# Chapter 1 : Complex Numbers

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

#### Complex Numbers arrow_upward

• 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 imaginary unit.
• i is the standard imaginary unit with the property .

• #### Imaginary Numbers arrow_upward

• 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
•      i2 = -1

i3 = -i

i4 = 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 Conjugate arrow_upward

• 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.

• #### Adding and Subtracting Complex Numbers arrow_upward

• 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

#### Multiplying Complex Numbers arrow_upward

• We multiply two complex numbers just as we multiply expressions of the form (x + y) together.
•      z1 × z2

= (a + ib)(c + id)

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

= ac + iad + ibc - bd

= ac – bd + i(ad + bc)

• So  z1 × z2 = 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)

•     = 6 + 4i + 9i - 6
•     = 13i

#### Dividing Complex Numbers arrow_upward

• 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 arrow_upward

• 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.

• #### Thank You from Kimavi arrow_upward

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