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(2+3i)(4-5i)

2 min read Jun 16, 2024
(2+3i)(4-5i)

Multiplying Complex Numbers: (2 + 3i)(4 - 5i)

This article will guide you through the process of multiplying two complex numbers: (2 + 3i) and (4 - 5i).

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1 (i² = -1).

Multiplying Complex Numbers

Multiplying complex numbers is similar to multiplying binomials. We use the FOIL method (First, Outer, Inner, Last).

  1. First: Multiply the first terms of each binomial: 2 * 4 = 8
  2. Outer: Multiply the outer terms: 2 * -5i = -10i
  3. Inner: Multiply the inner terms: 3i * 4 = 12i
  4. Last: Multiply the last terms: 3i * -5i = -15i²

Now we have: 8 - 10i + 12i - 15i²

Remember that i² = -1. Substitute this into the expression:

8 - 10i + 12i - 15(-1)

Simplify the expression:

8 - 10i + 12i + 15

Combine the real terms and the imaginary terms:

(8 + 15) + (-10 + 12)i

Therefore, the product of (2 + 3i) and (4 - 5i) is 23 + 2i.

Key Points

  • Remember that i² = -1.
  • Use the FOIL method to multiply the binomials.
  • Combine the real terms and the imaginary terms separately.

This method can be applied to any complex number multiplication. Practice with different examples to solidify your understanding.

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