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3.1 Rechnungen mit komplexen Zahlen

Aus Online Mathematik Brückenkurs 2

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Contents:

  • Real and imaginary part
  • Addition and subtraction of complex numbers
  • Complex conjugate
  • Multiplication and division of complex numbers

Learning outcomes:

After this section, you will have learned to:

  • Simplify expressions that are constructed using complex numbers and the four arithmetic operations.
  • Solve first order complex number equations and simplify the answer.


Introduction

The real numbers represent a complete set of numbers in the sense that they "fill" the real-number axis. Despite this, the real numbers are not enough to be solutions of all possible algebraic equations, in other words. there are equations of the type

  1. REDIRECT Template:Abgesetzte Formel

which do not have a solution among the real numbers. For example, the equation x2+1=0 has no real solution, since no real number satisfies x2=−1. However, if we can imagine −1  as the number that solves the equation x2=−1 and manipulate −1  as any other number, it turns out that every algebraic equation has solutions. The number −1  however is not a real number, we cannot go out into the world and measure −1  anywhere, or find something that is numerically −1 , but we can still have great use of this number in many real contexts.


Example 1

If we would like to find out the sum of the roots (solutions) of the equation x2−2x+2=0 we can first obtain the roots x1=1+−1  and x2=1−−1 . These roots contain the non-real number −1 . If we allow ourselves to do calculations containing −1  we see that the sum of x1 and x2 turns out to be 1+−1+1−−1=2 , which certainly is a real number.

In order to solve our problem we had to use a number that was not real in order to arrive at the real number solution.


Definition of complex numbers

One introduces the imaginary unit i=−1  and define a complex number as an object that can be written in the form

  1. REDIRECT Template:Abgesetzte Formel

where a and b are real numbers, and i satisfies i2=−1.

If a=0 then the number is "purely imaginary". If b=0 the number is real. We can see that the real numbers are a subset of the complex numbers. The set of complex numbers is designated by C.

For an arbitrary complex number one often uses the symbol z. If z=a+bi, where a and b are real, then a is the real part and b the imaginary part of z. One uses the following notation:

  1. REDIRECT Template:Abgesetzte Formel

When one calculates with complex numbers one treats them in principle like real numbers, but keeps track of the fact that i2=−1.


Addition and subtraction

To add or subtract complex numbers one adds (subtracts) the real and imaginary parts separately. If z=a+bi and \displaystyle w=c+di are two complex numbers then,

  1. REDIRECT Template:Abgesetzte Formel

Example 2

  1. \displaystyle (3-5i)+(-4+i)=-1-4i
  2. \displaystyle \bigl(\tfrac{1}{2}+2i\bigr)-\bigl(\tfrac{1}{6}+3i\bigr) = \tfrac{1}{3}-i
  3. \displaystyle \frac{3+2i}{5}-\frac{3-i}{2} = \frac{6+4i}{10}-\frac{15-5i}{10} = \frac{-9+9i}{10} = -0\textrm{.}9 + 0\textrm{.}9i


Multiplication

Complex numbers are multiplied in the same way as ordinary real numbers or algebraic expressions, with the extra condition that \displaystyle i^2=-1. Generally one has for two complex numbers \displaystyle z=a+bi and \displaystyle w=c+di that

  1. REDIRECT Template:Abgesetzte Formel

Example 3

  1. \displaystyle 3(4-i)=12-3i
  2. \displaystyle 2i(3-5i)=6i-10i^2=10+6i
  3. \displaystyle (1+i)(2+3i)=2+3i+2i+3i^2=-1+5i
  4. \displaystyle (3+2i)(3-2i)=3^2-(2i)^2=9-4i^2=13
  5. \displaystyle (3+i)^2=3^2+2\cdot3i+i^2=8+6i
  6. \displaystyle i^{12}=(i^2)^6=(-1)^6=1
  7. \displaystyle i^{23}=i^{22}\cdot i=(i^2)^{11}\cdot i=(-1)^{11}i=-i


Complex conjugate

If \displaystyle z=a+bi then \displaystyle \overline{z} = a-bi is called the complex conjugate of \displaystyle z (the opposite is also true, that \displaystyle z is conjugate to \displaystyle \overline{z}). One obtains the relationships

  1. REDIRECT Template:Abgesetzte Formel

but most importantly, using the difference of two squares rule, one obtains

  1. REDIRECT Template:Abgesetzte Formel

i.e. that the product of a complex number and its conjugate is always real.


