Download Complex numbers 2

1
Complex numbers 2.
Revision from last week’s session
Problem 1
Solve the quadratic equation
x2 - 6x + 34 = 0
Problem 2
Evaluate the following powers of j
a)
b)
Problem 3
j12
j24
Add the following complex numbers:
a)
(3 + j) + (2 + j3) =
b)
(2 + j5) + (5 – j2) =
c)
Problem 4
(4 + j3) + (4 – j2) =
Subtract the following complex numbers:
a)
(3 + j) - (2 + j3) =
b)
(2 + j5) - (5 – j2) =
c)
(4 + j3) + (4 – j2) =
Problem 5
Solve the expression:
(5 + j) + (2 +j3) – (3 + j2) =
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Multiplication and division of complex numbers
Multiplication
Complex numbers are multiplied together in the same way as you would
multiply the sum of two numbers together. An example should show you
how this works:
Example 1.
(3 + j4)(2 + j5) = 6 + j8 + j15 + j2 20 = 6 + j23 –20 {since j2 = -1} =
-14 + j23
The above result was found by doing the following:




Multiply the
Multiply the
Multiply the
Multiply the
two
two
two
two
left-hand terms
inner terms
outer terms
right-hand terms
If the expression contains more than two factors, we multiply them
together in stages.
Example 2.
(3 + j4)(2 – j5)(1 – j2) = (6 + j8 – j15 – j2 20)(1 – j2)
The first factor on the right hand side of the equal sign has been found
by carrying out the four steps shown in the bullet points above by
multiplying (3 + j4)(2 – j5). The next step is to simplify the first factor
on the right hand side, and it becomes (26 –j7). This means that we have
the following expression:
(3 + j4)(2 – j5)(1 – j2) = (26 –j7)(1 – j2)
We now carry out the operations given in the 4 bullet points above on this
new expression and get:
(3 + j4)(2 – j5)(1 – j2) = 12 – j59
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Division
Division of a complex number by a real number is simple and is done as
shown in the following example:
Example 3
(5 – j4)/3 = 5/3 – j4/3
You divide both the real and imaginary part by the real number in the
divisor.
It becomes more complicated if we have to divide a complex number by
another complex number. The method is to turn the denominator into a
real number. Multiplying both the numerator and the denominator by the
same complex number that converts the denominator into a real number
does this. An example will show how this works:
Example 4
(7 – j4)/(4 + j3) =
Looking at the above we can see that the complex number that will turn
the denominator into a real number is (4 – j3) {this is called the conjugate
number} Therefore:
(7 – j4)/(4 + j3) = (7 – j4)(4 – 3j)/(4 + j3)(4 – j3) =
If you now carry out the processes described in the 4 bullet points above
you will get:
(7 – j4)(4 – 3j)/(16 + 9) = (7 – j4)(4 – 3j)\25
Carrying out the same processes again but this time on the numerator
gives:
(7 – j4)/(4 + j3) = 16/25 – j37/25