Complex Numbers: Definition, Formulas & Calculations

A complex number is a number which is of the form a + ib, where a and b are real numbers, and “i” is called the imaginary number. It is also known as “iota”.

Let a complex number be of the form z= a+ib, then a is the real part which is denoted by Re z, and b is called the imaginary part, signified by the Im z of the complex number z.

For example, if z= 4+i7, then Re z=4 and Im z = 7.

Read Also: Algebra Formulas for Class 10

Two complex numbers of the form, z1 = a + ib and z²= c + id are called equal if a = c and b = d

a + ib

Key Terms: Real numbers, imaginary number,complex numbers ,quadratic equation.

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What are Real Numbers?

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Real numbers are the numbers that consist of all the numbers present in the number system i.e., fractions, irrational numbers, positive, rational numbers, negative, , integers, zero, etc. 

For example, 0. 7, -9, 10/15, 7.9, √-11

The video below explains this:

Complex Numbers Detailed Video Explanation:

Also Read: Square root

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What are Imaginary Numbers?

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All the numbers except real numbers are called an imaginary number. The square of imaginary numbers is a negative number. For example, √-48, √-23 etc

Some examples of the complex numbers are as follows:

Read More: Root 2 is Irrational

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Can 0 be a complex number?

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O is a real number, and all real numbers are a part of complex numbers. This means that 0 is a complex number of the form 0+0i. 

Algebra of Complex Numbers

Addition:

Let z1= a + ib and z2 = c + id be two complex numbers. 

Then, z_1 + z_2= (a + c) + i (b + d)

For example, z1= 3+i8; z2 = -5-8i

So,z1 + z2 = (3-5) + i (8-8)

z1 + z2 = -2+0i

Substraction:

Let z1= a + ib and z2 = c + id be two complex numbers. 

Then, z1 - z2 = z1 +(- z2)

For example, z1= 5+i7 and z2= -9+i4

So, z1 - z2 = 5+i7-(-9+i4)

=5+i7+9-i4

= (5+9) + I (7-4)

=14+i3

Check Important Notes for Euclid’s Division Lemma

Multiplication:

Let z1= a + ib and z2 = c + id be any two complex numbers. 

Then, z1z2= (ac – bd) + i(ad + bc)

For example, z1=7+i9 and z2=1+i6

So, z1z2= (7+i9)(1-i6)=(7×1-9×6) + i (7×6+9×1)

= (7-54) + i (42+9)

= -47 + i51

Division:

Let z1= a + ib and z2 = c + id be two complex numbers where z2 ≠ 0, then the quotient z¹/z² is

= (ac + bd)/ (c2 + d2) + i (bc – ad) / (c² + d²)

For example, 

Z1= 4+i5

Z2= 7+i9

So, z1/z2 = ( (4x7 + 5x9))/[(7)(7)+(9)(9)] + i [ ((5X7-4X9))/[(7)(7)+(9)(9)] ]

= [(28+45)/(49+81)]+i [(35-36)/(49+81)]

= [73/130]+ i[-1/130]

=1/130(73-i1)

Power of i (iota)

Read Further: Fundamental Theorem of Arithmetic

We know that, 

I3= i2i =(-1)i , i4= (i2)2=(-1)2=1

Also,

i-1=1/ixi/i = i/(-1)= -i

We can conclude that, for any integer k, i4k=1, i4k+1=i, i4k+2=1, i4k+3= -i

Square root of a negative real number:

Also, i2 = -1 and (-i)2 = i2= -1

Therefore, the square roots of -1 are i and -i

The symbol √(-1) would mean i only.

If a is a positive real number, thens √(-a) = √a √(-1) = √a i

Note that,

i2=√(-1) √(-1)=√((-1)(-1)) [Assuming that √a × √(b )= √ab for all real numbers)

= √1 = 1 which is a contradiction to the fact that i2 = -1

Therefore, √a × √(b ) ≠ √ab if both a and b are negative real numbers.

Identities

(z1 + z2)2 = (z1)2 + (z2)2 + 2 z1 × z2

(z1 – z2)2 = (z1)2 + (z2)2 – 2 z1 × z2

(z1)2 – (z2)2 = (z1 + z2) (z1 – z2)

(z1 + z2)3 = (z1)3 + 3(z1)2 z2 +3(z2)2 z1 + (z2)3

(z1 – z2)3 = (z1)3 – 3(z1)2 z2 +3(z2)2 z1 – (z2)3

Also Check: Complex Numbers and Quadratic Equations

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The Modulus and Conjugate of a Complex Real Number

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Let z = a + ib be a complex number. Then, the modulus of z which is denoted by | z |, is known as the non-negative real number √(a^2+ b^2 )

| z | = √(a^2+ b^2 )

and the conjugate of z which is denoted as ¯z is the complex number a – ib, is also written as ¯z = a – ib

Argand Plane and Polar Form

The complex number x + iy having the ordered pair (x, y) can be represented in the graph as the unique point T (x, y) in the XY-plane and vice-versa.

For example, Complex numbers such as 2+1i,0+3i,2+4i,2-3i,-6-5i and -3+4i which correspond to the ordered pairs (2,1), (0,3), (2,4), (2,-3),(-6,-5) and (-3,4) are represented in the graph by points A,B,C,D,E and F respectively.

The plane which has a complex number assigned to a point is called the Argand plane.

There are two axes in the Argand plane. They are as follows:

The real axis is called the horizontal axis.

The imaginary axis is called the vertical axis.

Let “Q” be the non-zero complex number x + iy. OQ is the line segment of length s and makes an angle θ with the positive direction of the X-axis.

Ordered pair (s, θ) is called the polar coordinates of the point Q, as the point “Q” is represented by the points (s, θ). The origin( 0,0) is called the pole and the positive X-axis is called the initial line.

Check Also: Degree of Polynomial

Then,

x = s cosθ

y = s sinθ

We can write z = x + iy as z = s cosθ + i s sinθ = r(cosθ + i sinθ), which is the polar form of complex numbers.

Here, s = |z|= √(\( x^2\)+\(y^2\) )is modulus of z and θ is known as the argument or amplitude of z denoted as arg z

For any non-zero complex number z, there is one value of θ corresponding to z, in the interval [0, 2π)

Read more: Zeros of Polynomial

In any other interval of length 2π, consider the interval -π < θ ≤ π, then the value of θ is called the principal argument of z.

Example,

Representing z=1+i√3 in the polar form.

Let 1 = r cos θ and √3 = r sin θ

By squaring and adding,

\(r^2\) (\(cos^2\) θ+\(sin^2\) θ) = 4

r=√4=2

Therefore, cosθ= 1/2 , sinθ= √3/2, which gives θ= π/3

So, required polar form is z=2(cos π/3+i sin[π/(3 )])

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Quadratic equations

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Let there be a quadratic equation of the form a\(x^2\)+bx+c=0 with coefficients a,b,c which are real numbers and a≠0

Also, let us assume \(b^2\)-4ac<0

So, the solution to the above equation is given by:

x=(-b±√(\(b^2\)-4ac))/2a = (-b±√(4ac-\(b^2\) ))/2a i

Also Read More: Cubic Polynomial

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Sample Questions