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Product of Negatives (2010)

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Re: Product of Negatives (2010)

#41

Post by mokus »

TheOtherHobbes wrote: post #28
stan_rogers wrote: post #6

Earlier quoted context omitted.

Complex numbers, the way they're used in most cases, is a tuple notation. They're a handy way of keeping your chocolate separate from your peanut butter, so to speak, as that little "times i " makes it difficult to accidentally get things mixed up. And that's the way I always explained it to my students: there are imaginary numbers in the original sense of fake roots that will go away if you ignore them long enough,…

They're the same thing in the sense they have the same roots. The most confusing thing about complex numbers is the language. First you're told negative numbers can't have roots, then you're told they so can too, but you have to call the roots "complex" or "imaginary." This sets up cognitive dissonance which can be harder to deal with than the math. (What even is an "imaginary number"? What are those words supposed t…

I found so much of math I had learned previously in the “it’s weird but this works if you take it on faith” sense was suddenly blatantly obvious after learning some abstract algebra. I wish I had learned that stuff way earlier.

In the case of complex numbers, I find the “paradox” disappears when you think of it in terms of fields abstractly. To put it maybe a bit overly simply - instead of focusing on the idea of “square roots of negative numbers”, instead step back and consider that number-like operations make sense for things that aren’t numbers at all in the traditional sense. One particularly useful example is 2d vectors, which you can add in the usual sense and “multiply” in polar form by multiplying “r” and adding “theta”. It turns out that these vectors with these operations act a LOT like numbers, and it also turns out that that weird multiply operation is actually super useful. One easy interpretation is combined scale and rotation transforms, with “multiplication” implementing composition.

Once you do that, it also turns out that solving equations like “what transformation composed with itself equals a 2x scaling with 180° rotation?” also make sense (i.e. “solve x^2 = -2”), and when you solve polynomials in this new system you get more solutions than you did for regular numbers. And that the thing you just invented IS the field of “complex numbers”.

[Sorry for the verbosity and probably poor organization, I’m in a bit of a hurry IRL and didn’t have time to edit it down. I did edit a bit for clarity and to fix typos, etc., though]

Re: Product of Negatives (2010)

#42

Post by _2d30 »

soVeryTired wrote: post #24
_2d30 wrote: post #22

Earlier quoted context omitted.

What does this analogy have to do with multiplication?

It shows that subtracting a minus one is equivalent to adding a plus one. The one logical leap that isn't explicitly spelled out is that subtracting X is the same as adding (-1)X. But I'm pretty sure that's the definition of integer multiplication.

I see how it’s intuition for addition/subtraction but that doesn’t tell us much about multiplication. You’re asserting that negative one times X is itself negative which is in fact what the article is attempting to prove in the first place so by explicitly supposing that, your analogy isn’t useful.

Re: Product of Negatives (2010)

#43

Post by robinzfc »

soVeryTired wrote: post #33
alipang wrote: post #26

Earlier quoted context omitted.

I've heard the objection about complex number not being real many times. I think the sensible answer is to argue that the natural numbers don't "actually exist" either. They're an abstraction just like the complex numbers. Arguably we might one day find out that the universe is discrete at which point we could begin to try to define the naturals as something that "exists", at least up to some maximum large number. Bu…

I have a much harder time beliving in the full set of real numbers than I do believing in the basic construction of complex numbers. The full set of real numbers requires me to accept things like the axiom of choice, and to believe that non-computable numbers 'exist' on the same level as computable ones. That doesn't sit right with everyone. Basic complex numbers, on the other hand, just require me to expand what I a…

The construction of real numbers does not need the axiom of choice in the sense that there are constructions of models of real numbers that do not need it. One example of such construction is described on the Wikipedia page [1], look for "Edudoxus reals" there.

[1] https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...

