Slide1 : Iota
Euler was the first mathematician to introduce the symbol i (iota) for the square root of - 1 with the property i2 = -1. He also called this symbol as the imaginary unit. i (iota) is not a real number and neither 0, nor > 0 and nor < 0.
Properties of Iota
(i) i0 = 1 i1 = i i2 = -1 i3 = i2 . i = - i i4 = 1 i5 = i4 . i = i i6 = i4 . i2 = - 1 i7 = i4 . i3 = - i i8 = i4 . i4 = 1 i9 = i8 . i = i i10 = - 1 i11 = - i
i4p = 1 i4p + 1 = i i4p + 2 = -1 i4p + 3 = -i
(ii) i0 is defined as 1.
(iii) The sum of four consecutive powers of i is zero
i.e. in + in + 1 + in + 2 + in + 3 = 0 n Î N. COMPLEX NUMBERS
Slide2 : (iv) The value of different integral powers of i are 1 or i or - 1 or - i.
The digit in the unit place of the value of a positive integral power of a digit also follows a sequence of digits.
Power of 2
21 = 2 22 = 4 23 = 8 24 = 16 25 = 32 26 = 64 27 = ......8 28 = ....6 29 = .....2 210 = ......4 211 = .....8 212 = .....6
24n + 1 = .....2 24n + 2 = ......4 24n + 3 = .....8 24n = .....6
Power of 3
31 = 3 32 = 9 33 = 27 34 = 81 35 = 243 36 = ....9 37 = ....7 38 = .....1 39 = .....3 310 = ....9 311 = ....7 312 = .....1
34n + 1 = ....3 34n + 2 = ....9 34n + 3 = ....7 34n = .....1
Slide3 : Power of 4
41 = 4 42 = 16 43 = 64 44 = .....6 45 = …....4 46 = .....6
42n + 1 = ….....4 42n = .....6
Power of 5
Last digit of every positive integral power of 5 is 5 i.e. 5n = .....5 positive integers n.
Powers of 6
Last digit of every positive integral power of 6 is 6 i.e. 66 = .....6 positive integers n.
Power of 7
71 = 7 72 = 49 73 = .....3 74 = ......1 75 = .....7 76 = .....9 77 = ......3 78 = ......1 79 = .....7 710 = .....9 711 = ....3 712 = .....1 74n + 1 = .....7 74n + 2 = ....9 74n + 3 = ....3 74n = .....1
Slide4 : Imaginary Quantities
The square root of a negative real number is called an imaginary quantity or an imaginary number.
e.g., etc. are imaginary quantities.
Note:
(i) For any two real numbers = is true only when at least one of a and b is either positive or zero. In other words, = is not valid if a and b both are negative.
(ii) For any positive real number a, we have
= ia.
Slide5 : Complex Number
A symbol of the form a + ib, where a and b are real numbers and i = -1 is called a complex number. It is denoted by z, i.e., z = a + ib.
A complex number may also be defined as an ordered pair of real numbers and may be denoted by the symbol (a, b). If we write z = (a, b) then a is called the real part and b the imaginary part of the complex number z and may be denoted by Re (z) = a and Im (z) = b respectively.
(i) A real number a can be written as a + i . 0 therefore every real number can be considered as a complex number whose imaginary part is zero. Thus the set of real numbers (R) is a proper subset of the complex numbers (C). i.e, R C.
(ii) A complex number z is said to be purely real, if Im(z) = 0 and is said to be purely imaginary if Re(z) = 0.
(iii) The complex number 0 = 0 + i. 0 is both purely real and purely imaginary.
(iv) A complex number is an imaginary number if and only if its imaginary part is non-zero. Here real part may or may not be zero. 3 + 2i is an imaginary number but not purely imaginary.
(v) All purely imaginary numbers except zero are imaginary numbers but an imaginary number may or may not be purely imaginary.
Slide6 : Conjugate of a Complex Number
The conjugate of complex number z = x + iy is denoted by =
and is defined as = = x - iy .
Note: (i) A complex number z is purely real if and only if = z.
(ii) A complex number z is purely imaginary if and only if = z.
(iii) Re(z) = Re( ) = and Im(z) = ( ) =
Equality of Complex Number
(i) Two complex numbers (z1 and z2) are said to be equal if and only if their real parts and imaginary parts are separately equal a + ib = c + id Û a = c and b = d OR Re(z1) = Re(z2) & Im(z1) = Im(z2).
