√ x when x < 0 was impossible using real numbers. Then, one day some guy decided that i = √ -1 and suddenly we can work with negative square roots. Which is still quite weird, but I suppose is valid (?)
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But there still stuff we can’t calculate in any of the number sets we have. Something we see much earlier than roots: division by 0.
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I get why x / 0 = x is false. Basically that would mean that 1 = 0, which is absurd (reductio ad absurdum). Ok. I don’t quite get why x/0 = 0 is false, but I’m sure there’s a contradiction somewhere that would make this impossible (if it was that simple, division by 0 would have been solved thousand of years prior). But if we can simply invent a number that doesn’t exists and it can fit maths no problem such as i = √-1 why can’t we make a number n = 0^(-1)?
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I also know the concept of limits but lim x->0 f(x) just means that x gets really close to the value of 0 but is never quite there so yeah, I don’t think that solves the issue.
In: Mathematics
The reason they made the square root of -1 the letter i is because complex numbers are very useful in many different applications. There is no real use for x/0, so we call it undefined, or not a number.
For what reason would someone do this? What’s the utility here?
They pretty much have, it’s called “ERROR” it represents either and infinitely positive or negative value.
Sure they could come up a symbol to represent that, but there isn’t much value to it
There is actually a construct in math for what you’re talking about, called the ‘projective real number line’. It’s a number line which extends in both directions to positive and negative infinity, but which adds an extra “point at infinity” which connects the positive and negative extensions of the line.
https://en.wikipedia.org/wiki/Projectively_extended_real_line
In this extended number system, the point at infinity *is* the reciprocal of zero. But there are certain other well-defined algebraic properties that this infinity simply doesn’t have. It can’t be factored, for instance, so you can’t deduce 2(∞) = 3(∞) therefore 2=3 or anything like that. But a number system which includes this concept of infinity still has its uses.
> Then, one day some guy decided that i = √ -1 and suddenly we can work with negative square roots. Which is still quite weird, but I suppose is valid (?)
This was quite controversial until people realised you could define complex numbers in a slightly different way. Basically you define a complex number as a pair of real numbers *a* and *b* and denote it *a* + *b*i. Then you define rules for performing addition and multiplication on these numbers. specifically (a+bi) + (c+di) = (a+c) + (b+d)i, and (a+bi)(c+di) = (ac-bd) + (ad+bc)i. From this you can see that complex numbers for which the “b” (the “imaginary part”) is zero are equivalent to real numbers, and that the complex number 0+1i, or just i, produces -1 when you square it.
And from there you can define all kinds of other operations on complex numbers and examine what properties they have. So instead of starting off with the idea that i is “the square root of -1”, we’ve just defined a structure made up of real numbers and shown that something in this structure happens to behave as if it’s the square root of -1.
> But if we can simply invent a number that doesn’t exists and it can fit maths no problem such as i = √-1 why can’t we make a number n = 0^(-1)?
You can do this, and there are a number of extensions of the real numbers in which this is done. For example the real projective line consists of the real numbers together with ∞, with a/0=∞ for nonzero a. These systems do have some uses, but they’re generally a bit awkward to work with as they don’t follow some of the standard rules of arithmetic that we’re used to. For example if a/c=b/c, it’s not necessarily the case that a=b, because c might be zero. And they’re nowhere near as interesting or as useful as the complex numbers, which have all kinds of incredible mathematical properties and widespread applications in quantum mechanics, electrical engineering, etc.
They can, and there’s probably a number of people that have done so and played around to see what happens when you define such a value.
The problem is that when you’re doing arithmetic with the most common sets of numbers, introducing such a definition causes a ton of other operations to become inconsistent nonsense.
Example:
let 1/0=j
let n be any real number
(n)(1)/0=nj
since 1n=n
n/0=nj multiply both sides by 0
n=0nj and 0n=0 so
n=j
So I just showed that j can be (and must be) literally any real number.
