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

3 min read Jun 16, 2024
(3-2i)(2+3i)/(1+2i)(2-i)

Simplifying Complex Fractions

This article will guide you through simplifying the complex fraction: (3 - 2i)(2 + 3i) / (1 + 2i)(2 - i)

Understanding Complex Numbers

Complex numbers are numbers 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).

Steps to Simplify

  1. Expand the Numerator and Denominator:

    • Numerator: (3 - 2i)(2 + 3i) = 6 + 9i - 4i - 6i² = 6 + 9i - 4i + 6 = 12 + 5i
    • Denominator: (1 + 2i)(2 - i) = 2 - i + 4i - 2i² = 2 - i + 4i + 2 = 4 + 3i

  2. Express the Result as a Single Complex Number: The fraction now becomes (12 + 5i) / (4 + 3i). To simplify this, we need to get rid of the imaginary unit in the denominator. We achieve this by multiplying both the numerator and denominator by the complex conjugate of the denominator.

  3. Multiply by the Complex Conjugate: The complex conjugate of (4 + 3i) is (4 - 3i). Multiplying both numerator and denominator by (4 - 3i):

    (12 + 5i) / (4 + 3i) * (4 - 3i) / (4 - 3i)

  4. Simplify the Result:

    • Numerator: (12 + 5i)(4 - 3i) = 48 - 36i + 20i - 15i² = 48 - 16i + 15 = 63 - 16i
    • Denominator: (4 + 3i)(4 - 3i) = 16 - 9i² = 16 + 9 = 25

  5. Final Result: The simplified form of the given fraction is (63 - 16i) / 25, which can be written as (63/25) - (16/25)i.

Conclusion

By applying the steps above, we have successfully simplified the complex fraction (3 - 2i)(2 + 3i) / (1 + 2i)(2 - i) to (63/25) - (16/25)i. This process involves expanding the expressions, multiplying by the complex conjugate, and simplifying the resulting expression to a single complex number.

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