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

3 min read Jun 16, 2024

Simplifying Complex Fractions

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

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

Understanding Complex Numbers

Before we begin, let's quickly review complex numbers. A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, where i² = -1.

Simplifying the First Fraction

We'll start by simplifying the first fraction, (2 + 3i) / (1 - 2i). To do this, we'll multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of 1 - 2i is 1 + 2i.

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

This gives us:

(2 + 4i + 3i + 6i²) / (1 + 2i - 2i - 4i²)

Simplifying further by remembering that i² = -1:

(2 + 7i - 6) / (1 + 4)

This results in:

(-4 + 7i) / 5

Simplifying the Second Fraction

Now let's simplify the second fraction, (i) / (3 + i). We'll use the same method as before, multiplying the numerator and denominator by the conjugate of the denominator, which is 3 - i.

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

This gives us:

(3i - i²) / (9 - i²)

Simplifying further:

(3i + 1) / (9 + 1)

This results in:

(1 + 3i) / 10

Combining the Simplified Fractions

Now we have:

(-4 + 7i) / 5 + (1 + 3i) / 10

To add these fractions, they need a common denominator. We can multiply the first fraction by 2/2:

((-8 + 14i) / 10) + ((1 + 3i) / 10)

Now we can add the numerators:

(-8 + 14i + 1 + 3i) / 10

Combining like terms:

(-7 + 17i) / 10

Final Result

Therefore, the simplified form of the given complex fraction is (-7 + 17i) / 10.