Complex numbers: Introduction to Complex numbers
Imaginary numbers
We know that #-1# is negative and that the square of a real number is non-negative. This implies that the equation \[x^2=-1\] with an unknown #x# has no real solution. We are going to expand the set of real numbers in such a way that this equation \(x^2=-1\) does have a solution. This new set is called the set of complex numbers and is denoted by the letter \(\mathbb{C}\)
Characteristics of complex numbers
Characteristic properties of \(\mathbb{C}\) are:
- The real numbers are part of it.
- The numbers in \(\mathbb{C}\) can be added, subtracted, multiplied and divided; The result of such an operation is another element of \(\mathbb{C}\)
- Many of the rules of calculation of the real numbers also apply to complex numbers.
- The equation \(x^2=-1\) has a solution in \(\mathbb{C}\)
- If we say that the real numbers are a sub-set of it, we mean that #\mathbb{R}# with a subset of #\mathbb{C}# can be identified and adding, subtraction, multiplication and division with elements of that subset correspond to the same operation for the real numbers.
- If we say that the rules of calculation of the real numbers apply as much as possible, we mean the associativity, commutativity, distributivity, the rules #0# and #1# etc.
- If we say that the equation \(x^2=-1\) has a solution in \(\mathbb{C}\), we do not mean that this solution is unique.
Hence, there is a complex number #x# for which \(x^2=-1\) holds. This number can be called \(\sqrt{-1}\). But there are good reasons to introduce a new symbol for this.
Imaginary unit
The imaginary unit \({\ii}\) is a number with the property \({\ii}^2 = -1 \)
Why do we actually prefer to write \({\ii}\) for the imaginary unit instead of \(\sqrt{-1}\) We provide two reasons:
- One reason is that the main rule of calculation \[\sqrt{x}\cdot\sqrt{y}=\sqrt{x\cdot y}\] for the roots of positive numbers does not apply to the roots of negative numbers. Suppose that the rules would apply. If we than take \(x=-1\) and \(y=-1\) we would get the following contradiction: \[ -1 = {\ii}^2=\ii \cdot \ii = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1 \] With the root notation the likelihood of these types of errors is bigger than with a tightly regulated formula manipulation with the imaginary unit \({\ii}\)
- A second reason is that with positive real numbers #\sqrt{y}# indicates the unique positive real number #x# by #x^2=y#, while we can not say what we actually mean with positive when it comes to complex numbers. If we have established that #x=\ii# is a solution to the equation #x^2=-1#, then #x=-\ii# is a solution as well. That is why we prefer to indicate the number #\ii# with the property that #\ii^2=-1#, and next define #\sqrt{-1}# in terms of #\ii#: see below.
Engineers often use the symbol \(\mathrm{j}\) for the imaginary unit because the letter #\ii# is already used as a symbol of current in electricity theory. In this course, we stick with the usual mathematical notation.
In the context of the third feature of the complex numbers we consider the following well-known law.
\[\sqrt{x}\cdot \sqrt{y} = \sqrt {x\cdot y }\text{ for positive real numbers }x\text{ and }y\tiny.\]
We have already seen that the law does not work when #x# and #y# are both negative. We will now assume that the rule above does work if \(x\) or \(y\) are a negative real number, but not when both are negative. If we take \(y=-1\) and \(x\) being a positive real number, then we arrive at the following definition:
The square root of a negative real number
For a positive real number \(x\) we write: \[\sqrt{-x}=\sqrt{x}\cdot \ii=\sqrt{x}\,\ii \tiny.\] Particularly, with this we define #\sqrt{-1} = \ii#
The right-hand side, #\sqrt{x}\cdot\ii# is a special number of the form #a\cdot\ii# It must be a number in #\mathbb{C}# because we want to allow products of complex numbers, and because the real number #a#, or #\sqrt{x}# belongs to #\mathbb{C}#.
Because we assume that \(\sqrt{x}\cdot \sqrt{y} = \sqrt {x\cdot y }\) holds if no more than one of \(x\), \(y\) is negative, we find: \[\sqrt{-x}=\sqrt{x\cdot -1} = \sqrt{x}\cdot\sqrt{-1}=\sqrt{x}\,{\ii}\tiny. \]
The natural intepretation of #\sqrt{-x}#, with #x\gt0#, is that it is a complex number that satisfies #\sqrt{-x}^2=-x#. This is indeed the case: \[\sqrt{-x}^2=\left(\sqrt{x}\,\ii\right)^2=\sqrt{x}^2\cdot {\ii}^2 = x\cdot (-1)=-x\tiny.\]
The construction of complex numbers is based on adding the imaginary unit \({\ii}\) to the real numbers. What does it mean that we add \({\ii}\) to the real numbers? We definitely want to be able to multiply \({\ii}\) by a real number #y#; this gives numbers of the form \(y\,{\ii}\).
Imaginary number
A number of the form \(y\cdot \ii\) with a real number \(y\) is called an imaginary number.
The number #-1\cdot\ii# is often abbreviated as #-\ii#.
The square of the imaginary number \(y\cdot \ii\) is \(\left(y\cdot \ii\right)^2=-y^2\), a non-positive real number.
Hence, for each negative real number #a# there are two imaginary numbers, that when squared yield #a#: #\sqrt{-a}\cdot\ii# and #-\sqrt{-a}\cdot\ii#. Later we will see that these are the only two complex solutions of the equation #z^2=a#.
Arithmetic is already possible, for example, exponentiation with integer exponents. We only need to apply the usual rules of calculation rules and the property \({\ii}^2=-1\) to simplify the result. The result will always be a real or an imaginary number.
After all,
\[ \begin{array}{rcll} \sqrt{-13} &=&\sqrt{13 \cdot (-1)} &\phantom{uvwxyz}\color{blue}{\text{product of real numbers}}\\
&=&\sqrt{13} \cdot \sqrt{-1} &\phantom{uvwxyz}\color{blue}{13 \text{ is positive}}\\
&=&\sqrt{13} \cdot\ii &\phantom{uvwxyz}\color{blue}{\sqrt{-1} = \ii} \end {array}\]
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