Equation of the Day #1: Complex Numbers

Most of us are used to the real numbers. Real numbers consist of the whole numbers (0, 1, 2, 3, 4, \ldots) , the negative numbers (-1, -2, -3, \ldots), the rational numbers (1/2, 2/3, 3/4, 44/7, \ldots) , and the irrational numbers (numbers that cannot be represented by fractions of integers, such as the golden ratio, \sqrt{2} , or \tau ). All of these can be written in decimal format, even though they may have infinite decimal places. But, when we use this number system, there are some numbers we can’t write. For instance, what is the square root of -1 ? In math class, you may have been told that you can’t take the square root of a negative number. That’s only half true, as you can’t take the square root of a negative number and write it as a real number. This is because the square root is not part of the set of real numbers.

This is where the complex numbers come in. Suppose I define a new number, let’s call it i , where

i^2 = -1.

We’ve now “invented” a value for the square root of -1 . Now, what are its properties? If I take i^3 , I get -i , since i^3 = i\cdot i^2 . If I take i^4 , then I get i^2\cdot i^2 = +1 . If I multiply this by i again, I get i . So the powers of i are cyclic through i, -1, -i, and 1 .

This is interesting, but what is the magnitude of i , i.e. how far is i from zero? Well, the way we take the absolute value in the real number system is by squaring the number and taking the positive square root. This won’t work for i , though, because we just get back i . Let’s redefine the absolute value by taking what’s called the complex conjugate of i and multiplying the two together, then taking the positive square root. The complex conjugate of i is obtained by negating the imaginary part of i . Since i is purely imaginary (there are no real numbers that make up i ), the complex conjugate is -i . Multiply them together, and you get that -i\cdot i = -1\cdot i^2 = 1 , and the positive square root of 1 is simply 1 . Therefore, the number i has a magnitude of 1 . It is for this reason that i is known as the imaginary unit!

Now that we have defined this new unit, i , we can now create a new set of numbers called the complex numbers, which take the form z = a + bi , where a and b are real numbers. We can now take the square root of any real number, e.g. the square root of -4 can be written as \pm 2i , and we can make complex numbers with real and imaginary parts, like 3 + 4i .

How do we plot complex numbers? Well, complex numbers have a real part and an imaginary part, so the best way to do this is to create a graph where the abscissa (x-value) is the real part of the number and the ordinate (y-axis) is the imaginary part. This is known as the complex plane. For instance, 3 + 4i would have its coordinate be (3,4) in this coordinate system.

What is the magnitude of this complex number? Well, it would be the square root of itself multiplied by its complex conjugate, or the square root of (3 + 4i)(3 - 4i) = 9 + 12i - 12i +16 = 25 . The positive square root of 25 is 5 , so the magnitude of 3 + 4i is 5 .

We can think of points on the complex plane being represented by a vector which points from the origin to the point in question. The magnitude of this vector is given by the absolute value of the point, which we can denote as r . The x-value of this vector is given by the magnitude multiplied by the cosine of the angle made by the vector with the positive part of the real axis. This angle we can denote as \phi . The y-value of the vector is going to be the imaginary unit, i , multiplied by the magnitude of the vector times the sine of the angle \phi . So, we get that our complex number, z , can be written as z = r(\cos\phi + i\sin\phi) . The Swiss mathematician Leonhard Euler discovered a special identity relating to this equation, known now as Euler’s Formula, that reads as follows:

e^{i\phi} = \cos \phi + i\sin\phi

Where e is the base of the natural logarithm. So, we can then write our complex number as z = re^{i\phi} . What is the significance of this? Well, for one, you can derive one of the most beautiful equations in mathematics, known as Euler’s Identity:

e^{i\tau} = 1 + 0

This equation contains the most important constants in mathematics: e , Euler’s number, the base of the natural logarithm; i , the imaginary unit which I’ve spent this whole time blabbing about; \tau , the irrational ratio of a circle’s circumference to its radius, which appears all over the place in trigonometry; 1 , the real unit and multiplicative identity; and 0 , the additive identity.

So, what bearing does this have in real life? A lot. Imaginary and complex numbers are used in solving many differential equations that model real physical situations, such as waves propagating through a medium, wave functions in quantum mechanics, electromagnetic phenomena, and fractals, which in and of themselves have a wide range of real life application.


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