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COMPLEX NUMBERS
Class 10 Notes
Introduction: 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 , b N 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.
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,
For 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.
For 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 – 4 i , 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:
to simplify , divide n by 4 and use the remainder to simplify it further.
Change the denominator to power with multiple of 4.
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