Product of Negatives (2010) • Hacker News Board

Re: Product of Negatives (2010)

#31

Post by abrax3141 » Sun, Mar 01, 2020, 3:14 PM UTC

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…

Unfortunately (or perhaps fortunately?) we depend upon the a+bi notation to do the right thing algebraically with complex numbers. I do agree, though, that we should always teach that ALL numbers are really complex numbers, although the phase is usually 0 or pi (and that a negative real part is just a shorthand to save us having pi all over the place in most simple math.)

Re: Product of Negatives (2010)

#32

Post by susam » Sun, Mar 01, 2020, 3:22 PM UTC

kidintech wrote: post #7

I strongly dislike these kinds of articles/posts due to one reason: if you're going to prove such a fundamental thing, can you please provide the axioms that we start from? I.e. "we know" that a - a = 0, multiplication is distributive, and a x - b = - a x b. These seem arbitrary properties and "equally" fundamental to -a x -b = ab. Either start from peano and prove everything along the way, or tell the reader your as…

> I highly doubt they were there when I wrote the comment.

Hi! I am the author of this blog post. The assumptions in the third paragraph were there at least since 16 Feb 2019. See https://github.com/susam/susam.in/commits/master/content/blo... for the change history of this post.

Re: Product of Negatives (2010)

#33

Post by soVeryTired » Sun, Mar 01, 2020, 3:25 PM UTC

alipang wrote: post #26

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…

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 accept as the solution of an equation. Note that I've already done this with fractions.

Fractions: Given integers a and b, ax + b = 0 has a meaningful solution

Complex numbers: The equation x^2 + 1 = 0 has a meaningful solution

Re: Product of Negatives (2010)

#34

Post by nathias » Sun, Mar 01, 2020, 3:40 PM UTC

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

Re: Product of Negatives (2010)

#35

Post by red_trumpet » Sun, Mar 01, 2020, 4:02 PM UTC

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…

At which point does the construction of the reals require the axiom of choice (AC)? I'm not familiar what exactly happens without AC, but defining R as the set of rational Cauchy-sequences modulo zero-sequences does not seem to use it?

Re: Product of Negatives (2010)

#36

Post by heinrich5991 » Sun, Mar 01, 2020, 4:08 PM UTC

red_trumpet wrote: post #35

soVeryTired wrote: post #33

Earlier quoted context omitted.

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…

At which point does the construction of the reals require the axiom of choice (AC)? I'm not familiar what exactly happens without AC, but defining R as the set of rational Cauchy-sequences modulo zero-sequences does not seem to use it?

The moment where you want a Cauchy sequence of real numbers to converge.

You have a Cauchy sequence of real numbers (a_i)_i. You pick a representative (b_ij)_j (a Cauchy sequence of rational numbers) for each sequence member a_i (axiom of choice!) and then produce the diagonal sequence (b_ii)_i which is the constructed limit of the Cauchy sequence (a_i)_i.

Re: Product of Negatives (2010)

#37

Post by enriquto » Sun, Mar 01, 2020, 4:41 PM UTC

nathias wrote: post #34

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/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.

Re: Product of Negatives (2010)

#38

Post by dorchadas » Sun, Mar 01, 2020, 5:14 PM UTC

throwfermat wrote: post #5

saagarjha wrote: post #2

An alternative, "common sense proof" would be that you're undoing the taking away of things, meaning you have more than you started (i.e. a positive result).

Here is a nice one I read sometime back. Create a video of your friend walking 3 metres. Now play the video 4 times. Your friend walks 12 metres in the video. Play the video in reverse 4 times. Your friend walks 12 metres backwards in the video. Create another video of your friend walking backwards 3 metres. Now play the video 4 times. Your friend walks 12 metres backwards in the video. Play the video in reverse 4 ti…

As a math teacher, I've found the best luck with teaching negatives by putting them in terms of direction. When kids start seeing "right" as "positive" and "left" as "negative", it makes a bit more sense to them. Then couple in positive/negative with forward/backward and it just generally clicks.

It also even helps explain division when you talk about it in terms of direction and how many times you have to move to get to 0. It also helps them understand why you can't divide by 0. e.g. 3/0 would be explained like "Starting at 3, how many times can you move 0 to get to 0," They can clearly see it's impossible, and it helps give them at least some basic intuition into it.

I've also found this extends to at least some properties of complex numbers as well, as you can easily extend from the number line to a coordinate plane.

Re: Product of Negatives (2010)

#39

Post by thaumasiotes » Sun, Mar 01, 2020, 5:17 PM UTC

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?

Re: Product of Negatives (2010)

#40

Post by thaumasiotes » Sun, Mar 01, 2020, 5:20 PM UTC

akalin wrote: post #12

throwfermat wrote: post #4

There is a discussion in this post's comments section⁽¹⁾ that this works for fields and rings too. I know there are precise definitions for fields and rings but can someone here give me some good examples of fields and rings? Being a non-mathematician, I find it easy to manipulate examples than manipulate definitions. Are the set of integers a field? I guess not because the multiplicative inverse of 2 is not present…

There's a subtle point to keep in mind when generalizing to rings/fields. The concept of 'positive' and 'negative' are defined in terms of an order relation, e.g., 'positive' means >0 and 'negative' means For example, the integers mod n is a ring, so (-a) * (-b) = a * b holds, but it doesn't make sense to call a number mod n positive or negative, since -a mod n effectively means n - a mod n. (posted an earlier versio…

> The concept of 'positive' and 'negative' are defined in terms of an order relation, e.g., 'positive' means >0 and 'negative' means I thought the concept of "negative" was defined by reference to an operation. "Negative 5" is whatever value Q satisfies the equation 5 + Q = 0.

That definition immediately tells you that the negative of a negative is a positive. Once we know 5 + Q = 0, we ask what the negative of Q is. It's the value V such that Q + V = 0. But by the definition of Q (and the commutativity of addition), we already know V = 5.

Once you define negatives this way, it's trivial to show that negatives obey the standard ordering. But that ordering wasn't necessary in order to define them.

Summing up, the product of negatives is positive because negation is a kind of inversion (additive inversion), and two successive inversions always cancel in any context.