(1%2B3i)(2%2Bi)%5E-1.jpeg "(1+2i)(1+3i)(2+i)^-1")
Simplifying Complex Expressions: (1+2i)(1+3i)(2+i)^-1
This article will guide you through the process of simplifying the complex expression (1+2i)(1+3i)(2+i)^-1. We will break down the calculation into manageable steps, highlighting key concepts in complex number manipulation.
Understanding the Basics
Before diving into the calculation, let's refresh our understanding of complex numbers:
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Complex Number: A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i.e., i² = -1).
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Complex Conjugate: The complex conjugate of a complex number a + bi is a - bi.
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Reciprocal of a Complex Number: The reciprocal of a complex number a + bi is (a - bi) / (a² + b²).
Step-by-Step Calculation
Let's now simplify the expression (1+2i)(1+3i)(2+i)^-1 step-by-step:
1. Simplify (2+i)^-1:
- Find the complex conjugate of (2+i): 2 - i
- Calculate the denominator: 2² + 1² = 5
- Therefore, (2+i)^-1 = (2 - i) / 5
2. Multiply (1+2i) and (1+3i):
- (1+2i)(1+3i) = 1 + 3i + 2i + 6i²
- Simplify using i² = -1: 1 + 5i - 6 = -5 + 5i
3. Multiply the result from step 2 with the result from step 1:
- (-5 + 5i) * [(2 - i) / 5] = (-10 + 5i + 10i - 5i²) / 5
- Simplify using i² = -1: (-10 + 15i + 5) / 5 = -1 + 3i
Therefore, (1+2i)(1+3i)(2+i)^-1 simplifies to -1 + 3i.
Conclusion
Simplifying complex expressions involves understanding the fundamental properties of complex numbers and applying the appropriate operations. By breaking down the problem into manageable steps, we can effectively manipulate complex numbers and arrive at the simplified result.