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Why don't we define “imaginary” numbers for every “impossibility”? (2012)

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Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#61

Post by marcosdumay »

nextaccountic wrote: post #55
ndsipa_pomu wrote: post #8

Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…

> The classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to "prove" that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful/interesting. There are multiple extensions to the real n…

> There are multiple extensions to the real numbers that allow division by zero.

Well, the gotcha is that they redefine the operations so that none of addition, subtraction, multiplication or division are total. Those operations just break in a different number than zero.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#62

Post by tshaddox »

comte7092 wrote: post #57
nextaccountic wrote: post #55

Earlier quoted context omitted.

> The classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to "prove" that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful/interesting. There are multiple extensions to the real n…

1 / 0 = +infinity Implies that 0 * +infinity = 1, so it does run into make of the same issues. There are instances that make it useful, but the extended real number line isn’t used heavily in practice.

Yeah, I don't really see what this gets you. With basic real number division you have to make the exception for zero in the definition:

    a/b = c if and only if a = c*b and b!=0
And with this infinity thing you just have to make essentially the same exception for multiplication and infinity:

    c*b = a if and only if a/b = c and b!=infinity and c!=infinity

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#63

Post by Grustaf »

ndsipa_pomu wrote: post #8

Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…

On the contrary, the extensions can be very useful and interesting. You do typically have to sacrifice something, like commutativity in the case of quaternions, but it will often be worth it.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#64

Post by tshaddox »

xigoi wrote: post #33
afiori wrote: post #14

Earlier quoted context omitted.

let ϴ = 0/0 then 1*ϴ = ϴ = 0/0 = (0*0)/0 = 0*(0/0) = 0*ϴ it follows 1 = 0 and thus x = x * 1 = x * 0 = 0 = y * 0 = y * 1 = y for all x and y

This is assuming that Θ interacts with arithmetic operations the usual way (that is, ℝ ∪ {Θ} is a field), which the person you're replying to did not say.

Sure, but the whole "problem" we were trying to solve was that zero doesn't interact with arithmetic operations the usual way.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#65

Post by Grustaf »

comte7092 wrote: post #57
nextaccountic wrote: post #55

Earlier quoted context omitted.

> The classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to "prove" that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful/interesting. There are multiple extensions to the real n…

1 / 0 = +infinity Implies that 0 * +infinity = 1, so it does run into make of the same issues. There are instances that make it useful, but the extended real number line isn’t used heavily in practice.

You might not have much use for the real projective line when tallying up prices in the grocery store, but projective geometry is definitely very useful. https://en.wikipedia.org/wiki/Projective_geometry

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#67

Post by thaumasiotes »

syzarian wrote: post #7
thaumasiotes wrote: post #3

Earlier quoted context omitted.

Those are all just normal imaginary numbers. The question is why, when we can't answer a question, we don't just invent a symbol, say it's the answer to the question, and call it a day. It's a stupid question, but it's not related to your response.

The question has 300+ upvotes. That’s a proxy for how “good” it is. A person is curious about an aspect of mathematics and posed a well stated question. It is not a stupid question. From their perspective mathematicians appear to do something and they wonder why it can’t be done in other situations. Such a question is the basis of understanding. It is by wondering such things that enables one to gain true understandi…

> It is not a stupid question. From their perspective mathematicians appear to do something and they wonder why it can’t be done in other situations.

I mean, you've already gotten it wrong. This can be done in other situations. Where it isn't done, it isn't done because doing it is pointless, not because there's some bar to giving names to opaque labels.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#68

Post by syzarian »

thaumasiotes wrote: post #67
syzarian wrote: post #7

Earlier quoted context omitted.

The question has 300+ upvotes. That’s a proxy for how “good” it is. A person is curious about an aspect of mathematics and posed a well stated question. It is not a stupid question. From their perspective mathematicians appear to do something and they wonder why it can’t be done in other situations. Such a question is the basis of understanding. It is by wondering such things that enables one to gain true understandi…

> It is not a stupid question. From their perspective mathematicians appear to do something and they wonder why it can’t be done in other situations. I mean, you've already gotten it wrong. This can be done in other situations. Where it isn't done, it isn't done because doing it is pointless, not because there's some bar to giving names to opaque labels.

How does your pedantry contribute meaningfully?

If something doesn’t behave like 0 in a ring or other algebraic structure then using that label is confusing and simply not done. You are free to use any symbol you want but mathematics is a human endeavor and as such communication is important. Using the symbol 0 signifies something to those with mathematical training. Zero can’t have an multiplicative inverse because anything you call 0 that has an multiplicative inverse makes it behave like something other than zero. So no one would use 0 to describe such an element. In a ring, or abelian group, the symbol 0 is reserved for the additive identity element.

Similarly, I could say snkwoo is what most people call a chair. A grammarian would say there is no word snkwoo even though I just defined it.

Your original comment was wrong and bad. Instead of just admitting it or moving on you’ve decided to double down and make another bad comment.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#69

Post by topaz0 »

syzarian wrote: post #54
iamerroragent wrote: post #51

Earlier quoted context omitted.

Okay so if you can get to a ring without a multiplicative inverse and then applying that operation to the ring forms it into a field then wouldn't it be fair to say that division is not really the opposite of multiplication the same way that subtraction absolutely is for addition?

The definition of division is multiplication by the multiplicative inverse. It may be the case that some elements don’t have such an inverse but the definition is analogous to that of subtraction. The analogy is not perfect because every element has an additive inverse while not every element had a multiplicative inverse.

What you're saying is that the analogy between subtraction and division is good as far as it goes. So why should "as far as it goes" end at zero not having an inverse, rather than division by zero producing something other than the multiplicative inverse of zero? The two choices end up having different structure, and so they end up being applicable to different things, but there is nothing wrong with either choice.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#70

Post by scotty79 »

comte7092 wrote: post #57
nextaccountic wrote: post #55

Earlier quoted context omitted.

> The classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to "prove" that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful/interesting. There are multiple extensions to the real n…

1 / 0 = +infinity Implies that 0 * +infinity = 1, so it does run into make of the same issues. There are instances that make it useful, but the extended real number line isn’t used heavily in practice.

In normal math 1/0 is undefined but in a math where 1/0 is defined to be inf the 0*inf is still undefined.
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