As a layperson who is interested in math, imaginary numbers always fascinated me. Like in the real world you taking the square of a negative makes no sense whatso ever, but in theoretical math you can just invent new imaginary numbers, make it so that *i*^2 = -1 and suddenly you have just revolutionized math. If this is useful, why can’t you break other rules and account for them with new imaginary symbols?
So let’s pretend that we call them made up numbers and use *m* to represent them. Why is *m*=1/0 impossible when something like *i*^2 = -1 is not?
In: 27
It’s been a while so this might not be the best way of going about it. But if we define m=1/0 then we are implicitly implying that m belongs to an algebraic structure in which multiplicative inverses are defined (otherwise / is meaningless), and that’s its inverse is therefore 0/1=0, and one in which the additive identity is defined (0). From that we end up with a contradiction. 1=(1/0)(0/1)=(1*0/0*1)=0/0=(0+0)/0=0/0+0/0=2.
Basically, i ends up useful because you can create an algebraically complete field (a field is basically everything that you can do with real numbers being defined, algebraically complete is why C is useful: every polynomial has an answer) with it as a member but m ends up useless because the moment you start trying to the binary operations to the set that contains you’ve used to define it then you start breaking things so you can’t use it for anything.
Dividing by 0 is perfectly possible in mathematics. The most common result is infinity although other results are possible. It is calculated by using limits.
Take this theoretical proof
Start
1) a = b
Multiply each side by a
2) a² = ab
Subtract b² from each side
3) a² – b² = ab – b²
Factor each side
4) (a + b)(a – b) = b(a – b)
Divide each side by (a – b)
5) (a + b) = b
Substitute b in for a (from what’s given at the start of the proof)
6) b + b = b
Condense
7) 2b = b
Divide by b
8) 2 = 1
So youll note something went wrong here for something to give us a blatantly incorrect statement. That thing is dividing by 0. On the step where you divide by (a – b), this equals to 0 and breaks everything
You could write a consistent set of rules where you’re allowed to “divide by zero,” whatever that means to you, and it’d still be math. I’ve taken a math class where we redefined addition and multiplication to mean something different and then investigated whether specific sets were vector spaces under those definitions. All math is made up by people.
It would then be up to you to persuade other people that your set of mathematical rules was interesting and worth studying. Many mathematicians have been persuaded that i^(2)=-1 makes for an interesting set of rules, both because it’s useful for electrical engineering and because it helps them achieve insights on problems they care about. None of the most popular systems of mathematical rules allows dividing by zero, but if you develop a good one and can argue for it persuasively, maybe you can change that.
Functionally, any number divided by zero = infinity.
You’ll notice that by dividing a number by anything less than one, you get a larger result, with the closer the denominator to zero, the larger the result.
So as you approach zero, the result of any number divided by that increases exponentially, to the point where it becomes essentially infinity. Which is as much a nonsense mathematical term as *i*, as it is a term that you can’t use in any equation in a similar way to zero without essentially making the whole equation meaningless.
A + 0 = A
A + (infinity) = infinity
A x 0 = 0
A x (infinity) = infinity
Yes, it can, and it’s essentially the idea of the so-called real projective line, which you can think of as basically all the real numbers plus one more thing which you call “1/0”.
I 2 am Spartacus. I can be whoever I want 2b.
But you can’t remove what isn’t there. 0/1 is taking nothing out of something. That’s just pulling air out of an empty hat.
But can you remove the rabbit out of your hat without vaporizing it? The 1 you want to take out of 0 can’t happen.
You can remove nothing from something, but can’t remove something from nothing.
In Calculus 1/0 is infinity. Its how we do the integral. Say ne need to travel from 0 to 1 in steps.
If we divide it into 10 step, you have 1/0.1 as the step size.
10000 steps, steps are 1/0.0001
1 million steps, steps are 1/0.000001
1 billion steps, steps are 1/0.000000001
What about an infinite number of steps? If we follow the same pattern, to get infinite step, you need 1/0.
Dividing by 0 can’t be done in pure mathematics period.
Also I believe you mean taking the square root of negative numbers .. you can square any negative number and it becomes positive because negative and negative multiplied make a positive by basic logic. The negation of a negation is a positive.
The imaginary numbers were created to handle these negative square roots which don’t have a real number solution. The square root of a negative number is as you know classed as imaginary. They’re not “real” numbers which are just numbers you can find along the traditional number line (integers, rational, and irrational numbers).
Simply as you know x^2 = -1 has the solution x = i. This new number system was invented to handle equations of this form which do in fact show up many times in engineering/physics problems and in other fields. Imaginary numbers were first invented though to handle solutions to certain polynomial equations as the real number system couldn’t account for the solutions to these equations. There simply wasn’t a set of numbers available to satisfy these equations. Imaginary numbers do allow us to solve equations in engineering, quantum mechanics, signals processing, etc that all have real world results/behavior.
Now with equations where we have division by 0 the story is a bit different. There doesn’t exist a number system for these cases because it makes no sense, in pure mathematics nor in the physical world. I’m pure mathematics 0 is null, nothing, the absence of value. If I say x/0 = y, then what is “y”? Well let’s try multiplying both sides by 0 to get x = 0*y and thus x must be 0. You see there is already a problem. The solution “y” can be any number. y is an infinite set actually that includes all numbers. All integers, rational, irrational and even imaginary numbers can satisfy this equation… if and only if x = 0. If x does not equal 0 then this equation is literally impossible and there are no solutions since 0 times any number must be 0. This equation makes no mathematical sense and no physical sense so there is no number system which can possibly satisfy this equation.
Let’s for the sake of argument say there was a number system that could satisfy this equation. We’ll denote them as “z” numbers and they’ll work similar to imaginary numbers. So z, 2z, 3z, etc. Such that z = 1/0. And then 2z = 2/0 and so on. So then from this numbering system obviously z + z = 2z. Does 1/0 + 1/0 = 2/0? Or more generally does az + bz have a unique solution of the form x/0 where a and b are some random factors of z?
So does az + bz = x/0?
-> (a+b)z = x/0
-> 0*(a+b)z = x
Again we run into the problem where a+b can be any value to make x = 0 or x cannot be non-zero to satisfy this equation. Thus there exists no number Mz (where M is any number non-zero number) where Mz = M/0. So the same logic makes this whole idea of a number system such as this fall apart.
For a physical example imagine you have 100 marbles and you want to evenly distribute them among zero people. How do you do that? If I have 2 people I give 50 each, if I have one person, I give 100, if I have 0 people, …. What about if I wanted to divide these 100 marbles amongst an unknown x amount of people in groups of 0 per person (100/x = 0 -> 100/0 = x)? Well if each person has 0 marbles, then I could have any number of people to satisfy this equation which there is no unique solution for.
Let me show you a cool algebra trick
Let’s start with:. a = b
Now let’s add a to both sides
2a = a + b
Now subtract 2b from both sides
2a – 2b = a – b
Factor the left side
2(a – b) = a – b
Now finally divide by (a-b)
2 = 1
Seems like a strange result! I was able to get this result because I was sneaky and I divided by 0 in the last step. If a = b, then (a-b) is 0. When you divide by 0, strange things happen. Once you prove 2 = 1, you can really just prove that anything = anything else. To avoid that, we say you can’t divide by 0.
But what’s stopping us from dividing by 0 and making up our own rules about how it works? Well nothing really. You can make up your own math if you’d like. But your biggest challenge for that made up math is going to be how you make sure that 2 and 1 are separate numbers. Cuz I say they’re equal if you can divide by 0.
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