Algebra of Complex Numbers Physics Notes | EduRev

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What is Complex Number?

Complex Number is an algebraic expression including the factor i = √-1. These numbers have two parts, one is called as the real part and is denoted by Re(z) and other is called as the Imaginary Part. Imaginary part is denoted by Im(z) for the complex number represented by ‘z’. 

Either of the part, real part or imaginary part, can be positive, negative, integer, fraction, decimal, rational, irrational or even zero. If only real part of any complex number ‘z’ is zero, i.e. Re(z) = 0, then these types of complex numbers are termed as ‘Purely Imaginary Number’. While if only imaginary part of any complex number ‘z’ is zero, that is. Im(z) = 0, then these are called as ‘Purely Real Numbers”.

Complex Number in its Cartesian form is expressed as z = a + ib or z = Re(z) + iIm(z).

Algebra of Complex Numbers Physics Notes | EduRev

For Example, for a complex number, z = 2 + 3i, a = Re(z) = 2  and b = Im(z) = 3.

Complex Number in its Cartesian form is expressed as z = a + ib or z = Re(z) + iIm(z). 
What are Imaginary Numbers in Math?

Imaginary Numbers are the real numbers multiplied with the imaginary unit ‘i’. ‘i’ (or ‘j’ in some books) in math is used to denote the imaginary part of any complex number. It helps us to clearly distinguish the real and imaginary part of any complex number. Moreover, i is just not to distinguish but also has got some value.

i = √-1

Main application of complex numbers is in the field of electronics. In electronics, already the letter ‘i’ is reserved for current and thus they started using ‘j’ in place of i for the imaginary part.

Algebra of Complex Numbers Physics Notes | EduRev


Explain Algebra of Complex Numbers?

Let’s understand the different algebras of complex number one by one below.

Equality of Complex Numbers

Two Complex numbers z1 and z2 are equal iff, 

Condition 1) Re (z1) = Re (z2)

Condition 2) Im (Z1) = Im(z2)

So If, z1 = x + 3i and z2 = -2 + yi are equal, then as per above conditions,
Re(z1) = x and Re(z2) = -2, so x = -2
And Similarly
Im(z1) = 3 and Im(z2) = y, so y = 3

Addition of Complex Numbers

Let z1 = a + ib and  z= c + id, then the sum of this two complex numbers that is z1+ zcalculated as:
z1+ z2 = (a + ib) + (c + id)
=(a + c) + i(b + d)
Therefore,
z1 + z2 = Re (z1+ z2) + Im(z1+ z2)

Addition of complex numbers can be another complex number.

Algebra of Complex Numbers Physics Notes | EduRev

Example
Let z1= -1 + 4i and z2 = 8 + 2i,
Then z1+ z2 = (-1 + 8) + i(4 + 2) =7+ i6

Addition of complex numbers satisfy the following properties:

  • Closure Law: The sum of two complex numbers is another complex number, that is. if z1 + z2 where  z1 and  z2 are complex numbers, then z will also be a complex number
  • Commutative Law: As per commutative law, for any two complex numbers z1 and  z2, z1 + z2 = z2 + z1.
  • Associative Law: For any three complex numbers say (z1+ z2 )+ z3 = z1+ (z2+ z3).
  • Existence of Additive Identity: Additive identity also called as zero complex number is denoted as 0 (or 0 + i0), such that, for every complex number z, z + 0 = z.
  • Existence of Additive Inverse: Additive inverse or negative of any complex number z, is a complex number whose both real and imaginary parts have the opposite sign. It is represented by –z and z + (-z) = 0

Difference of two Complex Numbers

Let  z1= a + ib and  z2 = c + id, then the difference of this two complex numbers that is. z1 -  z2 is calculated as:
z1- z2= (a + ib) - (c + id)
= (a – c) + i (b – d)
Therefore,
z1 - z2 = Re(z1 - z2 ) + Im(z1 - z2)
Difference of complex numbers can be another complex number

Example

Let z1= -1 + 4i and z2 = 8 + 2i,
Then z1- z2 =(-1 -8) + i(4 – 2) = -9 + i2
Difference of two complex numbers also satisfies the same properties as the addition of the two follows.

