Complex numbers: Construction of complex numbers and complex arithmetic
Complex numbers: addition and multiplication
We also want to be able to add complex numbers, in particular a real number, and an imaginary number, and we get in this way numbers of the form \(x+y\,\mathrm{i}\) with \(x,y\in\mathbb{R}\). The wish to be able to consider every real number as well as a complex number is thus satisfied: we agree that we let real numbers be elements of \(\mathbb{C}\) by representing them as \(x+0\,\mathrm{i}\) (with \(x\in\mathbb{R}\)), and we will just write \(x\) instead \(x+0\,\mathrm{i}\) .
The imaginary numbers \(y\,\mathrm{i}\) are written in short format instead of \(0+y\,\mathrm{i}\). Other simplified notations are \(x+\mathrm{i}\) and \(x-\mathrm{i}\) for \(x+1\,\mathrm{i}\) and \(x-1\,\mathrm{i}\), respectively. Instead of \(x+y\,\mathrm{i}\) we also write \(x+\mathrm{i}\,y\) when confusion is possible; For example, think of \(\mathrm{i}\sqrt{2}\) rather \(\sqrt{2}\mathrm{i}\).
Definitions The standard form of a complex number is \(x+y\,\mathrm{i}\) with \(x,y\in\mathbb{R}\).
\(x\) is called the real part of the complex number and \(y\) is called the imaginary part (note: \(y\) and not \(y\,\mathrm{i}\)).
Special notations are introduced herefor: \[\mathrm{Re}(x+y\,\mathrm{i})=x\qquad \mathrm{Im}(x+y\,\mathrm{i})=y\] In books you may also find: \[\Re(x+y\,\mathrm{i})=x\qquad \Im(x+y\,\mathrm{i})=y\]
Examples of complex numbers \[3-2\,\mathrm{i}\qquad \frac{1}{2}+\mathrm{i}\sqrt{3}\qquad \pi+\mathrm{i}\] The real and imaginary part of \(3-2\,\mathrm{i}\) are \(3\) and \(-2\), respectively. We then write: \[\mathrm{Re}(3-2\,\mathrm{i})=3\qquad \mathrm{Im}(3-2\,\mathrm{i})=-2\] Note that the imaginary part of \(3-2\,\mathrm{i}\) is not defined as \(-2\,\mathrm{i}\). The imaginary part of a complex number is thus not an imaginary number.
\(\phantom{x}\)
A welcome surprise is that this invented structure suffices, and that there are no more numbers needed to be able to calculate properly. Because the calculation rules of the real numbers should apply as much as possible, in the end only the following definitions of addition and multiplication are possible:
- Addition is done componentwise, where we consider \(x\) and \(y\) as components of the number \(x+y\,\mathrm{i}\): \[(x_1+y_1\,\mathrm{i})+(x_2+y_2\,\mathrm{i})= (x_1+x_2) + (y_1+y_2)\,\mathrm{i}\]
- To multiply, we must 'simply' expand brackets and use \(\mathrm{i}^2=-1\): \[\begin{aligned}(x_1+y_1\,\mathrm{i}) (x_2+y_2\,\mathrm{i}) &= (x_1 x_2) + (x_1 y_2+y_1\cdot x_2)\,\mathrm{i}+ y_1 y_2\,\mathrm{i}^2\\ &= (x_1 x_2- y_1 y_2) + (x_1 y_2+y_1 x_2)\,\mathrm{i}\end{aligned}\]
Look at examples of addition and multiplication until you understand calculating with complex numbers and understand that the usual calculation rules for exansion of brackets should be used in conjunction with the calculation rule \(\mathrm{i}^2=-1\).
(-3+4\,\mathrm{i})+(3+3\,\mathrm{i})&= (-3+3) + (4+3)\,\mathrm{i} \\ &\phantom{abcxyz}\color{blue}{\text{real and imaginary parts added as components}} \\
&=7\,\mathrm{i}
\\ &\phantom{abcxyz}\color{blue}{\text{simplification to standard form}}
\end{aligned}\]
The calculation rules for addition and multiplication of complex numbers can also be summarized as follows:
Rules for calculating real and imaginary parts Each pair of complex numbers #z#, #w# satisfies the following equations:
- #\mathrm{Re}(z+w)=\mathrm{Re}(z)+\mathrm{Re}(w)#
- #\mathrm{Re}(z\cdot w)=\mathrm{Re}(z)\cdot \mathrm{Re}(w)-\mathrm{Im}(z)\cdot\mathrm{Im}(w)#
- #\mathrm{Im}(z+w)=\mathrm{Im}(z)+\mathrm{Im}(w)#
- #\mathrm{Im}(z\cdot w)=\mathrm{Re}(z)\cdot \mathrm{Im}(w)+\mathrm{Im}(z)\cdot\mathrm{Re}(w)#
- If #z# is real, then applies #\mathrm{Re}(z\cdot w)=z\cdot \mathrm{Re}(w)#
Mathcentre videos
Adding and Substracting (8:17)
Multiplying (8:17)
