We’ll start with the definition of a complex number and its addition and multiplication rules. You may find the multiplication rule quite mysterious. Don’t worry. We’ll soon gets lots of practice using it and seeing how useful it is.
DefinitionB.1.1.
The complex plane is simply the \(xy\)-plane equipped with an addition operation and a multiplication operation. A complex number is nothing more than a point in that \(xy\)-plane. It is conventional to use the notation \(x+iy\) 1
In electrical engineering it is coonventional to use \(x+jy\) instead of \(x+iy\text{.}\)
to stand for the complex number \((x,y)\text{.}\) In other words, it is conventional to write \(x\) in place of \((x,0)\) and \(i\) in place of \((0,1)\text{.}\)
The first component, \(x\text{,}\) of the complex number \(x+iy\) is called its real part and the second component, \(y\text{,}\) is called its imaginary part, even though there is nothing imaginary 2
Don’t attempt to attribute any special significance to the word “complex” in “complex number”, or to the word “real” in “real number” and “real part”, or to the word “imaginary” in “imaginary part”. All are just names. The name “imaginary” was introduced by René Descartes in 1637. René Descartes (1596–1650) was a French scientist and philosopher, who lived in the Dutch Republic for roughly twenty years after serving in the (mercenary) Dutch States Army. Originally, “imaginary” was a derogatory term and imaginary numbers were thought to be useless. But they turned out to be incredibly useful!
about it.
The sum of the complex numbers \(\, z_1=x_1+i y_1 \) and \(\, z_2=x_2 +iy_2 \, \) is defined by
Do not memorise this multiplication rule. We’ll see a simple, effective memory aid for it shortly. The heart of that memory aid is the observation that the complex number \(i\) has the special property that
To remember how to multiply complex numbers, you just have to supplement the familiar rules of the real number system with \(i^2=-1\text{.}\) The previous sentence is the memory aid referred to in Definition B.1.1(d).
and solving these equations for \(a\) and \(b\text{.}\) We will see a much shorter derivation in Remark B.1.6 below. For now, we’ll content ourselves with just verifying that \(\frac{x}{x^2+y^2}+\frac{-y}{x^2+y^2}i\) is the inverse of \(x+iy\) by multiplying out
The complex conjugate of \(\, z=x+iy\, \) is denoted \(\bar z\) and is defined to be
\begin{equation*}
\bar z=\overline{x+iy}=x-i y
\end{equation*}
That is, to take the complex conjugate, one replaces every \(i\) by \(-i\) and vice versa.
The distance from \(z=x+iy\) (recall that this is the point \((x,y)\) in the \(xy\)-plane) to \(0\) is denoted \(\, |z|\, \) and is called the absolute value, or modulus, of \(\, z\, \text{.}\) It is given by
In general, the complex conjugate provides us with a trick for rewriting any ratio of complex numbers as a ratio with a real denominator. For example, suppose that we want to find \(\frac{1+2i}{3+4i}\text{.}\) The trick is to multiply by \(1=\frac{3-4i}{3-4i}\text{.}\) The number \(3-4i\) is the complex conjugate of the denominator \(3+4i\text{.}\) Since \((3+4i)(3-4i)=9-12i+12i-16i^2=9+16=25\)
The symbols \(\Re\) and \(\Im\) are the letters \(R\) and \(I\) in the Fraktur font, which was created in the early 1500’s and became common in the German-speaking world. A standard alternative notation is \(\text{Re}(z)\) and \(\text{Im}(z)\text{.}\)
\(\Re z\) and \(\Im z\) stand for the real and imaginary parts of the complex number \(z\text{,}\) respectively. If \(z=x+ iy\) (with \(x\) and \(y\) real) they are defined by
