Complex numbers: Construction of complex numbers and complex arithmetic
Conjugate, modulus, and division of complex numbers
Complex numbers are often denoted by the letter , so .
Conjugate The complex conjugate of , sometimes in short called conjugate of and commonly denoted with or , is defined as .
With certain combinations and something specialis going on and this you might already have noticed in the exercises of addition and multiplication of complex numbers; if not, we give some extra examples.
So
When you have seen enough examples of the penny dropped: the sum and the product of a complex number with its conjugate is a real number. In fact, for the product the following holds:
For any complex number holds
If is a complex number in standard form, then:
Modulus
For any complex number we call the root the modulus or absolute value of that number and denote this as . For any complex number holds: .
Now that we know the conjugate and modulus of a complex number, we can easily introduce division of complex numbers. First a simple example of writing in standard form. The trick is to multiply the numerator and denominator by , the conjugate of the denominator, and to use the rule in order to simplify the result:
For any complex number we act similarly, namely multiplying the numerator and denominator by the conjugate of the denominator.
If . then
For two complex numbers and we perform division as follows:
Fully written out, division of two complex numbers goes as follows:
Division of complex numbers as and are complex numbers in standard form, then division goes as follows:
You do not need to memorize this formula; it suffices to know the concept and be able to apply the method. In fact, it suffices in division of two complex numbers to multiply the numerator and denominator by the complex conjugate of the denominator, then to exapand brackets and make use of the rule that . Some sample calculations illustrate this.
Mathcentre videos
Complex Conjugate (5:23)
Division (7:23)