(9+5i)(4-2i)

2 min read Jun 16, 2024
(9+5i)(4-2i)

Multiplying Complex Numbers: (9 + 5i)(4 - 2i)

This article will guide you through the process of multiplying complex numbers, specifically focusing on the example of (9 + 5i)(4 - 2i).

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).

Multiplying Complex Numbers

To multiply complex numbers, we use the distributive property (also known as FOIL method) similar to multiplying binomials in algebra.

Step 1: Expand the product

(9 + 5i)(4 - 2i) = 9(4) + 9(-2i) + 5i(4) + 5i(-2i)

Step 2: Simplify each term

= 36 - 18i + 20i - 10i²

Step 3: Substitute i² with -1

= 36 - 18i + 20i - 10(-1)

Step 4: Combine real and imaginary terms

= 36 + 10 + (-18i + 20i)

Step 5: Final result

= 46 + 2i

Therefore, the product of (9 + 5i)(4 - 2i) is 46 + 2i.

Visualizing Complex Numbers

Complex numbers can be visualized on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Multiplying complex numbers can be seen as a rotation and scaling operation on this plane.

Conclusion

Multiplying complex numbers involves applying the distributive property and substituting i² with -1. The result is another complex number expressed in the form a + bi. Understanding complex number multiplication is fundamental in various fields like electrical engineering, quantum mechanics, and signal processing.

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