Complex Number Multiplication and Division Calculator

Complex Number Multiplication and Division

What are Complex Number Multiplication and Division?

Complex number multiplication and division are operations performed on complex numbers, which are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1).

Formulas

For complex numbers z₁ = a + bi and z₂ = c + di:

Multiplication: \[z₁ × z₂ = (a + bi)(c + di) = (ac - bd) + (ad + bc)i\]

Division: \[\frac{z₁}{z₂} = \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}\]

Calculation Steps

Let's multiply (3 + 2i) and (1 - 4i):

1. Apply the multiplication formula:
   (a + bi)(c + di) = (ac - bd) + (ad + bc)i
2. Substitute the values:
   (3 + 2i)(1 - 4i) = (3 × 1 - 2 × (-4)) + (3 × (-4) + 2 × 1)i
3. Calculate:
   = (3 + 8) + (-12 + 2)i
   = 11 - 10i

Example and Visual Representation

Let's visualize the multiplication of (3 + 2i) and (1 - 4i):

This visual representation shows:

• The first complex number 3+2i in blue
• The second complex number 1-4i in red
• The result of multiplication 11-10i in green