FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Throughout this module x means thepositive square root.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Note that throughout this module we usethe symbol x to represent the positivesquare root of x.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

The historical references included in thismodule are included only for interest andare not necessary for an understanding ofcomplex numbers.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

The terms integer, rational and real areexplained in more detail in Subsection 2.5.But briefly their meanings are as follows:

An integer is a positive or negative wholenumber such as 0, ±1, ±2, ±3, …

A rational number is a fraction of the formp/q where q ≠ 0 and p are integers.

A real number is a decimal number (whichdoes not necessarily terminate) such as1.345, −1.7, π or 2 . (See Subsection 2.5)

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Engineers often use j instead of i, but i ismore common in scientific work.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Notice that the normal rules of algebraapply to the symbol i so that −i = −1 × i.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

7i

6 or

7

6i or 7i/6 or (7/6)i are equally

acceptable, but not 7/6i, which is likely tobe interpreted as 7/(6i).

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

The parentheses are not necessary, but theyhelp to distinguish the pairs of imaginarynumbers.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Again notice that the order in which thefactors are multiplied does not affect thefinal result.

In particular, multiplying the imaginarynumber i by the real number 0 producesthe imaginary number 0i, which wenormally denote simply by 0 (and there isusually no need to distinguish between thereal number zero and the imaginarynumber zero).

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Notice that we may omit the multiplicationsigns, as in normal algebra, and we preferi × i × i or i 13 to i i i.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

These tricks for dividing imaginarynumbers are not worth remembering sincewe will introduce more general methods(for complex numbers) shortly.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Note that 1

i= −i , since

1

i= 1

i× i

i= i

−1= −i

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Notice the crucial distinction betweencomplex (meaning to involve i, wherei2 = −1) and complicated!

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

We may refer to ‘z as complex’, as analternative to ‘z is a complex number’.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

There is a tradition of writing z = x + iyrather than z = iy + x. The two expressionsare completely equivalent.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

It is possible to provide a soundmathematical definition of complexnumbers based on the notion of orderedpairs of real numbers, indeed this is theway most pure mathematicians would doit. The first element is the real part and thesecond element (a real number) theimaginary part. It is for this reason that theimaginary part of z = x + iy is y ratherthan iy.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Some authors use different symbols andtypefaces for Re and Im.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

It is not necessary to remember thisgeneral formula provided that you use theordinary rules of algebra and replace everyoccurrence of i2 by −1.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

A popular alternative to z* is z ,pronounced ‘z bar’.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

z + z* = (x + iy) + (x − iy) = 2x

z − z* = (x + iy) − (x − iy) = 2iy

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Notice that zw* means z times w* not(zw)*. Also notice that it is not necessaryto write z and w in terms of their real andimaginary parts in order to obtain thisresult.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Quantum theory provides the best accountcurrently available of many microscopicphenomena, including the behaviour ofatoms and molecules.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

You will be familiar with a calculation

such as 2

3= 2 × 5

3 × 5= 10

15.

Similarly, multiplying the numerator anddenominator of a complex fraction by acommon factor will leave the value of thefraction unchanged.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Question T16 is intended to provide youwith an opportunity to improve yourcomplex arithmetic if you feel that youneed to.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

The natural numbers are sometimes knownas the positive integers or the countingnumbers.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

The set of non-negative integers issometimes known as the set of wholenumbers.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Notice that the set, R is known as the realline even if we don’t actually draw a line.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

The notations N, Z, Q, R and C arecommonly used by mathematicians butless frequently by physicists.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

We have exhausted all the points on thereal line, so we would need somethingfairly innovative.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Consult polynomial equation in theGlossary for further details.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Solutions to polynomial equations arediscussed more fully elsewhere in FLAP.See the Glossary for details.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Notice that using x and an does notnecessarily imply that they are real.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

The solutions, or roots, of a polynomialequation are sometimes referred to as thezeros of the polynomial.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

The polynomial on the left of Equation 21is sometimes described as ‘a polynomial inx’. A function such as part (c) of QuestionR4 might be decribed as ‘a polynomial insin 1x’ by some authors (although it is notactually a polynomial).

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

The proof of the fundamental theorem ofalgebra is included in most elementarycourses on complex analysis (the calculusof complex functions).

