Complex numbers formula sheet for 11th or 12th grade PDF

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COMPLEX NUMBERS COMPLEX NUMBERS If ‘a’, ‘b’ are two real numbers, then a number of the form a + ib is called a complex number Set of complex Numbers : The set of all complex numbers is denoted by C. i.e. C = {a + ib | a,b  R } Equality of Complex Numbers : Two complex numbers z1 = a 1 + ib1 and z2 = a 2 + ib2 are equal if a 1 = a 2 and b1 = b2 i.e. Re (z1) = Re(z2) and Im (z1) = Im (z2)

FUNDAMENTAL OPERATIONS ON COMPLEX NUMBERS ADDITION : Let z1 = a 1 + ib1 and z2 = a 2 + ib2 be two complex numbers. Then their sum z1 + z2 is defined as the complex number (a 1 + a 2) + i (b1 + b2) Pr oper ties of addition of complex number s (i)

Addition is commutative : For any two complex numbers z1 and z2, we have z1  z2  z 2  z1

(ii) Addition is associative : For any three complex numbers z1, z2, z3 we have (z + z ) + z = z + (z + z ) (iii) Existence of a

ment for addition i.e.

(iv) Existence of

h that

The complex number –z is called the additive inverse of z. Substraction : Let z1 = a 1 + ib1 and z2 = a 2 + ib2 be two complex numbers. Then the subtraction of z2 from z1 is denoted by z1 – z2 and is defined as the addition of z1 and –z2. Thus, z1 – z2 = (a 1 – a 2) + i (b1 – b2) Multiplication : Let z1 = a 1 + ib1 and z2 = a 2 + ib2 be two complex numbers. Then, the multiplication of z1 with z2 is denoted by z1z2 and is defined as the complex number. (a 1a 2 – b1 b2) + i (a 1b2 + a 2b1) Pr oper ties of Multiplication : (i)

Multiplication is commutative. For any two complex numbers z1 and z2, we have z1 z2 = z2 z1 (ii) Multiplication is associative : For any three complex numbers z1, z2, z3 we have (z1 z2) z3 = z1 (z2 z3) (iii) Existence of identity element for multiplication. The complex number 1 = 1 + i0 is the identity element for multiplication i.e. for every complex number z, we have z.1=z (iv) Exitence of multiplicative inverse : Corresponding to every non-zero complex number z = a + ib there exists a complex number z1 = x + iy such that z . z1 = 1  z1 

1 z [1]

[2]

Complex Numbers

The complex number z1 is called the multiplicative inverse or reciprocal of z and is given by

a i( b)  2 2 a b a  b2 (v) Multiplication of complex numbers is distributive over addition of complex numbers : For any three complex numbers z1, z2, z3 we have z1 

2

(i)

z1 (z2  z3 )  z1z 2  z1z 3

(Left distributivity)

(ii)

(z 2  z 3 )z1  z 2 z1  z3 z1

(Right distributivity)

Division : The division of a complex number z1 by a non-zero complex number z2 is defined as the multiplication z1 of z1 by the multiplicative inverse of z2 and is denoted by z . 2

1  z1  z1 .z2 1  z1.   z2 z 2 

Thus,

Conjugate : Let z = a + ib be a complex number. Then the conjugate of z is denoted by z and is equal to a – ib. Thus, z = a + ib  z  a  ib Pr oper ties of Conjugate : If z, z1, z2 are complex numbers, then (i)

z  z  2Re(z)

(ii)

z  z  2 Im(

(iii) z  z  z is (iv) z  z  0  z (v)

zz  {Re(z)}

(vi) z1  z2  z1  z2 (vii) z1  z2  z1  z2 (ix) z1 z2  z1 z2 (x)

 z1   z2

 z1   ,z 2  0  z2

(xi) (z)  z

MODULUS OF A COMPLEX NUMBER Definition : the modulus of a complex number z = a + i b is denoted by |z| and is defined as | z |  a2  b2  {Re(z)}2  {Im(z)}2



The multiplicative inverse of a non-zero complex number z is same as its reciprocal and is given by Re(z) (  Im(z)) z i  | z |2 | z |2 | z |2

[3]

Complex Numbers

If b is positive  1  a  ib     2

then





2 2 a b a i

 1 2 2 { a  b  a} 2 

If b is negative then  1  1 a  ib    {| z |  Re(z)}  i {| z |  Re(z)} 2  2 

Ar gument or (amplitude) of a Complex Number 1

y x

(i)

If x and y both are positive, then the argument of z = x + iy is the acute angle given by tan

(ii)

x < 0 and y > 0, then the argument of z = x + iy is    , where  is the acute angle given by tan–1 |y/x|.

(iii) If x < 0 and y < 0 then the argment of z = x + iy is    where  is the acute angle given by tan 

y . x

y (iv If x > 0 and y 0, k  1 is a real number then | z  z |  k represents a circle. For 2 k = 1 it repre (iii) Let z1 and z2 (a) If k | z of majo (b) If k = |z1 – z2| represents the line segment joining z1 and z2.

z1) and B(z2) and length

(c) If k < |z1 – z2| then | z  z 1 |  | z  z 2 | k does not represent any curve in the argand plane. (iv) Let z1 and z2 be two fixed points, k be a positive real number. (a) If k < |z1 – z2|, then | z  z 1 |  | z  z 2 |  k represents a hyperbola with foci at A(z1) and B(z2). (b) If

k = (z1 – z2), then (z – z1) – (z – z2) = k represents the straight line joining A(z1) and B(z2) excluding the segment AB. (v) If z1 and z2 are two fixed points, then |z – z1|2 + |z – z2|2 = |z1 – z2|2 represents a circle with z1 and z2 as extremities of a diameter. (vi) Let z1 and z2 be two fixed points and  be a real number such that 0     then (a) If 0     and



 then arg 2

 z  z1      represents a segment of the circle passing through A(z1) and B(z2)  z  z2  P  A(z1)

B(z1 )

[9]

Complex Numbers

 z  z1   (b) If  2   / 2, then arg  z  z     2 represents a circle with diameter as the segment joining 2   A(z1 ) and B(z2 ) .

 z  z1  (c) if    then arg     represents the straight line joining A(z1) and B(z2) but excluding  z  z2  the segment AB. A(z 1)

B(z1 )

 z  z1  (d) If   0, then arg  z  z   ( 0)  2  A(z1)

represents the segment joining A(z1) and B(z2)

B(z 1)...