Class 10 – Complex-Numbers

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Complex-Numbers
  • Introduction to complex numbers
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Complex-Numbers
  • Algebraic Operations on complex numbers
  • Modulus and conjugate
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If a, b are natural numbers such that a > b, then the equation  is not solvable in N, the set of natural numbers i.e., there is no natural number satisfying the equation . So, the set of natural numbers is extended to form the set I of integers in which every equation of the form ; a, bN is solvable.
But equations of the form
, where  are not solvable in I. Therefore, the set I of integers is extended to obtain the set Q of all rational numbers in which every equation of the form  is uniquely solvable.

The equation of the form  etc. are not solvable in Q because there is no rational number whose square is 2. Such numbers are known as irrational numbers. The set Q of all rational numbers is extended to obtain the set R which includes both rational and irrational numbers. This set is known as the set of real numbers. The equation of the form  etc. are not solvable in R i.e., there is no real number whose square is a negative real number.

Euler was the first mathematician to introduce the symbol  (iota) for the square root of –1 with the property  He also called this symbol as the imaginary unit.

So, the necessity to study of COMPLEX NUMBERS arose.

Integral powers of iota (i): For positive integral powers of i:

We have

In order to compute  for n > 4, we divide n by 4 and obtain the remainder r. Let m be the quotient when n is divided by 4. Then,

Thus, the value of  for n > 4 is , where r is the remainder when n is divided by 4.

Negative integral powers of i:

By the law of indices, we have

           

If

 where r is the remainder when n is divided by 4.

Description: http://images.clipartpanda.com/bloc-clipart-note-md.png is defined as 1.

Example: Evaluate the following:

(i)                            (ii)                           (iii)                       (iv)

Solution:
(i) 135 leaves remainder as 3 when it is divided by 4. Therefore,

           

(ii) The remainder is 3 when 19 is divided by 4. Therefore,

           

(iii) We have,

           

On dividing 999 by 4, we obtain 3 as the remainder. Therefore,

So,

(iv) We have,

           

Example:Show that

(i)

(ii)

(iii)

Solution:(i) We have,

           

(ii) We have,

           

(iii) We have,

           

Imaginary quantities: The square root of a negative real number is called an imaginary quantity or an imaginary number.

For example,  etc. are imaginary quantities.

A useful result: If a, b are positive real numbers, then

                       

Proof: We have,

And

Therefore,

           

Description: http://images.clipartpanda.com/bloc-clipart-note-md.pngFor any two real numbers  is true only when at least one of a and b is either positive or zero.

In other words,  is not valid if a and b both are negative.

Description: http://images.clipartpanda.com/bloc-clipart-note-md.pngFor any positive real number a, we have .

Example:Compute the following:

(i)                    (ii)                        (iii)

Solution:(i) We have,

           

(ii) We have,

(iii) We have,

           

Example:A student writes the formula . Then he substitutes a = –1 and
b = –1 and finds 1 = –1. Explain where is he wrong?

Solution:Since a and b both are negative. Therefore,  cannot be written as . In fact for a and b both negative, we have

           

Example:Is the following computation correct? If not give the correct computation:

           

Solution:The said computation is not correct, because –2 and –3 both are negative and  is true when at least one of a and b is positive or zero. The correct computation is as given below:

           

Complex number: If a, b are two real numbers, then the number of the form a + ib is called a complex number.

For example 7 + 2i, –1 + i, 3 – 2i, 0 + 2i, 1 + 0i etc. are complex numbers.

Real and imaginary parts of a complex number: If z = a + ib is a complex number, then ‘a’ is called the real part of z and ‘b’ is known as the imaginary part of z. The real part of z is denoted by Re (z) and the imaginary part by Im (z).

For example, if z = 3 – 4i, then Re (z) = 3 and Im (z) = – 4

Purely real and purely imaginary complex numbers: A complex number z is purely real if its imaginary part is zero i.e., Im (z) = 0 and purely imaginary if its real part is zero i.e., Re (z) = 0.

E.g.: purely real  = 3; purely imaginary = – 4i

Set of complex numbers: The set of all complex numbers is denoted by C.

i.e.,

since a real number ‘a’ can be written as a + 0i, therefore every real number is a complex number. Hence, R C where R is the set of all real numbers.

Equality of complex numbers: Two complex numbers  and  are equal if  i.e.,  and .

Thus,  and .

Example:If  are equal, then find x and y.

Solution:We have,

                       

Example: If  find.

Solution:We have,

             Solving these eqautions, we get

Square root of a complex number:

 The square root of a negative number is called an imaginary number
Example:

Solution:


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to simplify , divide n by 4 and use the remainder to simplify it further.

Change the denominator to power with multiple of 4.