Example 4

  1. \displaystyle z=5+i\qquad then \displaystyle \quad\overline{z}=5-i\,.
  2. \displaystyle z=-3-2i\qquad then \displaystyle \quad\overline{z} =-3+2i\,.
  3. \displaystyle z=17\qquad then \displaystyle \quad\overline{z} =17\,.
  4. \displaystyle z=i\qquad then \displaystyle \quad\overline{z} =-i\,.
  5. \displaystyle z=-5i\qquad then \displaystyle \quad\overline{z} =5i\,.

Example 5

  1. If \displaystyle z=4+3i one has
    • \displaystyle z+\overline{z} = 4 + 3i + 4 -3i = 8
    • \displaystyle z-\overline{z} = 6i
    • \displaystyle z \cdot \overline{z} = 4^2-(3i)^2=16+9=25
  2. If for \displaystyle z one has \displaystyle \mathop{\rm Re} z=-2 and \displaystyle \mathop{\rm Im} z=1, one gets
    • \displaystyle z+\overline{z} = 2\,\mathop{\rm Re} z = -4
    • \displaystyle z-\overline{z} = 2i\,\mathop{\rm Im} z = 2i
    • \displaystyle z\cdot \overline{z} = (-2)^2+1^2=5


Division

For the division of two complex numbers one multiplies the numerator and denominator with the complex conjugate of the denominator thus getting a denominator which is a real number. Thereafter, both the real and imaginary parts of the numerator is divided by this number (the new denominator). In general, if \displaystyle z=a+bi and \displaystyle w=c+di:

  1. REDIRECT Template:Abgesetzte Formel

Example 6

  1. \displaystyle \quad\frac{4+2i}{1+i} = \frac{(4+2i)(1-i)}{(1+i)(1-i)} = \frac{4-4i+2i-2i^2}{1-i^2} = \frac{6-2i}{2}=3-i
  2. \displaystyle \quad\frac{25}{3-4i} = \frac{25(3+4i)}{(3-4i)(3+4i)} = \frac{25(3+4i)}{3^2-16i^2} = \frac{25(3+4i)}{25} = 3+4i
  3. \displaystyle \quad\frac{3-2i}{i} = \frac{(3-2i)(-i)}{i(-i)} = \frac{-3i+2i^2}{-i^2} = \frac{-2-3i}{1} = -2-3i