Re: Product of Negatives (2010)

#44

Post by carapace »

enriquto wrote: post #37
nathias wrote: post #34

Earlier quoted context omitted.

complex/negative numbers don't actually exist in contrast to natural numbers that don't actually exist in a different way

Complex numbers exist just as much as the Euclidean plane. Furthermore, they are also scalings and rotations of said plane.

The Euclidean plane also doesn't exist.

Re: Product of Negatives (2010)

#45

Post by ouid »

yunruse wrote: post #3

An important thing about numbers in general is that whenever somebody says “complex/negative numbers don’t actually exist”, they are somewhat right, in a sense. What exists is magnitude and phase Does that mean we should abandon them? Absolutely not. Encoding phase (or in a much more common subset, parity) is so absolutely useful it’s no wonder we bake 90° intervals (-, i) into our notations: they can be intuitively…

A number is just something that acts like a number. Polynomials are numbers. True and False are numbers.

Re: Product of Negatives (2010)

#46

Post by wsxcde »

thaumasiotes wrote: post #39
wsxcde wrote: post #25

This is not a proof of why the product of negative numbers is positive. The reason why the product of negative numbers is positive is that we define multiplication to be that way. Also, this post conflates the unary negation operator with negative numbers. The two are not the same. In so far as this post constitutes a proof (which IMO it does not), it is a proof about the behavior of the negation operator. A good que…

> If we'd chosen some other definition of multiplication, a lot of the "intuitive" properties of multiplication... would no longer be true Well, sure, if you change the definition of something, then it may end up having different properties. What's your point?

My point is that you cannot prove something that is true by definition. The OP trying to prove that the product of two negative numbers is positive is like asking to prove that 0 + 1 = 1 in Peano arithmetic.

The OP thinks that his "proof" is showing why multiplying negative values yields a positive result. But the proof is a load of nonsense because it assumes facts like distributivity of multiplication over addition and subtraction. It is literally impossible to prove that $\forall a, b, c \in Z. (a - b) * c = (a * c - b * c)$ -- distributivity of multiplication over subtraction -- without having already defined the meaning of a * b for all integers! This leads to a circular reasoning loop that the OP's "proof" can't get out of.

The thing to realize is that multiplication is not some magic operation handed down to us by god. It is just a binary total function defined over the integers. What the OP is trying to confusedly get at is the following:

1. There is an intuitive definition of multiplication as repeated addition over natural numbers.

2. It is not clear what the corresponding definition of multiplication over negative numbers is.

3. If we want to define multiplication as a total function over the integers, we need to define what the result should be when multiplying negative integers.

4. Specifically, with (3), we are taught in school that the result of multiplying two negative numbers should be positive, but it is not clear why this seemingly arbitrary choice was made.

Unfortunately, the OP is going about this all backwards. One cannot prove what the OP wants to prove. What one can instead do is argue that the specific (but seemingly arbitrary) definition that one has chosen for multiplication is a "good" choice because it has the same properties (distributivity etc.) as multiplication over natural numbers. At its core, this is a stylistic appeal about the "naturalness" of the definition.

Re: Product of Negatives (2010)

#47

Post by crdrost »

stan_rogers wrote: post #6
yunruse wrote: post #3

An important thing about numbers in general is that whenever somebody says “complex/negative numbers don’t actually exist”, they are somewhat right, in a sense. What exists is magnitude and phase Does that mean we should abandon them? Absolutely not. Encoding phase (or in a much more common subset, parity) is so absolutely useful it’s no wonder we bake 90° intervals (-, i) into our notations: they can be intuitively…

Complex numbers, the way they're used in most cases, is a tuple notation. They're a handy way of keeping your chocolate separate from your peanut butter, so to speak, as that little "times i " makes it difficult to accidentally get things mixed up. And that's the way I always explained it to my students: there are imaginary numbers in the original sense of fake roots that will go away if you ignore them long enough,…

You're describing some sort of type mismatch between two concepts I think, and I really don't understand it: and I feel like as someone who occasionally teaches these things I really would like to. Could you elaborate?

