhere anytime?

And if negative numbers really are contrived to help us deal with systems in isolation, then have we hamstrung our fundamental mathematics by treating them as real numbers instead of imaginary?

[twangist]

They're quite as real as nonnegative numbers, which are also abstract things. It's ironic that in your last line you use the terms "real" and "imaginary", in their colloquial meanings, as these have long since been adopted and coopted to denote... the real and complex numbers respectively :) which, some of us contend, are no less real than the integers. Face it: groups abound in physics. A group has a binary operation such that for every element has a (unique) inverse element w.r.t. that operation. When the group is Abelian we write the operation as "+", and then the "additive" inverse of an element x is -x, also an element of the group. It's silly to claim that, for example, rotation through an angle $\theta > 0$ should be considered more "real" than rotation through $-\theta$. One is counterclockwise, the other, clockwise, by the same amount.

[joehilton]

I was actually using the terms "real" and "imaginary" non-colloquially to denote real and complex numbers (5, -1, pi, e, -1^-2, etc.).

It's true that nonnegative numbers are just abstract ideas that we come up with to represent quantity, but that's just it: quantity is a real thing. Is there such a thing as negative quantity? That's my question.

I like the reference to the contention of what we currently denote as imaginary numbers and how they should be compared as just as real as other numbers. This is the same issue I'm questioning but from the other side. I'm questioning if we should count negative numbers as real as opposed to understanding them as a set that is at least imaginary, if not complex or irrational.

I also like the point about a counterclockwise rotation being just as real as a clockwise rotation. This is where the non/negative sign is used to indicate a vector. As long as you have a point of reference, this is always ok because it's simply referring to what type of amount of whatever quantity/magnitude/etc we're dealing with.

But I believe this is different than what we often do with negative numbers when working through many of our problems that involve unitless numbers/constants or take an abstraction and apply it as a physical theory - in these cases the negative numbers were treated as separate and real when in fact they should have been directions, unit movement, or other information about actual physical quantities. This is where I'm wondering if we should treat negative numbers as a different set altogether.

[twangist]

Fine point: in the case of "negative" (clockwise) rotations, the negative sign applies only to a scalar (an angle), not a vector. Surely even negative money is a real thing too: debt, or anyway debit. (Granted, debt can be less real than a pile of cash: if you're owed you might never be repaid, if you owe you might default. But that's a case of reality not matching theory well :)

In physical applications, relative to a coordinate system negative numbers have perfectly intuitive meanings. A negative spatial coordinate simply indicates a point in one of the quadrants or octants other than the all-positive-coords one. A negative acceleration means something is slowing down.

I see no good purpose served by special-casing "negative" integers, and many needless ensuing headaches.

[dozzie]

I think you don't understand the difference between amount of something and a number. Number is just an abstract concept, and it doesn't have any inherent properties like being "real" or "imaginary", whatever that means for you. On the other hand, amount can never be negative.

BTW, read a little more about number sets. There are such sets as "real numbers" and "imaginary numbers". I think you can find some material in mathematics textbooks for ground school and high school (probably not about imaginary numbers, though).

[joehilton]

The difference between the amount and a number is exactly what I'm afraid negative numbers misrepresent.

Real and imaginary number sets are the heart of the point (and I think my high school calc - if I can remember that many years ago - did include some work with imaginary numbers, certainly by early college work).

Since an amount can never be negative, then a negative number only makes sense to denote the movement of an amount from one system to another. If we say that negative numbers are real numbers, and especially if we use them as the same class as positive numbers when building blocks to formulate ideas and theories about physical properties and amounts, I'm wondering if there is a potential downside to our fundamental understanding of what our results actually mean.

One of the other replies points out the necessity of units paired with numbers, and I think this is correct. The problem I'm observing, and the heart of my question, is that we have many, many ideas/proofs/theorems in mathematics with unitless numbers (constants and so on, both positive and negative), and then we use the results of this work to jump back into the physical universe to "prove" something. Could there be a problem with our basic use of numbers when we do this?

[ttctciyf]

If this bothers you, you probably won't be pleased to learn that the overwhelming majority of "real numbers" can't be written down or even computed (ok, in principle you can enumerate the digits of at least one such number, but "Becher and Figueira proved in 2002 that there is a computable absolutely normal number, however no digits of their number are known.") [1]

1: https://en.wikipedia.org/wiki/Normal_number

[joehilton]

This is very interesting - thanks for the reply and link. I'll dive into Becher and Figueria.

[geekishmatt]

You are mixing up the mathematical universe with the physical (also known as real) universe.

For a mathematican 42 or -42 makes sense - there are no units necessary. The mathematical intepretation of a negative number is just being element of a set.

For a physicist 42 or -42 are pointless, because a unit is missing. The physical intepretation of a negative number is the quantified absence of a unit.

- by the way: https://xkcd.com/435/

[joehilton]

Yes this is exactly right: "The physical intepretation of a negative number is the quantified absence of a unit." In the physical world, the absence of a unit just means the perspective doesn't include whatever it is that is intended to be observed, although that thing may exist elsewhere. In other words, a negative number may be an abstraction for quantifying that a unit has left an isolated system, but I'm not sure our current usage of numbers as abstractions captures this - it seems like we often use negative numbers (in any unit) to denote the actual subtraction of that unit universally as opposed to its transfer to another isolated system. (This could be where mathematicians can get into trouble compared to physicists as perhaps they don't have the same habit of observing units.)

[geekishmatt]

Sometimes a subtraction of a unit absolutly legal and you are still in the same model or context. (Temperature loss)

Sometimes a subtraction of a unit is hidden-black-magic and you are switching between two models or contexts but still being in a meta-model or meta-context. (Debits and credits)

It just depends on your model or context how you interprete your value/unit relation, but this is a typical problem in applied physic, engineering or computer science :)