This is a big problem if I ever try to do math with my new number j. I could multiply both sides of an equation by j, but j is any real number, so I could do:
1=1
1j=1j
1(5)=1(20)
5=20
You end up in a situation where every number is equal to every other number. Such a mathematical system has little practical use, and honestly gets kind of boring to play with when everything breaks like this.
>√ x when x < 0 was impossible using real numbers. Then, one day some guy decided that i = √ -1 and suddenly we can work with negative square roots. Which is still quite weird, but I suppose is valid (?)
Not quite. Instead, mathematicians were trying to solve a certain class of problems. In doing so, they came across the square root of negative numbers. In the end, those square roots cancelled out, but the fact that they could exist without breaking everything indicated that there was something there.
Mathematicians still didn’t like it and these numbers were treated with derision for along while, but we’ve come to eventually accept them as a valid extension of the real numbers.
Point is, we were pretty much *forced* to acknowledge their existence. We didn’t just decide “hey, let’s make a new class of numbers.”
>But there still stuff we can’t calculate in any of the number sets we have. Something we see much earlier than roots: division by 0.
>
>
I get why x / 0 = x is false. Basically that would mean that 1 = 0, which is absurd (reductio ad absurdum). Ok. I don’t quite get why x/0 = 0 is false, but I’m sure there’s a contradiction somewhere that would make this impossible (if it was that simple, division by 0 would have been solved thousand of years prior). But if we can simply invent a number that doesn’t exists and it can fit maths no problem such as i = √-1 why can’t we make a number n = 0^(-1)?
Ok. Let’s say x/0 = 0. That means x = 0*0; x = 0. So it would only work for x = 0. If you tried it for any other value of x, you’d immediately get a contradiction (1/0 = 0 implies 1 = 0; 2/0 = 0 implies 2=0, etc.)
But there’s issues even with saying 0/0 = 0.
When we ask “What is a/b” we’re asking, “What can we multiply b by to get a?” This means “What is 0/0” we’re asking “What can we multiply 0 by to get 0.” And the answer is *anything*. So there isn’t a single answer to that question.
At the end of the day, trying to assign a value to x/0 ends up in contradictions or otherwise breaking fundamental mathematical concepts.
Compared to square root of negative numbers were shown to be necessary and have greatly expanded our understanding of math.
There was nothing special about the square root of negative numbers. We started with just counting numbers and then subtraction caused the invention of negative numbers, and division caused the invention of rational numbers (fractions). Imaginary numbers were just one more step.
The key is that all these expansions of the types of numbers still followed all the rules of arithmetic and produced consistent results. You can’t do that by trying to define infinity as a number. If you try, you allow all those proofs that 2=3 to work.
First, there’s no need to ask about “mathematicians”. You’re a mathematician if you do math. You’re a mathematician if you think about numbers. You’re a mathematician if you ask questions about numbers. You are a mathematician.
Second, as a mathematician, you can have any numbers you like. You can make up a number for 1/0. Go for it.
Third, if you talk to other mathematicians, you will find that they like some numbers more than others, and their criterion for liking numbers is that they should be interesting, which is to say, that they should lead to interesting mathematics.
There are number systems, like the projective line and the Reimann sphere, that have a number for infinity that interacts with zero in various ways. You could look into these if you are interested.
Or you could make up your own definition for 1/0. Depending on how you define it, you can end up in situations (as other commenters have explained) were every number equals every other number. The problem with that isn’t that it is wrong, the problem is that–in the opinion of most mathematicians–it is boring. But they’re your numbers, and you get to decide for yourself whether they are boring.
Beyond that, you get to decide for yourself what you want out of your numbers. Maybe you aren’t looking for interesting mathematics. Maybe you just want a number that you can put on your shelf and stare at, like a Necker cube or an Impossible Trident.
Enjoy 🙂
Numberphile did a great video on this topic, an basically 1/x when x=1 is just 1 but as we make x smaller and smaller, the result shoots add to infinity
Do the same with but start at -1 and the result is -infinity the contradiction is the problem
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