Multiplication of two Complex Numbers

Let z1= a + ib and z2 = c + id, then the multiplication of this two complex numbers that is. z1× z2  is calculated as:
z1× z2 = (a + ib) ×(c + id)
z1×z2= (ac – bd) + i(ad + bc)
Therefore,
z1 × z2 = [Re(z1) Re(z2) – Im(z1) Im(z2)]+ i[Re(Z1) Im(z2) + Im(z2) Re(z2)]

Algebra of Complex Numbers Physics Notes | EduRev

Example
Let z1= -1 + 4i and z2 = 8 + 2i,
Then, z1 × z2 = (-8 -8) + i(-2 + 32) =-16 + i30
If k is any constant, then
kz = k(a + ib) = ka + ikb
Also, if kand k2 are any real constant, then
k(z1 + z2 )= kz1+ kz2
k1 (k2 z)=(k1 k2 )z
(k1+ k2 )z=k1 z+k2z

Multiplication of two complex numbers also posses few properties, let’s list them all here below:

  • Closure Law: The product of any two complex numbers is another complex number, that is. if z = z1- z2 where z1 and zare complex numbers, then z will also be a complex number
  • Commutative Law: As per commutative law, for any two complex numbers z1 and z2, z1 – z2 = z2 z1.
  • Associative Law: For any three complex numbers say z1, z2 and z3. (z1 z2 ) z3 = z1 (z2 z3).
  • Multiplicative Identity: Multiplicative Identity  is denoted as 1 (or 1 + i0), such that, for every complex number z, z .1 = z.
  • Multiplicative Inverse: For any non- zero complex number z,1/z or z-1 is called as ssthe multiplicative inverse as z,1/z = 1 If z = x + iy, then

Algebra of Complex Numbers Physics Notes | EduRev

  • Distributive Law:  For any three complex numbers z1, z2 and z3 we have

z1 (z2+ z3 )= z1 z2+ z1 z3

(z1+ z2 ) z3 = z1 z3 + z2 z

Division of two Complex Numbers

Let z1 = a + ib and  z2 = c + id, then the division of this two complex numbers that is z1/z2 is calculated as:

Algebra of Complex Numbers Physics Notes | EduRev

On Rationalization:

Algebra of Complex Numbers Physics Notes | EduRev

Example

Let   z1 = -1 + 4i and z2 = 8 + 2i,

Algebra of Complex Numbers Physics Notes | EduRev

Algebra of Complex Numbers Physics Notes | EduRev

Power of i

Since i = √-1 or  i = -1 which means i can be assumed as the solution of the equation x2 + 1 = 0 .i is called as Iota in complex numbers.

We can further formulate as,

Algebra of Complex Numbers Physics Notes | EduRev

So we can say now, i4n where n is any positive integer.

Algebra of Complex Numbers Physics Notes | EduRev

Also, 

 Algebra of Complex Numbers Physics Notes | EduRev

Also note that  i + i2 + i3 + i4 = 0 or i4n+1 + i4n+2 + i4n+3 = 0 for any integer n.


Square root of a Complex Number

Let  z1 = (a + ib) then the square root of a complex number z1, that is √z1 can be calculated as follows:

Assume, √z1 = x + iy

that is √(a + ib) = x + iy, Now squaring both the sides,

On simplification we get

(a + ib) = (x2- y2 )+ 2xyi

Now comparing both sides real and imaginary parts, we get

a = (x2- y2 ) and b = 2xy 

Now using the below identity:

(x2- y2) = (x2 + y2) - 4xy, find the value of x2 + y2

 And then finally find the values of x2 and y2,we get

Algebra of Complex Numbers Physics Notes | EduRev

On further simplification, get the value of x and y by taking square root both sides,
Finally we get,

Algebra of Complex Numbers Physics Notes | EduRev

Example
Let’s find the square root of 8 – 6i.
Assume, √(8 – 6i) = x + iy On squaring and simplifying, we get
x2- y2 = 8 and 2xy = – 6
And finally we get, √(z1) = x+ iy = ± (3 – i)

What does the asterisk in Complex Numbers mean?

Asterisk (symbolically *) in complex number means the complex conjugate of any complex number.
Let z1 = x + iy is any complex number, then its complex conjugate is represent by Algebra of Complex Numbers Physics Notes | EduRev

We can also define the complex conjugate of any complex number as the complex number with same real part and same magnitude of imaginary part but with opposite sign as of given complex number.

Refer the below table to understand it more clearly

Algebra of Complex Numbers Physics Notes | EduRev
Algebra of Complex Numbers Physics Notes | EduRev

Also note few important properties of conjugate:

  • Algebra of Complex Numbers Physics Notes | EduRev
  • Algebra of Complex Numbers Physics Notes | EduRev
  • Algebra of Complex Numbers Physics Notes | EduRev
  • Algebra of Complex Numbers Physics Notes | EduRev
  • Algebra of Complex Numbers Physics Notes | EduRev
  • Algebra of Complex Numbers Physics Notes | EduRev
  • Algebra of Complex Numbers Physics Notes | EduRev

Algebra of Complex Numbers Physics Notes | EduRev

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