Example 7

  1. \displaystyle \quad\frac{2}{2-i}-\frac{i}{1+i} = \frac{2(2+i)}{(2-i)(2+i)} - \frac{i(1-i)}{(1+i)(1-i)} = \frac{4+2i}{5}-\frac{1+i}{2}
    \displaystyle \quad\phantom{\frac{2}{2-i}-\frac{i}{1+i}}{} = \frac{8+4i}{10}-\frac{5+5i}{10} = \frac{3-i}{10}\vphantom{\Biggl(}
  2. \displaystyle \quad\frac{1-\dfrac{2}{1-i}}{2i+\dfrac{i}{2+i}} = \frac{\dfrac{1-i}{1-i}-\dfrac{2}{1-i}}{\dfrac{2i(2+i)}{(2+i)} + \dfrac{i}{2+i}} = \frac{\dfrac{1-i-2}{1-i}}{\dfrac{4i+2i^2 + i}{2+i}} = \frac{\dfrac{-1-i}{1-i}}{\dfrac{-2+5i}{2+i}}
    \displaystyle \quad\phantom{\smash{\frac{1-\dfrac{2}{1-i}}{2i+\dfrac{i}{2+i}}}}{} = \frac{-1-i}{1-i}\cdot \frac{2+i}{-2+5i} = \frac{(-1-i)(2+i)}{(1-i)(-2+5i)} = \frac{-2-i-2i-i^2}{-2+5i+2i-5i^2}\vphantom{\Biggl(}
    \displaystyle \quad\phantom{\smash{\frac{1-\dfrac{2}{1-i}}{2i+\dfrac{i}{2+i}}}}{} = \frac{-1-3i}{3+7i} = \frac{(-1-3i)(3-7i)}{(3+7i)(3-7i)} = \frac{-3+7i-9i+21i^2}{3^2-49i^2}\vphantom{\Biggl(} \displaystyle \quad\phantom{\smash{\frac{1-\dfrac{2}{1-i}}{2i+\dfrac{i}{2+i}}}}{} = \frac{-24-2i}{58} = \frac{-12-i}{29}\vphantom{\Biggl(}

Example 8

Determine the real number \displaystyle a so that the expression \displaystyle \ \frac{2-3i}{2+ai}\ becomes real.

Multiply the numerator and denominator with the complex conjugate of the denominator so that the expression can be written with separate real and imaginary parts.

  1. REDIRECT Template:Abgesetzte Formel

If the expression is to be real , the imaginary part must be 0, ie.

  1. REDIRECT Template:Abgesetzte Formel


Equations

For two complex numbers \displaystyle z=a+bi and \displaystyle w=c+di to be equal, requires that both the real and imaginary parts are equal. In other words, that \displaystyle a=c and \displaystyle b=d. When you are looking for an unknown complex number \displaystyle z in an equation, you can either try to solve for the number \displaystyle z in the usual way, or insert \displaystyle z=a+bi in the equation and then compare the real and imaginary parts of the two sides of the equation with each other.

Example 9

  1. Solve the equation \displaystyle 3z+1-i=z-3+7i.

    Collect all \displaystyle z on the left-hand side by subtracting \displaystyle z from both sides
    1. REDIRECT Template:Abgesetzte Formel and now subtract \displaystyle 1-i
    2. REDIRECT Template:Abgesetzte Formel
    This gives that \displaystyle \ z=\frac{-4+8i}{2}=-2+4i\,\mbox{.}
  2. Solve the equation \displaystyle z(-1-i)=6-2i.

    Divide both sides by \displaystyle -1-i in order to obtain \displaystyle z
    1. REDIRECT Template:Abgesetzte Formel
  3. Solve the equation \displaystyle 3iz-2i=1-z.

    Adding \displaystyle z and \displaystyle 2i to both sides gives
    1. REDIRECT Template:Abgesetzte Formel
    This gives
    1. REDIRECT Template:Abgesetzte Formel
  4. Solve the equation \displaystyle 2z+1-i=\bar z +3 + 2i.

    The equation contains both \displaystyle z as well as \displaystyle \overline{z} and therefore we assume \displaystyle z to be \displaystyle z=a+ib and solve the equation for \displaystyle a and \displaystyle b by equating the real and imaginary parts of both sides
    1. REDIRECT Template:Abgesetzte Formel
    i.e.
    1. REDIRECT Template:Abgesetzte Formel
    which gives
    1. REDIRECT Template:Abgesetzte Formel
    The answer is therefore, \displaystyle z=2+i.


Study advice

Keep in mind that:

Calculations with complex numbers are done in the same way as with ordinary numbers with the additional information that \displaystyle i^2=-1.

Quotients of complex numbers are simplified by multiplying the numerator and denominator with the complex conjugate of the denominator.

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/3.1_Rechnungen_mit_komplexen_Zahlen“