Complex Numbers

Assume that $$R$$ represents a positive, real number.  In order to solve equations of the form $$!x^2+R=0$$ for the variable $$x$$, we would naturally write $$!x=\pm\sqrt{-R},$$ but how exactly do we compute the square root of a negative number?  As long as we use real numbers, there is no way to write the value of $$x$$ in this case.  In order to remedy this, mathematicians invented (or discovered) a type of number called “imaginary numbers.”  The idea is simple; we can write the square root as $$! \pm\sqrt{-R}=\pm\sqrt{-1}\sqrt{R}$$ and then define $$!i^2\equiv -1$$ where $$i$$ is the imaginary unit, analogous to the number 1 in the real number system. This allows us to write the solution to the equation above as $$!x=\pm i\sqrt R.$$ For example, solving $$!0=4+x^2$$ for $$x$$, we get $$!x=\pm 2i.$$ Another problem arises if we try to solve the quadratic $$!ax^2+bx+c=0$$ where $$a$$, $$b$$, and $$c$$ are arbitrary real numbers.  To solve for $$x$$, we use the quadratic formula, $$! \displaystyle x = \frac{-b\pm\sqrt{b^2 – 4ac}}{2a}.$$ If $$b^2 \geq 4ac$$, then $$x$$ is real, however, if  $$b^2\lt 4ac$$, then $$x$$ is a combination of real and imaginary numbers: $$! x = \alpha \pm i \beta$$ This combination, a real number plus an imaginary number, is called a complex number.  Complex numbers are normally written as $$!c=a+ ib.$$ Notice that, since $$i$$ is defined as $$!i^2\equiv -1\;\;\Rightarrow\;\;i=\pm \sqrt{-1},$$ complex numbers always come in pairs. These are known as conjugate pairs.  Suppose $$!c=a+ib.$$ Then the conjugate of $$c$$, which we denote as $$c^{*}$$, is $$!c^{*}=a-ib. $$ The quantity $$a$$ is called the real part of $$c$$ and the quantity $$b$$ is called the imaginary part of $$c$$.  These are denoted as $$!\displaystyle \Re\mbox{e}\left\{ c \right\} = a$$ and $$!\displaystyle \Im\mbox{m}\left\{ c \right\} = b$$ The real and imaginary parts of a complex number can be written in terms of the conjugate notation as $$!\displaystyle \Re\mbox{e}\left\{ c \right\} = \frac{c+c^{*}}{2}$$ and $$!\displaystyle \Im\mbox{m}\left\{ c \right\} = \frac{c-c^{*}}{2i}$$ The absolute value (or magnitude) of a complex number is given by $$!|c|=\sqrt{c c^{*}}=\sqrt{a^2+b^2}$$ This may remind you of the Pythagorean theorem, where $$|c|^2=a^2+b^2.$$  This is not a coincidence.  We’ll examine this more when we discuss Euler’s formula. Notice that $$c^2$$ and $$|c|^2$$ are not the same expression: $$!|c|^2=c\cdot c^{*} = a^2 + b^2$$ whereas $$!c^2=c\cdot c = \left( a+ib\right)\left( a+ib\right) = a^2+2iab-b^2$$ Until this point, we have limited the discussion to complex constants.  Variables and functions can be complex as well. The standard notation for a complex variable is $$z$$, $$!z=x+iy$$ and the standard notation for a complex function is $$!f(z)=u(x,y)+iv(x,y)$$ where $$x$$, $$y$$, $$u$$, and $$v$$ are real.

Next: Euler’s formula