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I have a 2 part question:

1) If they are imaginary, one may get asked, "Well, if imaginary, they don't exist! Why do ANYTHING with them??" The only thing I can think of is that while both $\sqrt{-9}$ and $\sqrt{-16}$ both don't exist, they don't exist in slightly different ways. Obviously a 3 is more relevant to one of those, and a 4 is more relevant to the other. So, they don't entire not exist, right?

2) When they are taught, you're just taught HOW ($\sqrt{-16}=4i)$ with no real explanation of WHY you'd ever bother with introducing this placeholder in the first place. How does one actually justify imaginary numbers? Can someone outline some sort of actual application of math (Physics? Electricity?) where you're trying to figure something out, and you run into a $\sqrt{-16}$ and is makes sense to turn this into $4i$.

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Yeah, you have your pretty well-defined and understandable natural numbers. And you can solve equations like $x+2=8$.

But after that you realize that the equation $x+3=1$ is unsolvable. OK, let's do some extra work and invent new numbers and new symbol "$-$". We have integers now and it is good.

There is, btw, another example of bad behaviour: $2x=1$. OK... Now we take this long baton $-$ and create a new type of numbers. OK, $\dfrac12$, whatever is it. This "fraction" is just a name for some thing which is equal to $1$ when it was taken twice.

You can guess now. $x^2=2$ — and we don't have any numbers for the possible solution. Let us create another name: $\sqrt2$. By doing some math, we understand that real numbers comprise a field, etc.

But still we have a problem. With $x^2+1=0$. There is no real numbers which may be a solution. We may think that $\sqrt{-1}$ will be a solution, but it isn't a real number. So, we did it many times — add some new number. Let it be $i$. And $i$ is the root of the equation $x^2+1=0$. OK, now we do some math, making a closure of field, doesn't really matter.

And that is the final step. Now all possible polynomial equations have their roots. Congrats.

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First of all, the word "imaginary" is used as a name for these numbers. Don't try to give this word the same meaning as you would in other contexts, such as your "imaginary" friend; the meaning doesn't cross over.

Second, one well known application of imaginary numbers is in Electrical Engineering. I'm sure there are others; this is the one I am most familiar with.

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An example is that imaginary numbers occur sometimes in differential equations solving. Differential equations appear in many fields of Physics and Mechanics, which are fields that have to do with real things.

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