(5-2i)+(5+3i)/(1+i)-(2-4i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Numbers: A Step-by-Step Guide

This article will guide you through simplifying the complex number expression: (5-2i)+(5+3i)/(1+i)-(2-4i).

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.

Simplifying the Expression

1. Simplify the division:

   • We need to get rid of the complex number in the denominator. We can do this by multiplying both the numerator and denominator by the complex conjugate of the denominator (1-i).
   • (5+3i)/(1+i) * (1-i)/(1-i)
   • (5-5i + 3i - 3i²)/(1² - i²)
   • (8 - 2i)/(1 + 1) (Since i² = -1)
   • (8 - 2i)/2
   • 4 - i

2. Substitute the simplified division back into the original expression:

   • (5-2i) + (4 - i) - (2-4i)

3. Combine like terms:

   • (5 + 4 - 2) + (-2 - 1 + 4)i
   • 7 + i

Final Answer

The simplified form of the complex number expression (5-2i)+(5+3i)/(1+i)-(2-4i) is 7 + i.