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Home » Algebra » Complex Numbers

Complex Numbers

There are a few sets of numbers which we are already familiar with. These numbers are called real numbers. The aggregate of the sets of rational and irrational numbers is called the set of real numbers. The basic and most important property of any real number is that its square is always positive or non-negative. Because of this property, it was realized that real numbers are not sufficient for solving all algebraic problems. Hence, there was the need to invent a new set of numbers. These numbers are  different from real numbers. This new set of a numbers are called complex numbers or imaginary numbers.

For example, the quadratic equation ax^2+bx+c=0 , where a,b,c \in \mathbb{R} , (where \mathbb{R} represents the set of real numbers), cannot be solved if b^2-4ac<0 , hence the need for complex numbers.

The equation x^2+1=0 cannot be solved in the real number system. Hence Euler introduced the symbol i to represent \sqrt{(-1)} and he said i^2=(-1) .

i is a solution to the equation x^2+1=0 .

Thus it has two solutions, x=\pm i where i=\sqrt{(-1)} .

This number i is called an imaginary number or complex number.

Definition of a Complex Number

If an ordered pair (x,y) of two real numbers x and y is represented by the symbol x+iy where i=\sqrt{(-1)} , then the ordered pair is called a complex or imaginary number.

Thus, a complex number z is defined as an ordered pair of real numbers and written as z=a+ib where a,b \in \mathbb{R} and i=\sqrt{(-1)}

For example, (3+5i), (2-3i) etc. are all complex numbers.

Real and Imaginary parts of Complex Number

If z=a+ib is a complex number, then the real part of z , is denoted by Re(z)=a and the imaginary part is denoted by Im(z)=b

In a complex number z=a+ib  when the real part Re(z)=a is zero or when a=0 , then the number is said to be purely imaginary.

Similarly, in a complex number, when the imaginary part, i.e., Im(z)=b is zero, or when b=0 , then the number is said to be purely real.

For example, 3i=0+i.3 and here a=0, b=3 .

So, 3i is a purely imaginary number.

Again, 5=5+i.0 and here a=5, b=0 .

So 5 is a purely real number.

Negative of Complex Number

If z=a+ib then (-z)=-(a+ib)

\therefore (-z) is the negative of the complex  number z , also called the additive inverse of z.

Powers of i

We already know that i=\sqrt{(-1)} . Similarly we can write the higher powers of  i as follows:

i^{4} =(i^2)^2=(-1)^2=1

i^{5}=i ^{4} \times i=i

i^{6}= i^{4} \times i^{2}=1 \times (-1) =(-1)

i^8=(i^{4})^{2}=1

i^9=i^8 \times i=i

This shows that the values repeat themselves in cycles of four.

Complex Number Operations

1. Addition:

Let there be two complex numbers, z_{1}=a+ib and z_{2}=c+id We add the real parts and imaginary parts separately.

\therefore z_{1} +z_{2}=(a+c)+i(b+d)

For example: Say, z_{1}=2+3i and z_{2}=5+6i

\therefore z_{1} +z_{2}=(2+5)+i(3+6)

\therefore z=z_{1} +z_{2}=7+9i

2. Subtraction:

We change one complex number into its additive inverse and then add the two numbers. Let there be two complex numbers, z_{1}=a+ib and z_{2}=c+id

\therefore z_{1} -z_{2}= z_{1}+ (-z_{2})=a+ib -(-c-id)=(a-c)+i(b-d)

\therefore z=z_{1} -z_{2}=(a-c)+i(b-d)

For example: Say, z_{1}=6+7i and z_{2}=3+2i

\therefore z_{1} -z_{2}=z_{1}+(-z_{2})=6+7i -(-3-2i)=(6-3)+i(7-2)=3+5i

\therefore z=z_{1} -z_{2}=3+5i

3. Multiplication:

Let there be two complex numbers, z_{1}=a+ib and z_{2}=c+id

\therefore z_{1}z_{2}=(a+ib)(c+id)

z_{1}z_{2}=ac+aid+cib+i^{2}bd \\ \vspace{5mm} \\    =ac+i(ad+cb)-bd \\ \vspace{5mm} \\    =(ac-bd) +i(ad+cb)

For example: Say, z_{1}=2+3i and z_{2}=5+4i

\therefore z_{1}z_{2}=(2+3i)(5+4i)

=10+8i+15i+12i^2 \\ \vspace{5mm} \\    =10-12+i(8+15) \\ \vspace{5mm} \\    =(-2+23i)

4. Division:

Let there be two complex numbers, z_{1}=a+ib and z_{2}=c+id . Division of these two complex numbers is defined as:

\dfrac{z_{1}}{z_{2}} =\dfrac{a+ib}{c+id}

We rationalise the denominator, \dfrac{z_{1}}{z_{2}} =\dfrac{(a+ib) \times (c-id)}{(c+id) \times (c-id)}

=\dfrac{(ac-aid+cib-i^{2}db)}{c^2-i^{2}d^{2}} \\ \vspace{5mm} \\    =\dfrac{(ac+bd)-i(ad-bc)}{c^2+d^2}