(ii) The complex number do not possess the property of order
i.e., (a + ib) < (or) > (a + ib) is not defined.
For example, the statement 9 + 6i > 3 + 2i makes no sense.
(iii) If a + ib > c + id it is possible only when a > c and b = d = 0.
Slide7 : Algebraic Operations with Complex Numbers
These are defined as follows.
(i) Closure law : Any algebraic operations of complex number results in a complex number.
(ii) Addition : (a + ib) + (c + ib) = (a + c) + i (b + d)
(iii) Subtraction : (a + ib) – (c + id) = (a – c) + i (b – d)
(iv) Multiplication : (a + ib) (c + id) = (ac – bd) + i(ad + bc)
(v) Division : (when at least one of c and d is non zero)
=
Slide8 : Properties of Addition of Complex Numbers
(i) Closure Law : Closure law for addition holds in the set of complex numbers i.e. if z1 and z2 are any two complex numbers then z1 + z2 is also a complex number.
(ii) Addition of complex numbers obeys Commutative law and Associative law.
(iii) Additive Identity : There exists a complex number 0 + i0 such that for every complex number x + iy,
(x + iy) + (0 + i0) = (x + 0) + i(y + 0) = x + iy.
(vi) Additive Inverse : For every complex number z the complex number - z is called additive inverse.
Slide9 : Properties of Multiplication of Complex Numbers
(i) Closure Property : Closure law for multiplication holds in the set of complex numbers i.e. if z1 and z2 are any two complex numbers then z1z2 is also a complex number.
(ii) Multiplication of complex numbers obeys Commutative law and Associative law also.
(iii) Multiplicative Identity : The complex number 1 is known as multiplicative identity of all complex number z.
(iv) Multiplicative Inverse : For every complex number
z = x + iy ¹ 0 + i . 0, there exists a complex number such that
(x + iy) = 1 + i . 0 = multiplicative identity.
(iv) Reciprocal : If z = x + iy be a complex number, then it’s reciprocal
= = .
Slide10 : The Square Root of a Complex Number
Approach
(i) If the square root of a + ib is to be evaluated, let a + ib = x + iy.
(ii) Squaring both the sides and equate real and imagining part which will give value of (x2 - y2) and xy.
(iii) Find x2 + y2 by (x2 + y2)2 = (x2 - y2)2 + 4x2y2.
(iv) From x2 - y2 and x2 + y2, we get the value of x.
(v) Put x in xy, we obtained corresponding value of y.
(vi) Now a + ib = x + iy.
Note :
(i) a b = ab is true only when at least one of a and b is non negative.
(ii) The square root of w are : ± w2
(iii) The square root of w2 are : ± w
(iv) The square root of i are : ± (1 + i/2)
(v) The square root of - i are : ± (1 - i/2)
Direct Formula
The square roots of z = a + ib are
for b > 0 and for b < 0
Slide11 : Modulus of a Complex Number
The modulus of a complex number z = x + iy is denoted by | z | = | x + iy | and is defined as | z | = = non-negative square root of (x2 + y2). Geometrically modulus of a complex number means that distance of the point (x, y) from the origin (0, 0) on the x-y plane
Argument or Amplitude of a Complex Number
The argument or amplitude of a complex number z = x + iy is the value of q which satisfies the two equations
cos q = and sin q = at the same time.
Slide12 : Argument of z is denoted by arg(z) or amp(z).
Geometrically argument or amplitude of a complex number means that the angle made by the line joining the point (x, y) from the origin (0, 0) in the anti-clockwise sense from the positive direction of the x-axis.
There will be infinite number of values of q satisfying the above equations since sin q and cos q are periodic and all these values will be argument of z but usually we take only that value of q for which 0 £ q < 2p.
There will be only one value of q Î [0, 2p) satisfying the above two equations. If this value q is a then arg z = a.
General value of arg z is denoted by Arg z and Arg z = 2np + a , n Î I.
Slide13 : Principal Value of arg (z)
There are infinite number of values of q satisfying the two equations
cos q = and sin q = .
But there will be a unique value of q such that - p < q £ p. The value of argument q satisfying the inequality -p < q £ p is called the principal value of the argument.
If z = x + iy, x, y Î R, and q = tan–1|y/x| then the argument (arg(z)) and principal argument (P.arg(z)) of the complex number is defined with the help of the figure given :
Slide14 : Notes. (i) Argument of the complex number 0 is not defined.
