1 EMTA 101: Complex Numbers
1.1 Introduction
The number system encompasses all numbers that we are likely to encounter. The real number system includes
rational and irrational numbers. However, at times non-real values appear, eg. in the solution of quadratic
equations. Such values cannot simply be ignored as they appear in the mathematics of real physical phenomena.
Applications in science and engineering include electrical alternating current theory and mechanical vector
analysis. Situations such as massess suspended from a spring generates non-real solutions.
1.2 A Complex Number Defined
Solve for x. x2 − 2x + 2 = 0. Using the quadratic formula,
√
−b ± b2 − 4ac
x=
√ 2a
2± 4−8
x=
√2 √ √ √
2 ± −4 2 ± 4 −1 2 ± 2 −1
x= = =
2
√ 2 2
x = 1 ± −1
√
Convention: Let j = −1 then x = 1 ± j
The value x = 1 ± j is known as a Complex√Number consisting of the REAL part, 1, and
IMAGINARY(NON-REAL) part, viz. j = −1
Thus a Complex Number consists of two parts: REAL and NON-REAL(IMAGINARY).
A Complex Number (z) is generally written in the form: z = a + jb, The REAL part is ’a’ and the
IMAGINARY(NON-REAL) part ’b’.
Examples: 2 + j3, −7j, 2.56 − j0.78, etc. √
N.B. In pure Mathematics the letter i is used, i.e. i = j = −1
1.3 Properties of Complex Numbers
1.3.1 Powers of ’j’
√ √
Remember j = −1, thus j 2 = ( −1)2 = −1
We find that j 3 = j 2 .j = −j and j 4 = (j 2 )2 = (−1)2 = +1
Summarising:j = j, j 2 = −1, j 3 = −j and j 4 = +1
Using these facts we can reduce any power of j
Examples:j 6 = j 4 .j 2 = (1)(−1) = −1; j 100 = (j 4 )25 = 125 = 1; j 159 = j 156 .j 3 = (j 4 )39 .j 3 = 139 .(−j) = −j
Exercise: Reduce to simplest form of ’j’.
i). j 56
ii). j 45
iii).j −120
iv).j 2548
1.3.2 Complex Conjugate
If z = a + jb, we define its Complex Conjugate z̄ to be z̄ = a − jb, i.e. simply change the sign of the
IMAGINARY part only
Find the Complex Conjugates:z = −7 + j5 therefore z̄ = −7 − j5 or z = −j2.56 then z̄ = +j2.56 etc.
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1.1 Introduction
The number system encompasses all numbers that we are likely to encounter. The real number system includes
rational and irrational numbers. However, at times non-real values appear, eg. in the solution of quadratic
equations. Such values cannot simply be ignored as they appear in the mathematics of real physical phenomena.
Applications in science and engineering include electrical alternating current theory and mechanical vector
analysis. Situations such as massess suspended from a spring generates non-real solutions.
1.2 A Complex Number Defined
Solve for x. x2 − 2x + 2 = 0. Using the quadratic formula,
√
−b ± b2 − 4ac
x=
√ 2a
2± 4−8
x=
√2 √ √ √
2 ± −4 2 ± 4 −1 2 ± 2 −1
x= = =
2
√ 2 2
x = 1 ± −1
√
Convention: Let j = −1 then x = 1 ± j
The value x = 1 ± j is known as a Complex√Number consisting of the REAL part, 1, and
IMAGINARY(NON-REAL) part, viz. j = −1
Thus a Complex Number consists of two parts: REAL and NON-REAL(IMAGINARY).
A Complex Number (z) is generally written in the form: z = a + jb, The REAL part is ’a’ and the
IMAGINARY(NON-REAL) part ’b’.
Examples: 2 + j3, −7j, 2.56 − j0.78, etc. √
N.B. In pure Mathematics the letter i is used, i.e. i = j = −1
1.3 Properties of Complex Numbers
1.3.1 Powers of ’j’
√ √
Remember j = −1, thus j 2 = ( −1)2 = −1
We find that j 3 = j 2 .j = −j and j 4 = (j 2 )2 = (−1)2 = +1
Summarising:j = j, j 2 = −1, j 3 = −j and j 4 = +1
Using these facts we can reduce any power of j
Examples:j 6 = j 4 .j 2 = (1)(−1) = −1; j 100 = (j 4 )25 = 125 = 1; j 159 = j 156 .j 3 = (j 4 )39 .j 3 = 139 .(−j) = −j
Exercise: Reduce to simplest form of ’j’.
i). j 56
ii). j 45
iii).j −120
iv).j 2548
1.3.2 Complex Conjugate
If z = a + jb, we define its Complex Conjugate z̄ to be z̄ = a − jb, i.e. simply change the sign of the
IMAGINARY part only
Find the Complex Conjugates:z = −7 + j5 therefore z̄ = −7 − j5 or z = −j2.56 then z̄ = +j2.56 etc.
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