# Chapter 1 : Complex Numbers

### Topics covered in this snack-sized chapter:

A Complex Number is a number consisting of a real and imaginary part.
It is denoted by letter z.
It can be written in the form z = a + bi
- Where a and b are real numbers.

i is the standard imaginary unit with the property .

An imaginary number is one that when squared gives a negative result.
Imaginary number is a complex number which has an imaginary part and zero real part.
We can split the negative numbers into positive number and –1.
We are defining √(-1) = i
i^{2
}= -1

i^{3
}= -i

i^{4
}= 1

Example: Find the imaginary number of √(-5)?
Solution: We know -5 = -1 × 5. So
√(-5) = √(-1 × 5) = √(-1) × √5

= √5 i

= 2.23i

So the imaginary number is 2.23i.
Complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs.
The complex conjugate of is .
Conjugate of a complex number is also denoted by .
Also,

A real number and a sum of two squares.
Let and be the two complex numbers.
The rule is to add or subtract the real and imaginary parts separately:

= a + ib + c + id

= a + c + i(b + d)

So = a + c + i(b + d).

= a + ib – c – id

= a – c + i(b - d)

So = a – c + i(b - d).
Example 1: Add the complex numbers (1 + i) and (3 + i).
Solution:
(1 + i) + (3 + i)
= 1 + 3 + i (1 + 1)

= 4 + 2i
Example 2: Subtract the complex numbers (2 + 5i) and (1 – 4i).
Solution:
(2 + 5i) – (1 – 4i)
= 2 + 5i – 1 + 4i

= 1 + 9i

We multiply two complex numbers just as we multiply expressions of the form (x + y) together.
z_{1
} × z_{2
}

= (a + ib)(c + id)

= ac + a(id) + (ib)c + (ib)(id)

= ac + iad + ibc - bd

= ac – bd + i(ad + bc)

So z_{1
} × z_{2
} = ac – bd + i(ad + bc)
Example: Multiply the complex numbers (2 + 3i) and (3 + 2i).
Solution:
(2 + 3i)(3 + 2i)
= 2 × 3 + 2 × 2i + 3i × 3 + 3i × 2i

(By substituting)

For dividing two complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator.

The denominator is now a real number.
Example:

The Complex Plane or z-plane is a geometric representation of the complex numbers established by real axis and orthogonal imaginary axis.
It can be modified as a Cartesian plane.
The real part of a complex number is represented by a displacement along the x-axis and the imaginary part by a displacement along the y-axis.
The multiplication of two complex numbers can be expressed easily in polar coordinates.
The magnitude or modulus of the product is the product of the two absolute value, or moduli.
The angle or argument of the product is the sum of the two angles, or arguments.
In particular, multiplication by a complex number of modulus 1 acts as a rotation.

Geometric representation of and its conjugate in the complex plane

The distance along the red line from the origin to the point z is the modulus or absolute value of z*.
*
The angle *
*is the argument of z*.
*