Understanding Engineering Mathematics

Any number of this form

z=a+jb j=

√

− 1

wherea,bare real numbers, is called acomplex number. We will usually writezrather
thanxfor such a number sincezalways signals that we are talking about complex numbers.

ais called thereal partofz, denoted Rez– it is a real number

b(notjb) is called theimaginary part, denoted Imz– also a real number

Real numbers are special cases of complex numbers with zero imaginary part.

Exercise on 12.1

Solve the following equations ina+jbform:

(i) x^2 + 4 = 0 (ii) x^2 +x+ 1 = 0 (iii) x^2 + 6 x+ 11 = 0

(iv) x^3 − 1 = 0

Answer

(i) ± 2 j (ii) −

1

2

±

√

3

2

j (iii) − 3 ±

√

2 j (iv) 1,−

1

2

±

√

3

2

j

12.2 The algebra of complex numbers

Complex numbers can be manipulated just like real numbers but using the propertyj^2 =
−1 whenever appropriate. Many of the definitions and rules for doing this are simply
common sense, and here we just summarise the main definitions.
Equalityof complex numbers:a+jb=c+jdmeans thata=candb=d.
To performadditionandsubtraction of complex numbers we combine real parts
together and imaginary parts separately:

(a+jb)+(c+jd)=(a+c)+j(b+d)

(a+jb)−(c+jd)=(a−c)+j(b−d)

Also note thatk(a+jb)=ka+jkbfor any real numberk.
At this point you may be noticing the similarity to our work on vectors in the previous
chapter.

Problem 12.2

Ifz= 1 Y 2 jandw= 4 −jevaluate 2z− 3 w.

We have 2z− 3 w= 2 ( 1 + 2 j)− 3 ( 4 −j)= 2 + 4 j− 12 + 3 j=− 10 + 7 j.
Tomultiplytwo complex numbers simply multiply out the brackets by ordinary algebra,
usej^2 =−1 and gather terms:

(a+jb)(c+jd)=ac+ajd+jbc+j^2 bd=ac−bd+j(bc+ad)