Algebra of Complex Numbers

Complex Numbers

A. Definition
Complex numbers are defined as expressions of the form a + ib where a , b ∈ R &   It is denoted by z i.e. z = a + ib . ‘a’ is called as real part of z (Re z) and ‘b’ is called as imaginary part of z (Im z).

Every Complex Number Can Be Regarded As

Purely real  -  Purely imaginary - Imaginary
if b = 0    -   if a = 0  -  if b ≠ 0

Remark :
(a) The set R of real numbers is a proper subset of the complex numbers . Hence the complete number system is
(b) Zero is both purely real as well as purely imaginary but not imaginary .
(c)  is called the imaginary unit . Also i² = - l ; i3 = -i ; i4 = 1 etc.
(d)  only if atleast one of either a or b is non - negative.

B. Algebraic Operations
The algebraic operations on complex numbers are similar to those on real numbers treating ‘i’ as a polynomial . Inequalities in complex numbers are not defined . There is no validity if we say that complex number is positive or negative.

e.g. z > 0 , 4 + 2i < 2 + 4 i are meaningless.

However in real numbers if a2 + b2 = 0 then a = 0 = b but in complex numbers,

z12 + z22 = 0 does not imply z1 = z2 = 0.

Equality In Complex Number :

Two complex numbers  are equal if and only if their real & imaginary parts coincide.

C.Conjugate Complex

If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by

Remark :
(i)

(ii)

(iii)  which is real
(iv)If z lies in the 1st quadrant then  lies in the 4th quadrant and -  lies in the 2nd quadrant.

Ex.1 Express (1 + 2i)2/(2 + i)2 in the form x + iy.

Sol.

Ex.2 Show that a real value of x will satisfy the equation

Sol.

We have

or   [by componendo and dividendo],

Therefore, x will be real, if

Ex.3 Find the square root of a + ib

Sol.

Let  = x + iy, where x and y are real. Squaring, a + ib =
Equating real and imaginary parts,

Now (x2 + y2)2 = (x2 – y2)2 + 4x2y2 = a2 + b2  or  x+ y2 = √(a2 + b2 )...(iii)

[ x and y are real, the sum of their squares must be positive]
From (i) and (iii),

If b is positive, both x and y have the same signs and in opposite case, contrary signs. [by (ii)].

The document Algebra of Complex Numbers | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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## Mathematics (Maths) for JEE Main & Advanced

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## FAQs on Algebra of Complex Numbers - Mathematics (Maths) for JEE Main & Advanced

 1. What is the algebraic representation of a complex number?
Ans. A complex number can be represented algebraically as a sum of a real part and an imaginary part. It is written in the form a + bi, where a is the real part and bi is the imaginary part.
 2. How do you add complex numbers?
Ans. To add complex numbers, simply add the real parts and the imaginary parts separately. For example, to add (3 + 2i) and (1 + 4i), you would add 3 + 1 = 4 for the real parts, and 2i + 4i = 6i for the imaginary parts. Therefore, the sum is 4 + 6i.
 3. How do you multiply complex numbers?
Ans. To multiply complex numbers, you can use the distributive property and the fact that i^2 = -1. For example, to multiply (3 + 2i) and (1 + 4i), you would multiply each term: (3 + 2i)(1 + 4i) = 3(1) + 3(4i) + 2i(1) + 2i(4i) = 3 + 12i + 2i + 8i^2 = 3 + 14i + 8(-1) = 3 + 14i - 8 = -5 + 14i.
 4. How do you find the conjugate of a complex number?
Ans. The conjugate of a complex number is found by changing the sign of the imaginary part. For example, the conjugate of 3 + 2i is 3 - 2i. The conjugate of a complex number is useful in various operations, such as dividing complex numbers or simplifying expressions.
 5. How do you represent complex numbers in polar form?
Ans. Complex numbers can also be represented in polar form, which involves expressing them in terms of a magnitude (or modulus) and an angle. The magnitude is the distance from the origin to the complex number in the complex plane, and the angle is the angle formed with the positive real axis. The polar form of a complex number is written as r(cosθ + isinθ), where r is the magnitude and θ is the angle.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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