(2+3i)(4-i)

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

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

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

Understanding Complex Numbers

A complex number is a number 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).

Multiplication Process

To multiply complex numbers, we use the distributive property (also known as FOIL - First, Outer, Inner, Last) just like we do with binomials.

  1. Expand: (2 + 3i)(4 - i) = 2(4) + 2(-i) + 3i(4) + 3i(-i)

  2. Simplify: = 8 - 2i + 12i - 3i²

  3. Substitute i² with -1: = 8 - 2i + 12i - 3(-1)

  4. Combine Real and Imaginary Terms: = (8 + 3) + (-2 + 12)i

  5. Final Result: = 11 + 10i

Conclusion

Therefore, the product of (2 + 3i) and (4 - i) is 11 + 10i. This process illustrates how complex numbers are manipulated through multiplication, maintaining the form a + bi.

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