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
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.
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