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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 w=c+di are two complex numbers then,

  1. REDIRECT Template:Abgesetzte Formel

Example 2

  1. (3−5i)+(−4+i)=−1−4i
  2. 21+2i−61+3i=31−i 
  3. 53+2i−23−i=106+4i−1015−5i=10−9+9i=−0.9+0.9i


Multiplication

Complex numbers are multiplied in the same way as ordinary real numbers or algebraic expressions, with the extra condition that i2=−1. Generally one has for two complex numbers z=a+bi and w=c+di that

  1. REDIRECT Template:Abgesetzte Formel

Example 3

  1. 3(4−i)=12−3i
  2. 2i(3−5i)=6i−10i2=10+6i
  3. (1+i)(2+3i)=2+3i+2i+3i2=−1+5i
  4. (3+2i)(3−2i)=32−(2i)2=9−4i2=13
  5. (3+i)2=32+23i+i2=8+6i
  6. i12=(i2)6=(−1)6=1
  7. i23=i22i=(i2)11i=(−1)11i=−i


Complex conjugate

If z=a+bi then z=a−bi is called the complex conjugate of z (the opposite is also true, that z is conjugate to 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. z=5+i then z=5−i.
  2. z=−3−2i then z=−3+2i.
  3. z=17 then z=17.
  4. z=i then z=−i.
  5. z=−5i then z=5i.

Example 5

  1. If z=4+3i one has
    • z+z=4+3i+4−3i=8
    • z−z=6i
    • zz=42−(3i)2=16+9=25
  2. If for z one has Rez=−2 and Imz=1, one gets
    • z+z=2Rez=−4
    • z−z=2iImz=2i
    • zz=(−2)2+12=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 z=a+bi and w=c+di:

  1. REDIRECT Template:Abgesetzte Formel

Example 6

  1. 1+i4+2i=(1+i)(1−i)(4+2i)(1−i)=1−i24−4i+2i−2i2=26−2i=3−i
  2. 253−4i=25(3+4i)(3−4i)(3+4i)=32−16i225(3+4i)=2525(3+4i)=3+4i
  3. i3−2i=i(−i)(3−2i)(−i)=−i2−3i+2i2=1−2−3i=−2−3i

Example 7

  1. 22−i−i1+i=2(2+i)(2−i)(2+i)−i(1−i)(1+i)(1−i)=54+2i−21+i
    =108+4i−105+5i=103−i
  2. 1−21−i2i+i2+i=1−i1−i−21−i(2+i)2i(2+i)+i2+i=1−i1−i−22+i4i+2i2+i=1−i−1−i2+i−2+5i
    =1−i−1−i2+i−2+5i=(−1−i)(2+i)(1−i)(−2+5i)=−2−i−2i−i2−2+5i+2i−5i2
    =3+7i−1−3i=(3+7i)(3−7i)(−1−3i)(3−7i)=32−49i2−3+7i−9i+21i2 =58−24−2i=29−12−i

Example 8

Determine the real number a so that the expression  2+ai2−3i  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 z=a+bi and w=c+di to be equal, requires that both the real and imaginary parts are equal. In other words, that a=c and b=d. When you are looking for an unknown complex number z in an equation, you can either try to solve for the number z in the usual way, or insert 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 3z+1−i=z−3+7i.

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

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

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

    The equation contains both z as well as z and therefore we assume z to be z=a+ib and solve the equation for a and 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, 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 i2=−1.

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

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