(ii) If z1 = z2 Û | z1 | = | z2 | and arg z1 = arg z2.
or Re(z1) = Re(z2) & Im(z1) = Im(z2)
(iii) If arg (z) = p/2 or – p/2, z is purely imaginary
(iv) If arg (z) = 0 or p, z is purely real.
(v) arg(z) = 0 or p Þ z =
(vi) arg(z) = ± p/2 Þ z + = 0 P.arg (z) = p - q arg (z) = p -q P.arg (z) = q arg (z) = q P.arg (z) = q - p arg (z) = p + q P.arg (z) = q arg (z) = 2p + q
Slide15 : Representation of a Complex Number
A complex number can be represented in the following forms:
(i) Geometrical form (ii) Vectorial form
(iii) Trigonometrical form or Polar form (iv) Eulerian form
The plane in which we represent a complex number geometrically is known as the Complex plane or Argand plane or the Gaussian plane or x-y plane or Cartesian plane. The point P, plotted on the Argand plane, is called Argand diagram. In the Argand plane x-axis is called Real axis and y-axis is called Imaginary axis or conjugate axis.
Slide17 : If a complex number is purely real, then its imaginary part is zero. Therefore, a purely real number is represented by a point on x-axis. A purely imaginary complex number is represented by a point on y-axis. That is why x-axis is known as the real axis and y-axis, as the imaginary axis.
Conversely, if P(x, y) is a point in the plane, then the point P(x, y) represents a complex number z = x + iy. The complex number z = x + iy is also known as the affix of the point P. Thus, there exists a one-one correspondence between the points of the plane and the members (elements) of the set C of all complex number , i.e., for every complex number z = x + iy there exists uniquely a point (x, y) on the plane and for every point (x, y) of the plane there exists uniquely a complex number z = x + iy.
Slide18 : Trigonometric or Polar or Modulus Argument form of a Complex Number
Let z = x + iy, z is represented by P(x, y) in the Argand plane.
By geometrical representation.
OP = = | z |
ÐPOM = arg (z)
In the triangle OPM,
x = OP cos (ÐPOM) = | z | cos (arg z)
y = OP sin (ÐPOM) = | z | sin (arg z)
z = x + iy
\ z = | z | {cos q + i sin q}
If | z | = r and arg z = q then z = r(cosq + i sinq) = reiq (Euler’s form). Here we should take the principal value of q. Y X X’ M O Y’ q P(x, y) y x
Slide19 : Eulerian form of a Complex Number
For any non zero complex number z = x + iy = r(cos q + i sin q) can be represented in exponential or Eulerian form as
z = reiq = r(cos q + i sin q)
where , r is the modulus and q is the argument of z.
Properties of cis
(i) cis (q) = eiq = cos q + i sin q
(ii)
(iii) cis q + cis (-q) = eiq + e-iq = 2cos q
(iv) cis q - cis (-q) = eiq - e-iq = 2i sin q
(v)
(vi)
Slide20 : Algebraic Operations
Cartesian Form Polar Form
Complex Numbers Z = a + ib, w = c + id Z = reiq , w = seif
Addition Z + w = (a + c) + i(b + d) Z + w = r cos q + ir sin q + s cos f + is sin f
= (r cos q + s cos f) + i(r sin q + s sin f)
Subtraction Z - w = (a - c) + i(b - d) Z - w = (r cos q - s cos f) + i(r sin q - s sin f)
Multiplication Zw = (ac - bd) + i(ad + bc) Zw = rs ei(q + f)
Division
Note : Addition and subtraction are easier to perform in cartesian form while multiplication and division are easier to perform in polar form.
Slide21 : Conjugate Complex Numbers
The complex numbers z = (a, b) = a + ib and = (a, –b) = a – ib, where a and b are real numbers, i = and b ¹ 0 are said to be complex conjugate of each other. (The complex conjugate is obtained by just changing the sign of i.)
Properties of Conjugate
(i) is the mirror image of z along real axis or x-axis.
(ii) = z. (iii) z = Û z is purely real.
(iv) z = – Û z is purely imaginary. (v) Re(z) = Re( ) = .
(vi) Im(z) = = - (vii)
(viii) (ix)
(x) where a, b Î R
(xi) where a, b Î C (xii)
(xiii) (xiv)
(xv) (xvi) If z = f(z1), then = f( )
Slide22 : Properties on Argument
These properties are valid for general value arguments.
