(5-i)%3D15-3i%2B10i-2i%5E2.jpeg "(3+2i)(5-i)=15-3i+10i-2i^2")
Complex Number Multiplication: A Step-by-Step Guide
This article will explore the process of multiplying complex numbers, using the example: (3+2i)(5-i) = 15-3i+10i-2i^2.
Understanding Complex Numbers
Complex numbers are numbers that consist of a real part and an imaginary part. The imaginary unit is denoted by 'i', where i² = -1. They are typically written in the form a + bi, where 'a' and 'b' are real numbers.
Multiplication of Complex Numbers
Multiplying complex numbers is similar to multiplying binomials. We use the distributive property, multiplying each term in the first complex number by each term in the second complex number.
Step-by-Step Solution
Let's break down the multiplication of (3+2i)(5-i):
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Distribute: (3+2i)(5-i) = 3(5-i) + 2i(5-i)
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Simplify: = 15 - 3i + 10i - 2i²
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Substitute i² = -1: = 15 - 3i + 10i - 2(-1)
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Combine like terms: = 15 + 2 + 10i - 3i
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Final Result: = 17 + 7i
Therefore, the product of (3+2i) and (5-i) is 17 + 7i.
Key Takeaways
- Complex number multiplication follows the distributive property.
- Substitute i² = -1 to simplify the expression.
- Combine real and imaginary terms to arrive at the final result in the form a + bi.
Understanding the process of multiplying complex numbers is crucial in various mathematical fields, including algebra, calculus, and linear algebra.