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Complex numbers (en)
Refresher, complex numbers ...
en pdf complex_en.pdf
en Basic properties of complex numbers Definitions A common, real number is usually illustrated as a point on the so-called number line. The magnitude is represented by the distance from the point in question to zero.
numberline.bmp
A complex number z consists of two components. It can be written as a + jb. Here, a and b are real numbers. j is the square root of -1 and is called the imaginary unit. a is the complex number real part Re (Z). b is the imaginary part, Im (Z).
Every complex number can be represented as a point in a two-dimensional coordinate system, the complex plane.
komplexplan.bmp
Number z is represented by a point with coordinates a and b.
The distance from the point to the origin represents the amount or number value |z|.
belopp.jpg
or
belopp2.jpg
The angle α is called the argument of z, arg(z) and as seen in the figure
tana.jpg
or
argz.jpg
We can also express z in polar form, eg with |z| and α. As seen in the figure
polarz.jpg
One can then imagine that it's the connecting line between the point and the origin that represents the number. We can see this as a pointer (vector) with the length |z| och en direction that is defined by the angle α.
Basic properties Complex numbers can be treated algebraically, the following rules apply.
Addition komplexadd.jpg
z1z2add.jpg
The figure shows what the addition means in the complex plane. The pointer of z equals the geometric sum of the pointers of z1 and z2. For |z| and arg(z) applies the previously mentioned general terms.
Subtraction z1z2sub.jpg
In the complex plane z equals the geometric difference between the z1 och z2.
Multiplication The multiplication rule, we demonstrate most easily with an example.
z1z1mult.jpg
The multiplication can also be implemented with the numbers expressed in polar form.
komplexmult.jpg
polarmult.jpg
This means that
polmulti.jpg
Division Algebraic the division is implemented like this:
div.jpg
Now, one often wants to have the results in the form a+jb and if so, one extends the denominator with the conjugates quantity a2 - jb2. Then one gets
konjugat.jpg
If numbers are expressed in polar form, the division rule that look like this:
argdiv.jpg
Some memory rules
* If z = z1 + z2, so is generally |z| ≠ |z1| + |z2|( only if arg(z1) = arg(z2) then |z| = |z1| + |z2| )
* When calculating the amount of a product or a quotient of two complex numbers z1 ochz2 it is generally unnecessary to first calculate the complex result and then form the amount. One calculates instead |z1| and |z2| separately, for as we have seen applies
abs.jpg
Example Example 1 Redo the expression 2+3/j to form a+jb.
ex1.jpg
Example 2 Write expression z = 6 + jA + 1/(jB) in the general form of complex numbers, and write a expression for the amount.
ex2.jpg
Example 3 Determine |z| and arg(z) when z = z1·z2 and z1 = j and z2 = -1 -j
ex3fig.jpg
Algebraic ex3.jpg
Polar ex3polar.jpg
Example 4 z1 = 3 + j5, z2 = 5 + j7. Calculate
ex4q.jpg
ex4.jpg
If instead multiplied with conjugate quantity the calculations had been
ex4konjug.jpg
If one compares the above one can see that complex conjugation involves much more work!
Exercises Question 1 In which direction points the complex pointer z = -2 + j2 ?
rutat.jpg
[ Answers and solutions ]
Question 2 What is the sum of z1 and z2 if z1 = 1 + j2 and z2 = 2 - j ?
[ Answers and solutions ]
Question 3 How long is the pointer 3 + j4 ?
[ Answers and solutions ]
Question 4 Draw the pointer z = z1 - z2 if z1 = 1 + j and z2 = 2 + j ?
rutat.jpg
[ Answers and solutions ]
Question 5 How large is Im(z) if z = z1 + z2 ?z1 = 3(1+j) and z2 = 2(1-j) .
[ Answers and solutions ]
Question 6 How large is |z| if z = z1·z2 ?z1 = 2 + j and z2 = -(2 + j) .
[ Answers and solutions ]
Question 7 What will be |3+j4|· |j2| ?
[ Answers and solutions ]
Question 8 Determine |z| and arg(z) if z = z1·z2 and z1 = 1 + j and z2 = -1 + j .
rutat.jpg
[ Answers and solutions ]
Question 9 What will be z = z1·z2 if z1 = j and z2 = 1 - j .
rutat.jpg
[ Answers and solutions ]
Question 10 What is |z| ?
q10.jpg
[ Answers and solutions ]
Question 11 Calculate z.z1 = 2 + j3 and z2 = 1 + j .
q11.jpg
rutat.jpg
[ Answers and solutions ]
This exercise booklet has been given to me by Per-Erik Lindahl. It has been used as an aid in courses of basic circuit theory at KTH.