(i) General value of arg (z1 z2) = arg (z1) + arg (z2). In general arg (z1 z2 z3 ..... zn) = arg (z1) + arg (z2) + arg (z3) + ..... + arg (zn)
(ii) arg (zn) = n arg z
(iii) arg (z1/z2) = arg z1 – arg z2
These properties are valid for principle value of arguments.
(i) arg = – arg z.
(ii) arg (z ) = 0
(iii) arg = 2 arg z
Slide23 : Properties of Modulus
Modulus of complex number is also defined as the distance between the point representing complex number z and the origin. If z1 and z2 are two complex numbers, then the distance between z1 and z2 is | z1 – z2 |.
(i) | z | > 0 Þ | z | = 0 iff z = 0 and | z | > 0 iff z ¹ 0.
(ii) – | z | < Re(z) < | z | and – | z | < Im(z) < | z |.
(iii) | z | = || = | –z | = | - |.
(iv) | z1 z2 | = | z1 | | z2 |
(v) | z1/z2 | = |z1| /|z2|
(vi) | zn | = | z |n
(vii) z = | z |2
(vii) z = | z |2
(viii) | z1 ± z2 |2 = (z1 ± z2) = | z1 |2 + | z2 |2 ±
= | z1 |2 + | z2 |2 ± 2Re (z1 2)
(ix) z1 + z2 = 2| z1 | | z2 | cos (q1 – q2) where q1 = arg (z1) and q2 = arg (z2).
Slide24 : (x) | z1 + z2 |2 + | z1 – z2 |2 = 2{| z1 |2 + | z2 |2}.
Geometrical significance of the above result in that in parallelogram the sum of the squares of the diagonals is twice the sum of the squares of the sides.
(xi) If z is unimodular then | z | = 1. In case of a unimodular complex number z is taken as z = cosq + i sinq, q Î R.
Note: z / |z| is always a unimodular complex number if z ¹ 0.
(xii) | z1 ± z2 | < | z1 | + | z2 |.
In general | z1 ± z2 ± z3 ± ... ± zn | < | z1 | + | z2 | + | z3 | + ... + | zn |.
(xiii) | z1 ± z2 | > || z1 | – | z2 ||.
(xiv) || z1 | – | z2 || < | z1 + z2 | < | z1 | + | z2 |.
Thus | z1 | + | z2 | is the greatest possible value of | z1 + z2 | and ||z1| – | z2 || is the least possible value of | z1 + z2 |.
Slide25 : De Moivre’s Theorem
(i) Case I : If n is an integer then (cos q + i sin )n has only one value
i.e. = ( cos nq + i sin nq)
Case II : If n is any Rational Number i.e. n Î Q such that
(a) n = p/q (b) p, q Î I (c) q > 0 and
(d) p and q have no common factors ,
then z = (cos + i sin )n = (cos + i sin )p/q
has ‘q’ distinct values one of them is (cos n + i sin n) , where n = p/q.
(ii) (cos - i sin )n = cos n - i sin n .
(iii) 1/cos + i sin = (cos + i sin )-1 = cos - i sin
(iv) (sin ± i cos )n ¹ sin n ± i cos n .
(v) (sin + i cos )n = [cos (p/2 - ) + i sin (p/2 - q)]n = cos (np/2 - nq) + i sin (np/2 - nq).
(vi) (cos + i sin f)n ¹ cos n + i sin nf.
(vii) If z = (cosq1 + i sinq1) (cosq2 + i sinq2) (cosq3 + i sinq3) .............. (cosqn + i sinqn),
Þ z = cos (q1 + q2 + q3 + ... + qn) + i sin (q1 + q2 + q3 + ........... + qn).
or
(viii)
Slide26 : Cube Roots of Unity
Let x = , Þ x3 – 1 = 0, Þ (x – 1) (x2 + x + 1) = 0
\ x = 1,
If second root represented by w then the third root has the value w2.
\ Cube roots of unity are 1, w, w2.
Properties of Cube Roots of Unity
(i) 1 + w + w2 = 0. (ii) w3 = 1. (iii) w = w3
(iv) w3n = 1, w3n + 1 = w, w3n + 2 = w2 ; (v) = w2 and ( )2 = w = 2
(vii) a + bw + cw2 = 0 Þ a = b = c if a, b, c are real.
(viii) 1p + wp + (w2)p =
Slide27 : (v) If a is any (+)ve number, than a1/3 has roots a1/3(1), a1/3(w), a1/3(w2).
