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

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Fractions: (1+i)(2+i)/(3+i)

This article will guide you through the process of simplifying the complex fraction (1+i)(2+i)/(3+i).

Understanding Complex Numbers

Before we begin, let's quickly recap what complex numbers are. A complex number is a number 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).

Simplifying the Expression

1. Multiply the numerators: (1+i)(2+i) = 1(2) + 1(i) + i(2) + i(i) = 2 + i + 2i + i² = 2 + 3i - 1 = 1 + 3i

2. Multiply the denominators: (3 + i) = 3 + i

3. Divide the simplified numerator by the simplified denominator: (1 + 3i) / (3 + i)

4. Rationalize the denominator: To get rid of the imaginary term in the denominator, we multiply both numerator and denominator by the complex conjugate of the denominator. The complex conjugate of (3 + i) is (3 - i).

   • Numerator: (1 + 3i) (3 - i) = 1(3) + 1(-i) + 3i(3) + 3i(-i) = 3 - i + 9i - 3i² = 3 + 8i + 3 = 6 + 8i
   • Denominator: (3 + i)(3 - i) = 3(3) + 3(-i) + i(3) + i(-i) = 9 - 3i + 3i - i² = 9 + 1 = 10

5. Final simplification: (6 + 8i) / 10 = (6/10) + (8/10)i = 3/5 + 4/5i

Therefore, the simplified form of (1+i)(2+i)/(3+i) is 3/5 + 4/5i.