For example: Say, z_{1}=2+3i and z_{2}=3+4i

\dfrac{z_{1}}{z_{2}}=\dfrac{2+3i}{3+4i} \\ \vspace{5mm} \\    =\dfrac{(2+3i) \times (3-4i)}{(3+4i) \times (3-4i)} \\ \vspace{5mm} \\    =\dfrac{6-8i+9i-12i^2}{9-16i^2} \\ \vspace{5mm} \\    =\dfrac{(6+12)+i(9-8)}{(9+16)} \\ \vspace {5mm} \\    =\dfrac{13+i}{25} \therefore \dfrac{z_{1}}{z_{2}}=\dfrac{13}{25}+\dfrac{i}{25}

5. Conjugate of Complex Number:

When two complex numbers only differ in the sign of their complex parts, they are said to be the conjugate of each other. The conjugate is denoted as \bar{z} .

If z=a+ib, then \bar{z}=a-ib

For example, if z=3+4i , the conjugate of z is \bar{z}=3-4i

Properties of Conjugates:

\overline{(\overline{z})}=z , i.e., conjugate of conjugate gives the original complex number

\overline{(z_{1}+z_{2})}=\bar{z_{1}}+\bar{z_{2}}

\overline{(z_{1}-z_{2})}=\bar{z_{1}}+\bar{z_{2}}

\overline{(z_{1}z_{2})}=\bar{z_{1}} \times \bar{z_{2}}

\overline{(\dfrac{z_{1}}{z_{2}})}=\dfrac{\bar{z_{1}}}{\bar{z_{2}}}

6. Modulus of a Complex Number:

The absolute value or modulus of a complex number, z=a+ib is denoted by \lvert z\rvert\ and is defined as:

\lvert z\rvert\ =\sqrt{a^2+b^2}=\sqrt{{Re(z)}^2+{Im(z)}^2}

Here, a=Re(z), b=Im(z)

For example: If z=2+3i \lvert z\rvert\ =\sqrt{2^2+3^2}=\sqrt{4+9}=\sqrt{13}

\therefore \lvert z\rvert\ =\sqrt{13}

Properties of Modulus:

\lvert -z\rvert\ = \lvert z\rvert\

\lvert z\rvert\ =0 only if z=0 when Re(z)=Im(z)=0

\lvert z_{1}z_{2}\rvert\ = \lvert z_{1}\rvert\  \lvert z_{2}\rvert\

\lvert \dfrac{z_{1}}{z_{2}}\rvert\ =\lvert z_{1}\rvert\ \div \lvert z_{2}\rvert\

\lvert \overline{z}\rvert\ = \lvert z\rvert\

7. Equality of Complex Numbers:

Two complex numbers z_{1}=a+ib, z_{2}=a-ib are said to be equal if and only if a=c and b=d

Illustrations:

1. If x+iy=\dfrac{a+ib}{a-ib} , then prove that x^2+y^2=1

Solution:

x+iy=\dfrac{a+ib}{a-ib}

\therefore \lvert x+iy\rvert\ =\dfrac{\lvert a+ib\rvert\ }{\lvert a-ib\rvert\ }

\Rightarrow \sqrt{x^2+y^2}=\dfrac{\sqrt{a^2+b^2}}{\sqrt{a^2+b^2}}

\Rightarrow \sqrt{x^2+y^2}=1

\Rightarrow x^2+y^2=1 (Proved)

2. Find the real numbers x and y if (x-iy)(3+5i) is the conjugate of (-6-24i)

Solution:

(x-iy)(3+5i)=3x+5xi-3iy-5i^2y=3x+5y+i(5x-3y)

The conjugate of (-6-24i) is (-6+24i)

According to the problem,

3x+5y+i(5x-3y) =-6+24i

\therefore 3x+5y=-6...(1)

\therefore 5x-3y=24...(2)

Solving equations (1) and (2) we get,

x=3 and y=-3 (Answer)

3. Find the values of x and y if the complex numbers z_{1}=6y+5i and z_{2}=12+10xi are equal

Solution:

z_{1}=z_{2}

\Rightarrow 6y+5i=12+10xi...(1)

\therefore Re(z_{1})=Re(z_{2})

and Im(z_{1})=Im(z_{2})

\therefore from (1) we have,

6y=12 \Rightarrow y=2

10x=5 \Rightarrow x=\dfrac{1}{2}

Answer: x=\dfrac{1}{2} \: y=2

4. Find the modulus of the following:

  1. \dfrac{3}{5} +\dfrac{4}{5}i
  2. \dfrac{\sqrt{7}}{2} -\dfrac{3}{2}i

Solutions:

4a) z_{1} =\dfrac{3}{5}+\dfrac{4}{5}i

\therefore modulus of z_{1} is given by \lvert z_{1}\rvert\ =\lvert \dfrac{3}{5}+ \dfrac{4}{5}i\rvert\ =\sqrt{(\dfrac{3}{5})^2+(\dfrac{4}{5})^2}=\sqrt{\dfrac{9}{25}+\dfrac{16}{25}}

