A complex number is a number that can be written in the form z = a + ib, where a is the real part, b is the imaginary part, and i is the imaginary unit defined by i² = −1.
In simple terms, a complex number is the combination of a real number and an imaginary number written in standard form. Operations on complex numbers are similar to operations on polynomials.
Addition and Subtraction of Complex Numbers
The addition and subtraction of complex numbers are fundamental operations. Similar to polynomials, we combine like terms. That is, we add or subtract the real parts together and the imaginary parts together.
Let z₁ = a + ib and z₂ = c + id, where a, b, c, and d are real numbers.
Real parts: 12 = 12 Imaginary parts: −2 = −2 Since both parts are equal, z₁ = z₂
Example: Find x and y if x + yi = 2y − (3x − 7)i.
Solution:
Since two complex numbers are equal, their real parts and imaginary parts must be equal.
Compare real parts: x = 2y … (1)
Compare imaginary parts: y = −(3x − 7) … (2)
Substitute x = 2y into (2): y = −(6y − 7) y = −6y + 7 7y = 7 ⇒ y = 1
From (1): x = 2y = 2
So, x = 2 and y = 1.
Dividing complex numbers
You probably already know that you can write an expression like 2 / (3 - √2) as a fraction with a rational denominator by multiplying the numerator and denominator by 3 + √2.
Because z\bar{z} is always real, you can use a similar method to write an expression like 2 / (3 - 5i) as a fraction with a real denominator, by multiplying the numerator
and denominator by 3+ 5i(as 3 + 5i is the complex conjugate of 3 - 5i)
This is the basis for dividing one complex number by another.
Example: Find the real and imaginary parts of 1/(3 + i).
Solution
Multiply the numerator and denominator by 3 - i
(As 3 - i is the conjugate of the denominator 3 + i)
1/( 3 + i) = ( 3 - i) / [( 3 + i)( 3 - i]
= ( 3 - i) / (9 + 1)
= ( 3 - i) / 10
The real part is 3/10 and the imaginary part is -1/10.
Modulus and Argument of Complex Numbers
The below figure shows the point representing z = x + iy on an argand diagram.
The distance of this point from the origin is √x² + y².
This distance is called the modulus of z and is denoted by |z|.
So, for the complex number z = x + yi,
|z| = √x² + y² .
Notice that z\bar{z} = (x + iy)(x - iy) = x² + y², then |z| = z×\bar{z}
The argument i.e angle θ is measured anticlockwise from the positive real axis. By convention, the argument is measured in radians.
Arg z (θ) = tan¯1(b/a), The argument should lie between −π to π,
Modulus-argument form / Polar form of a complex number
In the below figure, you can see the relationship between the components of a complex number and its modulus and argument.
Using trigonometry, you can see that sinθ = y/r and so y = rsinθ.
Similarly, cosθ = x/r so x = rcosθ.
Therefore, the complex number z = x + yi can be written as:
z = r (cosθ + i sinθ)
This is called the modulus-argument form / Polar form of the complex number and is sometimes written as (r, θ).