If a is any (-)ve number, than a1/3 has roots - | a |1/3, - | a |1/3w, - | a |1/3 w2.
(vi) The cube roots of unity when represented on complex plane represents the vertices of an equilateral triangle inscribed in a unit circle, having centre at the origin and with one vertex being on positive real axis. X X’ Y Y ’ w2 w 1 i -i O -1 X X’ Y Y ’ w2 (1)1/4 1 i -i O -1
Slide28 : Geometrical interpretation of Cube Roots of Unity
Roots of Unity 1 w w2
z1 = 1 + i0
Quadrants positive x-axis II Quadrant III Quadrant
Modulus | z1 | = 1 | z2 | = 1 | z3 | = 1
arg(z) = q arg(z) = 00 arg(z) = 2p/3 arg(z) = - 2p/3
Distances | z2 - z1 | = 3 | z3 - z2 | = 3 | z3 - z1 | = 3
Slide29 : S.I.M.
(i) x2 + x + 1 = (x – ) (x – 2)
(ii) x2 – x + 1 = (x + ) (x + 2)
(iii) x2 + xy + y2 = (x – y) (x – y 2)
(iv) x2 – xy + y2 = (x + y) (x + y 2)
(v) x2 + y2 = (x + iy) (x – iy)
(vi) x3 + y3 = (x + y) (x + y ) (x + y2)
(vii) x3 – y3 = (x – y) (x – y ) (x – y2)
(viii) x2 + y2 + z2 – xy – yz – zx = (x + y + z2) (x + y2 + z)
(ix) x3 + y3 + z3 – 3xyz = (x + y + z) (x + y + 2z) (x + 2y + z)
(x) a2 + b2 + c2 - ab - bc - ca = [(a - b)2 + (b - c)2 + (c - a)2]
(xi) a4 + a2b2 + b4 = (a - b) (a + b) (a - b 2) (a + b 2)
(xii) If a + b + c = 0 then a3 + b3 + c3 = 3abc
(xiii) an - bn = (a - b) (an - 1 + an - 2b + an - 3b2 + ........ + abn - 2 + bn - 1)
(xiv) a4 + a2 + 1 = (a2 + a + 1) (a2 - a + 1)
Slide30 : The nth Roots of Unity
The equation xn = 1 has n roots which are called as the nth roots of unity.
\ xn = 1 = cos 0 + i sin 0 = cos 2kp + i sin 2kp [k is an integer]
Þ x = (cos 2kp + i sin 2kp)1/n =
where k = 0, 1, 2, 3, ..., n – 1
Let a = cos (2p/n) + i sin (2p/n)
Then the nth roots of the unity are at (where t = 0, 1, 2, 3, ..., n – 1).
i.e., The nth roots of unity are 1, a, a2, ..., an – 1 which are in G.P.
Slide31 : Properties of nth Roots of Unity
(i) nth roots unity form a G.P. with common ratio ei(2p/n).
i.e., if a is an imaginary nth root of unity, then other imaginary roots are given by a2, a3, a4, ..., an - 1.
(ii) Sum of nth roots of unity is always zero.
i.e., 1 + a + a2 + ... + an – 1 =
(iii) 1 + a + a2 + ... + an – 1 = 0.
Þ
Þ
(iv)
(v) Sum of pth powers of nth roots of unity is zero, if p is not a multiple of n.
i.e., 1p + ap + (a2)p ..................+ (an - 1)p = 0 or
Slide32 : (vi) Sum of pth powers of nth roots of unity is n, if p is a multiple of n.
i.e., 1p + ap + (a2)p ..................+ (an - 1)p = n or
(vii) Product of nth roots of unity is (-1)n - 1 or (-1)n + 1
i.e., 1 . a . a2 . a3 ......... an - 1 =
(viii) xn - 1 = (x - 1) (x - a) (x - a2) (x - a3) ............................ (x - an - 1)
(ix) If nth roots of unity are plotted on the argand plane then they are representing the vertices of a regular plane polygon of n sides inscribed in a circle of radius one having centre at origin and one vertex being on positive real axis.
Slide33 : Logarithm of a complex number a + ib
\ ln(a + ib) = ln| z | + i (arg z) where z = a + ib (principle argument)
e.g., (a) ln(1 - i) = ln| 1 - i | + i arg (1 - i) = ln2 + i(-p/4)
(b) ii = e - p/2
(c) ln ii = i ln i = i ln(0 + i) =