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

For some quadratic equations you mayneed to evaluate more complicated squareroots, such as 2 + 3i . This is explainedelsewhere in FLAP. See the Glossary fordetails.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Remember that the real axis is alwayshorizontal while the imaginary axis isalways vertical.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

You will find that in ‘advanced’mathematics there is a tradition of usingthe term complex plane rather thanArgand diagram.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

a.c. is an abbreviation for alternatingcurrent.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

ω is the lower-case Greek letter omega.

henry = ohm second = Ω1s

farad = second per ohm = s1Ω−1

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

The point to note here is that complexarithmetic can often be done in severalways, and some methods may be shorterthan others. Also note that we write out allthe steps; if you try to do it in your headyou will get it wrong.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Notice that when we calculate the modulusit is always the positive square root.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Note that the imaginary part ofz = x + iy is y and not iy.

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

α + β = 5 + i 15 + 5 − i 15 = 10

αβ = (5 + i 15 )(5 − i 15 )

= 52 − (i 15 )2

= 25 −15i2 = 25 +15 = 403

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

(a) (3i)(2i) = 3 × i × 2 × i = 3 × 2 × i × i

= 3 × 2 × (−1) = −06

(b) i(3i) = −1 × 3 = −3

(c) (0i)(4i) = 0 × i × 4 × i = 0 × 4 × i × i

= 0 × 4 × (−1) = 0

(d) i4 = (i × i) × (i × i) = −1 × −1 = 13

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

(a) 3(2i) = 6i,4(b) 2.1(−3.6i) = −7.56i.3

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

i5 − 5(i + i3) + (3i)3 − i = i − 5(0) − 27i − i

= −27i.3

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

(i +1 i)7 = i + i

i × i

7

= (i − i)7 = 03

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

(a) Re(3 − i) = 3,4(b) Im(5i − 1) = 5,4(c) Re(i(i − 1)) = Re(−1 − i) = −1,

(d) Im((a + i0b)(a − i 0b)) = Im(a2 + b2)

= 0.3

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

(a) (2 + 3i) + (4 + i) = 6 + 4i

(b) (3 − 2.5i) + (7 + 0.1i) = 10 − 2.4i

(c) (3.4 − 2i) − (0.4 + i) = 3 − 3i

(d) (−1 + i) − (−1 − i) = 2i 3

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

(a) (2 + 3i) (4 + i)= 2 × 4 + 3i × i + 3i × 4 + 2 × i= 5 + 14i

(b) 5i(7 − 3i) = 5i × 7 + (5i ) × (−3i)= 15 + 35i

(c) 2(5 + 3i) = 10 + 6i

(d) 2i(1 − i)2 = 2i(1 − i)(1 − i)

= 2i(1 − i − i + i02)

= 2i(−2i) = 43

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

We simplify the product of the first twobrackets

(1 + i)(2 + i) = 1 + 3i

then the product of the last two brackets

(3 + i)(4 + i) = 11 + 7i

Then we multiply these two together

(1 + 3i)(11 + 7i) = −10 + 40i3

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

(a) (3 + 4i)* = 3 − 4i3

(b) (2.5 − 7.3i)* = 2.5 + 7.3i3

(c) (z*)* = z.3

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Let z = x + iy and w = u + iv

then zw = (xu − yv) + i(xv + yu)

and (zw)* = (xu − yv) − i(xv + yu)

also z*w* = (x − iy)(u − iv)

= (xu − yv) − i(xv + yu)4

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Using Equation 17

(zw)* = z*w* (Eqn 17)

we have

(z*w)* = (z*)*w* = zw*

and it follows that

z*w + zw*

is the sum of a complex number (z*w) andits conjugate (zw*), and from Equation 15this must be real.3

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

(a) (3 + 4i)(3 + 4i)* = 32 + 42 = 25

(b) (1 + i)*(1 + i) = 12 + 12 = 2

(c) i*i = 12 = 1.3

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

(a) |13 + 4i1| = 32 + 42 = 25 = 5

(b) |12 + i1| = 22 +12 = 5

(c) |13 − 4i1| = 32 + 42 = 25 = 5

(d) |1−041| = 42 = 4

(e) |12i1| = 22 = 4 = 2

(f) |101| = 02 + 02 = 03

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

Since wz = (3 + 2i)(x + iy)= (3x − 2y) + (2x + 3y)i

we have(3x − 2y) + (2x + 3y)i = 1

The right-hand side of this equation is real,so that, equating real and imaginary parts,we obtain the pair of simultaneousequations

3x − 2y = 1

2x + 3y = 0

Solving these equations (by the techniquesof ordinary algebra) we obtain

x = 3/13 and y = −2/13.3

FLAP M3.1 Introducing complex numbersCOPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1

3 + 4i

2 + i= 3 + 4i

2 + i× 2 − i

2 − i= 6 + 4 + 8i − 3i

5= 2 + i

3