(4-i)/(3+4i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Fractions: (4-i)/(3+4i)

In mathematics, complex numbers often appear in fractions, requiring us to simplify them. One common method is to multiply both the numerator and denominator by the complex conjugate of the denominator. Let's demonstrate this with the example of (4-i)/(3+4i).

Understanding Complex Conjugates

The complex conjugate of a number of the form a + bi is a - bi. The key property of complex conjugates is that when multiplied, they result in a real number:

(a + bi)(a - bi) = a² - (bi)² = a² + b²

Applying the Method

1. Identify the complex conjugate of the denominator: The complex conjugate of 3 + 4i is 3 - 4i.

2. Multiply both numerator and denominator by the conjugate:

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

3. Expand the multiplication:

   [(4 - i)(3 - 4i)] / [(3 + 4i)(3 - 4i)]

4. Simplify using distributive property (FOIL):

   (12 - 16i - 3i + 4i²) / (9 - 12i + 12i - 16i²)

5. Substitute i² with -1:

   (12 - 19i - 4) / (9 + 16)

6. Combine real and imaginary terms:

   (8 - 19i) / 25

7. Express in standard form (a + bi):

   (8/25) - (19/25)i

Conclusion

Therefore, the simplified form of (4 - i) / (3 + 4i) is (8/25) - (19/25)i. This method of multiplying by the complex conjugate is a crucial technique for simplifying fractions involving complex numbers and expressing them in standard form.