On: Why don't we define “imaginary” numbers for every ...

[nextaccountic]

> 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 numbers that allow division by zero. One is a real projective line, which has only one infinity so that 1 / 0 = -1 / 0 = infinity

https://en.wikipedia.org/wiki/Real_projective_line

Another is the extended real number line which has positive infinity and negative infinity, so 1 / 0 = +infinity and -1 / 0 = -infinity and they are different from each other

https://en.wikipedia.org/wiki/Extended_real_number_line

Those are all perfectly fine but they still can't define 0 / 0, which is a harder problem.

[marcosdumay → nextaccountic]

> 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.

[nextaccountic → marcosdumay]

Addition can be total if you have a single infinity, just make infinity + n = infinity for all n

Subtraction, multiplication and division is harder

But making further operations partial isn't that of a big deal; in fields the division is already partial due to division by zero not being defined.

[comte7092 → nextaccountic]

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.

[Grustaf → comte7092]

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

[tshaddox → comte7092]

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

[hgsgm → comte7092]

What is the "issue"?

"/" means "* reciprocal of".

If "infinity" is defined as "reciprocal of 0", what is the problem?

Yes it is an exception to 0*n=0.

It won't work in every setting, but it works in some settings, like inversive geometry.

[scotty79 → comte7092]

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.