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Product of Negatives (2010)

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Re: Product of Negatives (2010)

#21

Post by kidintech »

yori wrote: post #14
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…

Like mentioned in another comment on this thread, the assumptions are well known field axioms. They form a good starting point. And why start with Peano axioms? They seem like a bad starting point because it would take pages upon pages of proof and it won't easily extend to other algebraic structures like rings and fields.

> the assumptions are well known field axioms. They form a good starting point.

I gave Peano as an example. I don't mind the assumptions, as long as they're reasonable and presented before the proof. Another comment pointed me to the fact that they were mentioned in an earlier paragraph, so my issue is resolved.

Re: Product of Negatives (2010)

#22

Post by _2d30 »

soVeryTired wrote: post #18

I've always thought the best way to explain this was by analogy with the '90s TV show "The Crystal Maze" [0]. Contestents are put in a dome filled with gold and silver tickets being blown around by fans. For every gold ticket they collect, they get a point. For every silver ticket, they lose a point. If they collect enough points, they win a prize. Sorting through the team's collection of tickets and throwing away a…

What does this analogy have to do with multiplication?

Re: Product of Negatives (2010)

#24

Post by soVeryTired »

_2d30 wrote: post #22
soVeryTired wrote: post #18

I've always thought the best way to explain this was by analogy with the '90s TV show "The Crystal Maze" [0]. Contestents are put in a dome filled with gold and silver tickets being blown around by fans. For every gold ticket they collect, they get a point. For every silver ticket, they lose a point. If they collect enough points, they win a prize. Sorting through the team's collection of tickets and throwing away a…

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.

Re: Product of Negatives (2010)

#25

Post by wsxcde »

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 question to ask is why we made this specific choice of definition. Why should multiplication be defined such that -2*-3 = 6? This is a question that the post does shed some light on. 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.

Re: Product of Negatives (2010)

#26

Post by alipang »

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. But even then the numbers are probably still best thought of as just a helpful abstraction.

Re: Product of Negatives (2010)

#27

Post by gaze »

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…

what does “exists” mean? We made the whole thing up.

Re: Product of Negatives (2010)

#28

Post by TheOtherHobbes »

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,…

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 to mean?)

In reality complex numbers are a way of moving from the number line to a number circle. (Which eventually generalises to a 3-sphere when you get to quaternions.)

That's all they are. Instead of linear arithmetic - which is about combining magnitudes in one dimension - you can now do arithmetic that combines magnitudes with rotations.

The extra dimension makes it possible to solve equations with solutions that don't exist on the basic number line. It also makes it easier to do calculations that combine magnitude with phase - which includes pretty much anything that rotates or processes linear combinations of sine waves, and which a straight vector tuple can't handle.

If someone had told me this when I was learning complex numbers the cognitive dissonance wouldn't have hurt quite as much.

Re: Product of Negatives (2010)

#29

Post by empath75 »

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’m not sure that I would rely on a physical manifestation to show that a number ‘exists’. They exist because we’ve defined them as part of the rule set we use for the game we call math. Regardless of their applicability to nature, they still ‘exist’ in that sense.

Re: Product of Negatives (2010)

#30

Post by empath75 »

b0rsuk wrote: post #19

I've seen a better explaination in this Mathologer video. In a bizarre twist it is now private (?!). Maybe it will work for you. But I suspect it was a takedown notice because he used a short clip from a movie famous among teachers. https://www.youtube.com/watch?v=ij-EK-MZv2Q The first number represents the amount of something. If it's negative, you have a debt. The second number represents either a gain (if it's pos…

He had a dispute with his original camera man that resulted in some of his earlier videos being taken down.
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