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Compute with complex numbers

In the article on complex numbers it was written that the calculation rules for real numbers should also apply to complex numbers. This article describes how to calculate complex numbers.

 

Addition and subtraction

As first consider the addition of complex numbers to an example. Suppose we want to add 3 + i and 1 - 2i. We are looking for:

Addition of complex numbers

According to the principle of permanence, the calculation rules of real numbers should continue to be valid. This means that we are doing what we would do with real numbers. We would combine the two real expressions (3 and 1), as well as the two imaginary expressions (i and -2i).

That's exactly how we calculate here:

complex number addition

The result of the calculation is 4 - i.

 

When subtracting a complex number, the real expressions and the imaginary expressions are combined in the same way. It only needs to be noted that the minus sign changes the sign of the second number. You know that about calculating with real numbers.

As an example we subtract the numbers from the example above:

subtracting of a complex number

The result of the calculation is 2 + 3i.

 

To summarise, we can say:

Complex numbers are added by adding the real parts and the imaginary parts separately. The same applies to the subtraction. Complex numbers are subtracted by subtracting the real parts and the imaginary parts separately.

 

Multiplication

This paragraph describes how to multiply two complex numbers. As an example we again use the two numbers 3 + i and 1 - 2i. So it should be calculated:

complex number multiplication

According to the principle of permanence, the calculation rules of real numbers should continue to apply. We will therefore first multiply the parenthesis as normal. So we write:

Multiplication of complex number

Additional to expressions with i, in the formula appears i2. We can easily replace this i2. By definition i2 = -1. So we replace i2 with the number -1 and continue to calculate with the result from above as usual.

Multiplication of complex numbers

The result of the calculation is 5 - 5i.

This article describes the multiplication of complex numbers in normal form. Easier to calculate is the Multiplication of complex numbers in polar form.
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Conjugate complex numbers

Before we get to the division of complex numbers, we introduced a new concept. Every complex number has a so-called complex conjugate number. This conjugate complex number is needed here in the division, but we will come back to it later in other chapters.

As an example we take the number komplexe Zahl. The complex number conjugated to komplexe ZahlisKonjugiert komplexe Zahl The real parts of the two numbers are the same, the imaginary parts of the two differ only by the sign.

Let's take a look at the product of the two numbers:

multiplicate conjugate complex number

The product of the complex numbers and their conjugates is real. This is a special property of conjugate complex numbers that will prove useful again and again.

For the conjugate complex number konjugierte komplexe Zahlwe write konjugierte Zahl
In the example above, then: konjugierte
 

Division

Now let's talk about dividing complex numbers. In the next example we will divide the number 3 + i by the number 1 - 2i. We are looking for:

Division komplexer Zahl

According to the principle of permanence, the calculation rules of the real numbers should be valid here. It bothers us that in the denominator of the break, the i occurs. Division by a real number would be very easy.

This is where the complex conjugate comes into play. The fraction is extended by the conjugate complex number 1 + 2i of the denominator. This allows the i to be truncated in the denominator and the denominator becomes a real number. Only in the numerator remains a complex number, which can be easily multiplied out.

The division looks like this:

Komplexe Zahl dividieren

The result is:

dividieren komplexer Zahl

This article describes the division of complex numbers in normal form. Easier to calculate is the Division of complex numbers in polar form.
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Complex numbers

Basics of complex numbers
Computing with complex numbers
Addition and subtraction
Multiplication of complex numbers
Conjugate complex numbers
Division of complex numbers
Quadratic equations, complex numbers
Geometric representation
Geometric addition and subtraction
Absolute value of a complex number
The polar form of complex numbers
Calculate polar form
Convert polar form to normal
Multiplication in polar form
Division in polar form
   


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