(10 – 4i) ÷ (5 + I)

Jasonbradley

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Jun 16, 2024

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Dividing Complex Numbers: (10 – 4i) ÷ (5 + i)

This article will guide you through the process of dividing complex numbers, using the example of (10 – 4i) ÷ (5 + i).

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 (i² = -1).

Dividing Complex Numbers

Dividing complex numbers involves a process similar to rationalizing the denominator of a fraction. Here's how it works:

1. Multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of a complex number a + bi is a - bi.
2. Simplify the resulting expression.

Let's apply this to our example:

(10 – 4i) ÷ (5 + i)

1. Multiply by the conjugate of the denominator:

(10 – 4i) ÷ (5 + i) * (5 - i) ÷ (5 - i)

2. Expand the numerator and denominator:

[(10 – 4i) * (5 - i)] / [(5 + i) * (5 - i)]

3. Simplify using the distributive property and the fact that i² = -1:

[(50 - 10i - 20i + 4i²)] / [(25 - i²)]

[(50 - 30i - 4)] / [(25 + 1)]

[(46 - 30i)] / [26]

4. Express the result in standard complex number form:

(46/26) - (30/26)i

5. Simplify the fractions:

(23/13) - (15/13)i

Therefore, the result of (10 – 4i) ÷ (5 + i) is (23/13) - (15/13)i.