Complex numbers: Construction of complex numbers and complex arithmetic
Imaginary numbers
We have seen before that adjunction of a square root to the set of rational numbers yields a new set of numbers where a certain quadratic equation has a solution. For example, the equation \(x^2=2\) has a solution in \(\mathbb{Q}(\sqrt{2})\).
In a similar way, we are going to expand the set of real numbers such that the equation \(x^2=-1\) has a solution. This new collection is called the set of complex numbers and is denoted by the calligraphic letter \(\mathbb{C}\).
Wish list of properties of complex numbers
- The real numbers must be part of \(\mathbb{C}\) (or at least one must be able to identify them with a subset of \(\mathbb{C}\)).
- You can add, subtract, multiply and divide numbers in \(\mathbb{C}\); the result of such an operation must be another element of \(\mathbb{C}\).
- The calculation rules of the real numbers should apply as much as possible.
- The equation \(x^2=-1\) must have a solution in \(\mathbb{C}\).
Thus, there is in the complex numbers \(\mathbb{C}\) a number for which \(x^2=-1\). This number we could call \(\sqrt{-1}\). But mathematicians use this symbol \(\mathrm{i}\), as given in the following definition:
Imaginary unit
The imaginary unit \(\mathrm{i}\) is a number with the property that \(\mathrm{i}^2 = -1\).
Why do we write actually prefer \(\mathrm{i}\) rather \(\sqrt{-1}\) ? One reason for this is that the important calculation rule for the roots of positive numbers \[\sqrt{x}\cdot\sqrt{y}=\sqrt{x\cdot y}\] does not apply to the roots of negative numbers. Suppose that this rule would apply and take \(x=-1\) and \(y=-1\) , we would get the following contradiction: \[ -1 = \mathrm{i} \cdot \mathrm{i} = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1 \] With the square root notation, the probability of this kind of error is greater than in a tightly controlled formula manipulation of the imaginary unit \(\mathrm{i}\).
Once we have stated that \(\mathrm{i}\) is a number with the property that \(\mathrm{i}^2 = -1\), then the number has \(-\mathrm{i}\) the same property.
We want to remark that engineers often use for the complex unit the symbol \(\mathrm{j}\), because the letter i in electrical theory has already been reserved as a symbol of current. In this course we stick to the usual mathematical notation.
The third wish on our wish list is that the calculation rules of the real numbers should apply as much as possible. We apply this to the following:
\(\sqrt{x}\cdot \sqrt{y} = \sqrt {x\cdot y }\) for positive real numbers \(x\) and \(y\).
We imagine that the above rule also works well in case \(x\) or \(y\) is a negative real number (not both negative!). We assume \(y=-1\) and that \(x\) is a positive number, then we get at the following definition:
The square root of a negative real number
For a positive real number \(x\) holds: \[\sqrt{-x}=\sqrt{x}\,\mathrm{i} \]
For a positive real number \(x\) holds: \[\sqrt{-x}=\sqrt{x}\,\mathrm{i}\]
Explanation Because we assume that \(\sqrt{x}\cdot \sqrt{y} = \sqrt {x\cdot y }\) holds in case \(x\) or \(y\) is positive, we find: \[\sqrt{-x}=\sqrt{x\cdot -1} = \sqrt{x}\cdot\sqrt{-1}=\sqrt{x}\,\mathrm{i} \]
Examples show how you can apply this theorem:
\sqrt{-3} &=\sqrt{3 \cdot -1} &\color{blue}{\text{product of real numbers}}\\
&=\sqrt{3} \cdot \sqrt{-1} &\color{blue}{3\text{ is positive}}\\
&=\sqrt{3} \,\mathrm{i} &\color{blue}{\sqrt{-1}=\mathrm{i}}
\end{aligned}\]
The construction of complex numbers goes by adjoining the imaginary unit \(\mathrm{i}\) to the real numbers. What does it mean that we adjoin \(\mathrm{i}\) to the real numbers? We certainly want to be able to multiply a real number with \(\mathrm{i}\) and this gives numbers of the form \(y\,\mathrm{i}\).
Imaginary number
A number of the form \(y\,\mathrm{i}\) with \(y\) a real number and \(\mathrm{i}^2=-1\) is called an imaginary number.
With the imaginary numbers we can already compute, such as exponentiation with positive integer exponents. We only need to handle the usual calculation rules and to apply the property \(\mathrm{i}^2=-1\) to simplify a result.
\mathrm{i}^{8} &= (\mathrm{i}^2)^{4} &\phantom{abcxyz}\color{blue}{\text{calculation rule for powers}}\\
&= (-1)^{4} &\phantom{abcxyz}\color{blue}{\mathrm{i}^2=-1} \\
&=1 & \phantom{abcxyz}\color{blue}{\text{standard representation}}
\end{aligned}\]
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