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Can someone help me by explaining what imaginary numbers actually are, please?

I know that it is defined as $i = \sqrt{-1}$ and I can find umpteen sources telling me how to calculate with them but I cannot get my head around what they actually represent. I've seen a number of argand graphs but I just cannot relate what the lines are describing.

Can anyone help?

Thanks

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marked as duplicate by Rahul, RghtHndSd, user147263, Michael Albanese, Jonas Meyer Sep 18 '14 at 1:54

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  • $\begingroup$ Reading the history of mathematics as to why they needed $\mathrm{i}^2:=-1$ will make it a lot more clear. $\endgroup$ – UserX Sep 17 '14 at 19:27
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    $\begingroup$ In the context of the cartesian plane, complex numbers tell you how far away you are from the x-axis. ie, the number $2+5i$ is the same as the $(x,y)$ coordinate $(2,5)$. $\endgroup$ – graydad Sep 17 '14 at 19:28
  • $\begingroup$ How is the canonical representation of imaginary numbers as pairs of real numbers with the suitable operations, not sufficient to answer your question? $\endgroup$ – Did Sep 17 '14 at 19:36
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    $\begingroup$ What about negative numbers ? How do you understand that you have $-5$ apples ? Personally, for the time being, I would suggest you to accept the concept (well illustrated by graydad). If you don't, let us continue speaking about this ! By the way, welcome to Math SE ! Cheers :-) $\endgroup$ – Claude Leibovici Sep 17 '14 at 19:41
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    $\begingroup$ I dont think that they "represent " anything. They result from extending some operations (square roots) to a larger domain (negative numbers). In a real sense they are abstractions. $\endgroup$ – Rene Schipperus Sep 17 '14 at 20:06
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I think you're going in the wrong direction. In mathematics we don't understand things by what they are, but by what they do. Philosophers can argue for days about what is the number 1, but ultimately what gives the number 1 its mathematical identity is what it does and how it relates to the other numbers: It has the property that $1\cdot x = x$ for any other number $x$, for example, and it is the only number with this property. You would be amazed at the variety of bizarre objects that mathematicians identify as being the number 1. Most of them look nothing like the 1 you know and love. But they all have the required properties: each one belongs to some system of numbers in which $1\cdot x = x$ holds, among other things. As people sometimes say, if it walks like a duck and quacks like a duck, it must be a duck.

What do imaginary numbers do? They were first introduced to solve certain kinds of equations. Italian mathematicians of the 16th century had a method for solving certain equations of the form $$x^3 = ax + b$$ but they found that under certain circumstances, these methods produced seemingly nonsensical results involving the square root of $-1$; seemingly nonsensical because $-1$, being negative, was thought to have no square root.

For example, the same method that correctly solved many equations, when used to solve $x^3 = 15x + 4$, would produce the peculiar result $$\sqrt[3]{2+\sqrt{-121}}+ \sqrt[3]{2-\sqrt{-121}}.$$

This is a problem, because the equation $x^3=15x+4$ has an obvious solution, namely $x=4$. Why doesn't the method work? Why didn't it emit the number 4?

A mathematician named Bombelli observed that if you pretend that the solution makes sense, then it does work. Because if you calculate $(2+\sqrt{-1})^3,$ following all the usual rules, you find $$(2+\sqrt{-1})^3 = 2+\sqrt{-121},$$ and similarly $$(2-\sqrt{-1})^3 = 2-\sqrt{-121}$$ so the nonsense result above simplifies to $$(2+\sqrt{-1}) + (2 - \sqrt{-1})$$ and the nonsensical $\sqrt{-1}$ parts cancel out, leaving you with the correct answer, $4$.

It seemed like nonsense, but it solved the problem and got the right answer, so there must be something to it. Eventually people got over their prejudice against the seemingly nonsensical numbers (the name “imaginary” still speaks to that ancient prejudice) and we now accept that these numbers work just as well as real numbers.

We have mathematical models of imaginary numbers; these are mathematical objects that, when operated on in certain ways, behave the way we want the imaginary numbers to behave. For example we can model a complex number $a+bi$ as a pair $\langle a,b\rangle$, as a certain matrix, or as a certain family of polynomials. But the reason we accept these as models of imaginary numbers is because they do the things we want imaginary numbers to do.

What do we want imaginary numbers to do?

  1. We want them to contain a subset that behaves just like the real numbers.
  2. We want to be able to add and multiply them, and the addition and multiplication, when restricted to the subset that behaves like the real numbers, should give the answers we expect for real numbers. Whatever we identify as $2$ and $4$ in our system, we should still have $2+2=4$.
  3. Most of the usual rules of addition and multiplication should still apply. For example, $1\cdot x = x$ should hold when $x$ is imaginary just as it does when $x$ is real.
  4. There should be some number, which we call $i$, which should have the property that $i^2 = -1$. (Actually one can show that there are always two such numbers; you can't have one without the other.)

That's what they do; if you can make them do what you want, then it shouldn't matter what they are. This is why the umpteen sources tell you how to calculate with them but not what they are: once you know how to calculate with them, that is the whole story. There is nothing more!

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  • $\begingroup$ Thanks very much for your explanation. How did you get the result $$\sqrt[3]{2+\sqrt{-121}}+ \sqrt[3]{2-\sqrt{-121}}.$$ I think I am with you until there :) $\endgroup$ – Thomas Orange Sep 20 '14 at 20:29
  • $\begingroup$ @ThomasOrange I didn't explain it; I only said that the italians had a method that got that result, and I left out the details because they weren't important for the question you asked. The details are in the Wikipedia article on cubic functions if you want to see them. $\endgroup$ – MJD Sep 20 '14 at 21:55

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