=\sqrt{\dfrac{25}{25}}=1

4b) z_{2} =\dfrac{\sqrt{7}}{2}+\dfrac{3}{2}i

\therefore modulus of z_{2} is given by \lvert z_{2}\rvert\ =\lvert \dfrac{\sqrt{7}}{2} +\dfrac{3}{2}i\rvert\ =\sqrt{(\dfrac{\sqrt{7}}{2})^2+(\dfrac{-3}{2})^2}=\sqrt{\dfrac{7}{4}+\dfrac{9}{4}}

=\sqrt{\dfrac{16}{4}}=\sqrt{4}=2

5. If z=x+iy and \lvert z-1\rvert\ + \lvert z+1\rvert\ =4 , show that, 3x^2+4y^2=12

Solution:

\lvert z-1\rvert\ + \lvert z+1\rvert\ =4 or, \lvert x+iy-1\rvert\ +\lvert x+iy+1\rvert\ =4

or, \lvert (x-1)+iy\rvert\ +\lvert (x+1)+iy\rvert\ =4

or, \sqrt{(x-1)^2+y^2} +\sqrt{(x+1)^2+y^2} =4

or, \sqrt{(x-1)^2+y^2} = 4- \sqrt{(x+1)^2+y^2}

or, (x+1)^2+y^2=16+(x-1)^2+y^2-8\sqrt{(x-1)^2+y^2}

or, 8\sqrt{(x-1)^2+y^2} =16+(x-1)^2+y^2-(x+1)^2-y^2=16-4x

[\because (a+b)^2-(a-b)^2=4ab ]

or, 2\sqrt{(x-1)^2+y^2} =4-x

or, 4[(x-1)^2+y^2]=(4-x)^2 [ squaring  both sides ]

or, 4(x^2-2x+1+y^2) =16+x^2-8x

or, 3x^2+4y^2=12

Exercise:

  1. If a+ib=\dfrac{(x+i)^2}{2x-i} , then prove that a^2+b^2=\dfrac{(x^2+1)^2}{4x^2+1}
  2. If x+iy=\sqrt{\dfrac{1+i}{1-i}} , then prove that x^2+y^2=1
  3. Express \dfrac{1+7i}{(2-i)^2} in the form a+ib . Also find the conjugate and modulus of it.
  4. Find the modulus of 2\sqrt{2}-\dfrac{13}{\sqrt{7}}i
  5. Find the absolute value of (-3+4i)
  6. Find x and y if (3x-7)+5iy=2y+3-4(1-x)i
  7. Simplify 1+i^2+i^4+i^6
  8. Show that [i^{37}-(\dfrac{1}{i})^41]^3=-8i
  9. For any complex number, z , prove that: Re(z)=\dfrac{z+\overline{z}}{2} and Im(z)=\dfrac{z-\overline{z}}{2i}
  10. For any two complex numbers z_{1} and z_{2} , prove that: Re(z_{1}z_{2})=Re(z_{1})Re(z_{2})-Im(z_{1})Im(z_{2})
  11. Find the real values of x and y for which \dfrac{x-1}{3+i} +\dfrac{y-1}{3-i}=i
  12. If z_{1}=2-i and z_{2}=1+i find \lvert \dfrac{z_{1}+z_{2}+1}{z_{1}-z_{2}+i}\rvert\

 

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Filed Under: Algebra Tagged With: Complex Numbers, imaginary numbers, numbers, real numbers

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Table of Content

  • Complex Numbers
  • Quadratic Equations
  • Logarithm
  • Permutation
  • Combination
  • More on Complex Numbers
  • Classification of Numbers
  • Positive and Negative Quantities
  • Understanding Simple Algebraic Formulas With Examples
  • Integers
  • Linear Inequalities
  • An Introduction to Fundamental Algebra
  • Basic Number Properties – Commutative, Associative and Distributive
  • Algebraic Multiplication and Division
  • Simple Equations in One Variable
  • Simple Formulae and their Application
  • Rational and Irrational Numbers
  • Problems Leading to Simple Equations
  • Simultaneous Equations
  • Mathematical Induction
  • Different Type of Sets
  • Indices
  • Framing Formulas
  • Sequences
  • Introduction to Matrices
  • Addition Of Matrices
  • Subtraction Of Matrix
  • Multiplication of Matrices
  • Determinant of Matrices
  • Co-factor of Matrices
  • Minor of Matrices
  • Transpose and Adjoint of Matrices
  • Inverse of a Matrix
  • System of Linear Equations in Matrices
  • Introduction to Polynomials
  • Classification of Polynomials
  • Addition and Subtraction of Polynomials
  • Multiplication of Polynomials
  • Factoring Polynomials
  • Zeroes of Polynomial
  • Remainder Theorem of Polynomials
  • Factor Theorem of Polynomial
  • Simplifying Polynomial Fractions
  • Roots of a Polynomial
  • Addition of Polynomial Fractions
  • Subtraction of Polynomial Fractions
  • Multiplying polynomial fractions
  • Division of Polynomial Fractions

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