For my part, I do like to think of adjoining numbers onto an existing system, but that immediately becomes matrices.

So you decide to adjoin an ε such that ε² = 0. Your numbers are now vectors (a, b) and the action of ε is to map this to (b, 0) so that it is represented by the matrix

    [ 0, 0 ]
    [ 1, 0 ]
And thus the number a + b ε is perfectly encoded in the matrix algebra as a I + b ε =

    [ a, 0 ]
    [ b, a ]
This sort of trick is really helpful for programmers because we often have a field of bigints or rationals that we want to adjoin an irrational number to. So for example when you want to do Fibonaccis using exponentiation by squaring, it helps (although this is not obvious at first) to adjoin the golden ratio φ satisfying φ² = 1 + φ to your bigints, so that your numbers look like a + b φ =

    [ a,   b   ]
    [ b, a + b ]
Then F_n = [1, 0] . φ^(n+1) . [1, 0] and you can exponentiate by squaring straightforwardly. So you start from [0,1;1,1] and square that to [1,1;1,2] and square that to [2,3;3,5] and square that to [13,21;21,34] and square that to [610, 987; 987, 1597], so you get to skip ahead past 55, 89, 144, 233, and 377, at the cost that multiplications are slower than additions but also you potentially get to allocate less memory.

When you adjoin to the reals an i such that i² = -1 these matrices have the shape of the 2D scaled rotations, so that is what complex numbers just “are” to me. The representation as a tuple is to me the same as representing the above matrix as (a, b) or (a, b, a+b) to save time or space... The whole thing is a matrix but the entries are indeed redundant and so you don't need to store all of them at once.

So that's why I do not understand what you mean by these being separate concepts. Can you elaborate?

Re: Product of Negatives (2010)

#49

Post by yori »

wsxcde wrote: post #46
thaumasiotes wrote: post #39

Earlier quoted context omitted.

> If we'd chosen some other definition of multiplication, a lot of the "intuitive" properties of multiplication... would no longer be true Well, sure, if you change the definition of something, then it may end up having different properties. What's your point?

My point is that you cannot prove something that is true by definition. The OP trying to prove that the product of two negative numbers is positive is like asking to prove that 0 + 1 = 1 in Peano arithmetic. The OP thinks that his "proof" is showing why multiplying negative values yields a positive result. But the proof is a load of nonsense because it assumes facts like distributivity of multiplication over addition…

Not every arithmetic property needs to be proved from Peano axioms. One can but it is tedious and unnecessary. A much better starting point is the set of field axioms where the distributivity property is already available as an axiom.

The assumptions made in the article are perfectly fine as per field axioms. Granted it would have been nicer if distributivity over addition was used instead of distributivity over subtraction. But it is not a big leap to derive distributivity over substraction from field axioms by distributing multiplication over a positive number and the additive inverse of another positive number.

Wherever you see an assumption made about negative number, just mentally replace it with additive inverse of a positive number and you would be fine.

Re: Product of Negatives (2010)

#50

Post by yori »

wsxcde wrote: post #25

This is not a proof of why the product of negative numbers is positive. The reason why the product of negative numbers is positive is that we define multiplication to be that way. Also, this post conflates the unary negation operator with negative numbers. The two are not the same. In so far as this post constitutes a proof (which IMO it does not), it is a proof about the behavior of the negation operator. A good que…

> If we'd chosen some other definition of multiplication, a lot of the "intuitive" properties of multiplication that hold over the natural numbers (such as the distributivity of multiplication over addition and subtraction) would no longer be true over the integers.

This is backward reasoning. The chosen definition of multiplication is not to keep things "intuitive". If you start with the field axioms, the chosen definition of multiplication is pretty much dictated by the axioms. If you choose another definition of multiplication, you would end with contradictions like 1 = 0 and such nonsense! And mathematicians abhor contradictions!

"Product of additive inverses of two elements is equal to the product of the two elements" is dictated by the field axioms in all fields.

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