(1-2i)(2+i)/3-2i

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Numbers: A Step-by-Step Guide

This article will guide you through the process of simplifying the complex number expression:

(1 - 2i)(2 + i) / (3 - 2i)

Understanding Complex Numbers

Complex numbers are numbers 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).

Simplifying the Expression

1. Expand the numerator:

   • (1 - 2i)(2 + i) = 1(2) + 1(i) - 2i(2) - 2i(i)
   • = 2 + i - 4i - 2i²
   • = 2 - 3i - 2(-1) (Since i² = -1)
   • = 4 - 3i

2. Multiply the numerator and denominator by the conjugate of the denominator:

   • The conjugate of (3 - 2i) is (3 + 2i).
   • [(4 - 3i) / (3 - 2i)] * [(3 + 2i) / (3 + 2i)]

3. Expand the resulting expression:

   • [(4 - 3i)(3 + 2i)] / [(3 - 2i)(3 + 2i)]
   • = [12 + 8i - 9i - 6i²] / [9 + 6i - 6i - 4i²]
   • = [12 - i - 6(-1)] / [9 - 4(-1)] (Since i² = -1)
   • = [18 - i] / [13]

4. Express the result in standard form:

   • (18 - i) / 13 = (18/13) - (1/13)i

Conclusion

The simplified form of the complex number expression (1 - 2i)(2 + i) / (3 - 2i) is (18/13) - (1/13)i. This demonstrates how to perform operations involving complex numbers and express the result in standard form.