Complex Numbers

Like many stories, the story of complex numbers begins with a lie. A lie which you were told many years ago, by your math teachers, no less. (On behalf of all math teachers everywhere, I sincerely apologize.) And the lie was this: you can’t find the square root of a negative number.

Well, OK, it’s not exactly a lie. It just depends on your point of view.

The problem, of course, is that any positive real number multiplied by itself yields a positive real number — but any negative real number multiplied by itself yields a positive real number, too. So there are no real numbers which can be the square root of any negative real numbers.

But here is where, as mathematicians, we decide that we are undaunted by this fact, and that we will invent a new sort of number so that negative real numbers can have square roots after all. (The strange thing is, usually when we “invent” something in this way, it ends up corresponding so nicely with so many other things that we begin to wonder if we really invented it after all…)

So, we define the imaginary number i to be the square root of negative one:

i = \sqrt{-1}

Now we can express the square root of any negative number in terms of i, since

\sqrt{-n} = \sqrt{-1 \cdot n} = \sqrt{-1} \cdot \sqrt{n} = i \sqrt{n}.

So, for example, \sqrt{-4} = i \sqrt{4} = 2i.

In general, we call any such multiple of i (like 2i, -5i, \pi i) an imaginary number.

Now then, a complex number is simply a number of the form a + bi, that is, a real number plus an imaginary number. For example, 2 + 3i or 9/2 - i \sqrt{3}. One neat thing about complex numbers is that they are closed under the four normal arithmetic operations. In other words, if you take any two complex numbers and add, subtract, multiply, or divide them, you get another complex number. Here’s how:

  1. To add or subtract two complex numbers, just collect like terms; that is, add or subtract the real parts separately and add or subtract the imaginary parts separately. Like this:

    (2 + 3i) + (4 - 6i) = 2 + 4 + 3i - 6i = 6 - 3i


    \begin{array}{rcl}(3 - i) - (4 - 2i) & = & 3 - i - 4 + 2i \\ & = & 3 - 4 - i + 2i = -1 + i \end{array}

  2. To multiply two complex numbers, use FOIL to multiply them like any two binomials, keeping in mind the fact that since i = \sqrt{-1}, it is the case that i^2 = -1. Like this:

    \begin{array}{rcl}(2 + 3i)(4 - 5i) & = & 8 - 10i + 12i - 15i^2 \\ & = & 8 + 2i + 15 = 23 + 2i. \end{array}

14 Responses to Complex Numbers

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  4. MathBeginer says:

    My doubt in complex number is: any number can be quantified i.e. we can imagine how big or small it is. So is it possible to imagine or quantify the value of i?

    • Marcus Neal says:

      While the complex numbers can be given a partial order with respect to normal arithmetical operations, they cannot be totally ordered. This means that they do lose a sense of “size” in a way. But what you gain by going from real to complex numbers greatly outweighs what you lose! You are able to get a much better view of the behavior and characteristics of polynomials and many other algebraic operational using complex numbers. Viewing the world with complex numbers allow you to see more cleanly, the structure and design of many mathematical structures – from number theory, to integration, to geometry, to trigonometry, and on and on. The fact that real number represent visualizable quantities is a wonderful thing, but there is more to mathematics than its study of quantities.

  5. Brent says:

    Hi MathBeginner, it depends what you mean! It is not necessarily true that it is possible to imagine “how big or small” every number is. For example, how big is -6? Maybe you think you have a good idea of how big or small -6 is, but for a long time mathematicians thought it did not make any sense. At any rate, one thing we could sensibly mean by “how big or small” a number is, is “how far away from zero is it”? The distance of a number from zero is its “magnitude” or “absolute value”. So for example 6 is six units away from zero, so we write |6| = 6. -6 is also six units away from zero, so |-6| = 6 as well. This also applies to complex numbers if we think of them as being in a 2D plane, so |i| = 1 since i is one unit away from zero (the origin).

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  8. Tim says:

    In my own studies on complex numbers, I was never satisfied until I came across an explanation by the University of Toronto and an undergraduate Complex Analysis book which defined complex numbers as ordered pair of real numbers. To define i by an operation which produces it but never telling you what number you perform the operation with seems extremely unhelpful. The ordered pair (0, 1) is the answer to: which number squared gives us -1.

    • Brent says:

      I’m glad you found the definition with a pair of real numbers helpful — it is often the case that seeing multiple viewpoints on the same concept can really help solidify it. However, I don’t think the ordered pair definition is really any better an “answer” than defining i, because a pair of numbers by itself doesn’t really mean anything in this context until you define how to do addition and multiplication of such pairs. It might seem more familiar since it is made up of familiar parts, but it is really no different than the situation with i.

      • Tim says:

        Granted, you have to define operations on those ordered pairs. What makes it powerful for me is that there is no answer in the other number systems; the number system must be extended. Just as there is no answer to the question “what number squared is 3?” within the natural numbers until you extend it to provide fractions, there is no i until you extend further to create complex numbers. And once you do that, and define valid operations and so forth, it all comes together. Telling someone “let’s just pretend a number exists which provides this answer” and calling it “imaginary” (I think it was Euler or Gauss that wanted to call i “lateral” instead?) isn’t effective and is demonstrably false when we know that there certainly is an object which does that. But, I suppose they don’t want to do that in a Precalc text.

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  10. Pablo says:

    by chance I found your website and it seemed incredible to me, I have gone from topic to topic. Congratulations for what you’ve done in so long

  11. Jamal uddin says:

    How can you define something without describing its meaning? You have defined $i$ to be the square root of $-1$. But that is meaningless. Square roots were defined only for positive numbers and by convention we consider only the positive result. For example, $\sqrt{4}$ is either $2$ or $-2$ but by convention we went with $2$. So my question is, which square root, if that even makes sense, would you take for the value of $i$?

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