Is there a mathematical or physical, real world use for numbers passed I? Who determines what can be conceptualized or not?

Question

I is the Square root of -1, such that I * I = -1. Through this, I can be considered like a "Half Negative."

Why hasn't this been taken further? Why don't we make a quantity such that I^3 is -1, such that I is a third of a negative?

This can also be thought of as adding a new dimension. Positive numbers go to the right, negative numbers go to the left, imaginary numbers go up and down, and now, this new type of I goes into the third dimension.

In the end, nothing in math really exists. The concept of negatives were made up to make math work; however, this is hard to really understand. Why can't I just say that dividing by zero now produces some actual new concept? How can we just keep creating new numbers whenever we need? Who says that, for example, 8 / 0 is undefined. Why can't it be equal to 8j, or something of that nature, such that 8j * 0 = 8?

Why can't we just keep making up new math concepts?

In the realm of complex, we already have three solutions for the equation $x^3=-1$: $x_1=-1$, $x_2=\omega=\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}\text{i}$ and $x_3=\omega^2=\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}\text{i}$. Really no need for a new "j" where $j^3=-1$.
Plus, you might be looking for Quaternion? — CaldariNavyFleet, Jan 04 '24 at 06:25
To fill in the last comment more concretely: The polynomial $x^2+1=0$ (as well as $x^3+1=0$) cannot be factorized without going over to complex numbers; hence, the real numbers are not algebraically closed. But every complex polynomial can be factorized within complex numbers---this is the fundamental theorem of algebra---and so the complex numbers are algebraically closed: in a genuine sense, they are already as flexible as possible when it comes to roots. — Semiclassical, Jan 04 '24 at 07:21
Does this answer your question? Why don't we define "imaginary" numbers for every "impossibility"? — Hans Lundmark, Jan 04 '24 at 09:37
For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — Henrik supports the community, Jan 04 '24 at 10:03

freakish · Answer 1 · 2024-01-04T18:14:34.253

Why hasn't this been taken further? Why don't we make a quantity such that I^3 is -1, such that I is a third of a negative?

It is taken further. In fact inside the complex numbers every equation of the form $x^n=-1$ has a solution. Or more generally every polynomial has a root. So you don't have to invent new numbers, complex numbers will work just fine for that.

This can also be thought of as adding a new dimension.

No, not really. At least not in precise meaning of the word dimension. But we can and do actually add dimensions in different way, e.g. quaternions or more generally Cayley-Dickson algebra.

In the end, nothing in math really exists.

Well, it is a philosophical statement: what do you mean by "exists"? But I kind of agree with you.

The concept of negatives were made up to make math work;

Yes. Entire math is made up. So what?

Why can't I just say that dividing by zero now produces some actual new concept?

You can. Nothing stops you from doing this. However math is an art of logic. And so if I want to have a structure that satisfies some axioms, say a ring, then a logical consequence is that zero is not invertible. But there's nothing preventing us from considering different axioms in which indeed dividing by zero is possible.

How can we just keep creating new numbers whenever we need?

Yes, imagination is the limit. Isn't this the most beautiful thing about maths?

Why can't we just keep making up new math concepts?

We can, and we do that all the time. There's nothing wrong with that.

However the most important mathematical concepts are those that have applications. For example differential calculus is way more important than for example homological algebra. And so not all concepts are equal in this sense. Those that serve to solve real world problems will be valued highly. Other not necessarily. And that's why you will often hear "you cannot divide by zero". It is not a universal truth, it is just that those useful concepts don't allow it logically.

Logic is a limit though. Trying to work beyond it may lead to chaos, where everything is true and false at the same time. Beyond logic are unknown waters, forbidden territory. :) Again, maybe someone does consider it, but it is likely to be of little use.

Is there a mathematical or physical, real world use for numbers passed I?

Yes, complex numbers have huge applications, especially in quantum mechanics.

Who determines what can be conceptualized or not?

Absolutely no one. You can think of any concept you want. But life is life, some will be accepted, some rejected. And some will be accepted only after you die.

Toby Bartels · Answer 2 · 2024-01-04T22:22:15.697

We do indeed keep making up new math concepts! As the comment by @CaldariNavyFleet notes, we don't need to make up anything new to get another cube root of $ - 1 $, since we already have enough of those once we have $ \mathrm i $. However, as the same comment hints, we can make up additional square roots of $ - 1 $ if we want to get more dimensions. Or instead (or additionally), we can make up a number $ \infty $ to be the result of $ 8 / 0 $; this gives you the so-called extended real (or complex, quaternionic, etc) number system.

That said, we can't make things up willy-nilly; we have to see what properties our new number systems have. Extending from the real numbers to the complex numbers is not too bad, since the usual properties of addition and multiplication (the so-called field axioms: associativity, commutativity, distributivity, and the existence of identities and inverses, except for the multiplicative inverse of zero) continue to apply. However, there are subtler properties that fail; for example, in the real numbers, it's valid to conclude $ x = y = 0 $ whenever $ x ^ 2 + y ^ 2 = 0 $, while this is no longer valid in the complex numbers.

If you make up additional square roots of $ - 1 $ to get additional numbers, then you find that you really need to have three independent square roots (so four total dimensions including the real one), or else it's impossible to figure out how to multiply the different square roots together. William Hamilton spent a month in 1843 trying to figure this out! Even when he hit upon the solution, he found that he had to drop commutativity of multiplication; it's just not compatible with having multiple independent square roots of $ - 1 $.

Similarly, if you admit $ \infty $ into your number system, many of the familiar rules almost hold but no longer quite. There are actually different ways to do this; personally, I prefer the system of arithmetic where $ \infty + \infty = \infty $ and $ 0 \infty = 0 $, but some people prefer to leave these undefined. Either way, $ 0 $ and $ \infty $ aren't really multiplicative inverses (because that would require $ 0 \infty = 1 $ instead), and so division is no longer a simple matter of multiplying by the multiplicative inverse. Distributivity also has a few exceptions. So this is more complicated than it might appear at first.

But yes, you can make up whatever you like in math, as long as it's logically consistent; just don't assume that all of the previous results continue to apply.

Is there a mathematical or physical, real world use for numbers passed I? Who determines what can be conceptualized or not?

2 Answers2