The Idea behind ‘Imaginary Numbers’
The set of natural numbers is
This set of numbers does not have a solution to equations of the form
Thus, we defined the set of whole numbers as
and the set of integers as
Now, we do not have solution to the equations of the kind , when is not completely divisible by . Hence, we defined rational numbers. Their set is given by
Still, we do not have solutions to equations of the form , when is not a perfect power. Thus, we defined irrational numbers. Combining the rational and irrational numbers, we get the set of real numbers, .
However, when and is an even power, we do not have solutions to such equations in . Therefore, we define a new , which is equal to . It is denoted by . Given this, we can solve any equation.
This number is an imaginary number. Thus, .
A number of the form , where and is a complex number. The set of complex numbers is denoted by .
Defining the complex numbers completes the system of numbers; i.e. given an equation in a single variable, we are sure that its solution will always be there in .
is the real part and is the imaginary part.
Algebra of Complex Numbers
Consider 2 complex numbers and .
I) Equality : They are equal only when and
II) Addition and Subtraction :
III) Multiplication :
IV) Division : is done by multiplying numerator and denominator by complex conjugate of .
Complex conjugate of is .
V) Addition and multiplication are commutative and associative. Also, multiplication distributes over addition.
On plane, complex number is represented by a point whose coordinates are . The diagram representing the complex numbers is known as the Argand diagram.
In polar coordinates, and .
is known as the absolute value (or modulus) and is known as the amplitude or argument of the complex number.
The exponential representation of is due to Euler.
Since is periodic, we will consider as the principal value of argument.
Thus, and , and .
Let’s solve the following problem :
Show that for any complex number ,
De Moivre’s Theorem
For any real number ,
Applications of de Movire’s Theorem
In general, if , then . We say that is the th root of . Further, th root of any number has values (which may not be always real).
For example, has 2 roots , and .
The theorem can be used to get th roots of any number. This can be done by considering the periodicity of trigonometric functions. We know that
By substituting successive values of from till , we get different values of
This trick can be used to obtain the roots of an equation.
Complex Cube Roots Of Unity
In general, if the highest power of the variable in an equation is , there will be roots. Consider the equation
Clearly, is a root. The other 2 roots are given by solving the quadratic equation below:
Note, and and .
When plotted on Argand diagram, the cube roots of unity produce an equilateral triangle, with as a vertex.
In general, if we are finding th roots of unity, we will get a regular polygon of sides and will always be a vertex. This is because is always a root of .
Solved Example : Use de Moivre’s theorem to solve the equation .