Introduction to the basics of complex numbers
Why complex numbers?
In the article natural up to real numbers, the extension from natural to integer numbers is described, which has been extended to rational and then to real numbers. Many tasks could only be solved with real numbers.
But even with the real numbers, not all tasks can be solved. For example, the equation
\(x^2+1=0\) oder \(x^2 = -1\)
The equation is not solvable with real numbers because the square of a real number not equal to zero is always positive.
Such equations can still be calculated. For this purpose, the range of real numbers must be extended so that the equation is solvable. To solve such equations but also other mathematical problems complex numbers were introduced.
A complex number \(z\) consists of a real part \(a\) and an imaginary part \(b\). The imaginary part is marked with the letter \(i\).
The real part and the imaginary part are real numbers.
In the introduction it has already been stated that the equation \(x^2 = -1\) with the set of real numbers is not solvable because the square of a real number not equal to zero is always positive. So, we need to expand the range of real numbers so that the equation is solvable.
For this we need a new number, which makes the equation solvable. This new number is called imaginary part and is denoted by the symbol i. It has the property that if it multiplies by itself the result \(-1\).
\(i·i = -1\)
What can one imagine under \(i\)? Certainly \(i\) is not a real number, because the square of a real number is never negative. But \(i^2 = -1\), because the requirement was a solution of the equation \(x^2 = -1\). Although it is not yet clear what \(i\) looks like, we can already calculate with \(i\).
The calculation rules of real numbers should continue to be valid even for complex numbers. This is called principle of permanence. So we can also calculate
\(i^3 = i^2 · i = (-1) · i = -i\)
What is a complex number? The word complex is derived from the Latin word complexus = intertwined. A complex number is the connection of the imaginary part \(i)\ with a real number.
Examples of complex numbers
\(1 + 3i\), \(-1 + 3i\), \(2-3i\), \(2^2 – 5^2 i\)
The numbers are all composed of a real part and an imaginary part. For example, \(1 + 3i\), this is the real number \(1\) and the imaginary part \(3i\).
The \(+\) sign in \(1 + 3i\) is part of the complex number. It will later be considered as a addition. The part \(3i\) is understood as \(3 · i\), finally the calculation rules of the real numbers for the numbers \(3\) and \(i\) continue to be valid
A complex number is defined as
\(z = a + bi\)
This term is a complex number. \(a\) and \(b\) are real numbers and \(i\) stands for the imaginary part.
Another notation for complex numbers is the pair spelling: The real and imaginary part is written as a pair of numbers
\(z = (Re, Im)\)
For \(z = 2 + 3i\) this would be \(z = (2, 3)\). For the real number \(5\) the pair notation \(z = (5, 0)\) and for \(2i\) we write \(z = (0, 2)\).
As an example we take the complex number \(2 - 5i\). The complex number \(2 - 5i\) is uniquely determined by the real numbers \(2\) and \(-5\). Generally, any complex number \(a + bi\) is uniquely defined by the real numbers \(a\) and \(b\).
To describe complex numbers in this way, we introduced two new terms
The real part of a complex number is the purely real part of the number. The real part of the complex number \(2 - 5i\) is therefore \(2\). We also write \(Re (2 - 5i) = 2\).
The imaginary part of a complex number is the part of the number that precedes the imaginary part \(i\). The imaginary part of the complex number \(2 - 5i\) is thus \(-5\). We also writes Im \(Im (2 - 5i = -5\).
Note that the imaginary value of \(2 - 5i\) is the real number \(- 5\). It is not equal to the imaginary part of the number \(2 - 5i\). The imaginary part is namely \(- 5i \).
A complex number is defined as \(z=a+bi\), where \(a\) and \(b\) are real numbers
The real part of \(z\) is called \(a\); We write \(a = Re (z)\)
The imaginary part of \(z\) is called \(b\) We write \(b = Im (z)\)
Like the expansion from natural to integer numbers, which have been extended to rational and then to real numbers, complex numbers are an extension of real numbers. Real numbers are thus a subset of the complex numbers. A real number is identical to a complex number with the imaginary part \(0\).
Since a complex number consists of a pair of numbers \(Re\) and \(Im\), it can neither be represented as real numbers on a number-ray nor can complex numbers be compared with each other as \(greater\,than\), or \(less\,than\).
However, due to the number pair, complex numbers can be represented in a special coordinate system - a complex plane, the Gaussian plane of numbers. The real part here corresponds to the x-coordinate, the imaginary part of the y-coordinate. Read more about this in the article on the geometric representation of complex numbers.