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Chapter 8: Problem 31

Combine the following complex numbers. \([(3+2 i)-(6+i)]+(5+i)\)

Short Answer

Expert verified
The combined complex number is \(2 + 2i\).

Step by step solution

01

Expand and Simplify Inside the Brackets

Start by focusing on the expression inside the brackets, \((3 + 2i) - (6 + i)\). To simplify, distribute the negative sign across the second complex number:\((3 + 2i) - 6 - i \). Now combine the real and imaginary parts separately: \(3 - 6 + 2i - i = -3 + i\). This simplifies to \(-3 + i\).
02

Add the Simplified Expression to the Second Complex Number

Now that we have simplified the expression in the brackets to \(-3 + i\), we need to add it to the second complex number, \(5 + i\). Combine the real parts and the imaginary parts separately:Real part: \(-3 + 5 = 2\).Imaginary part: \(i + i = 2i\).So, the result of this addition is \(2 + 2i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Complex Numbers
Complex numbers can sometimes look puzzling, with their real and imaginary components. Simplifying them, however, is as simple as following basic mathematical operations. Given a complex number in the form of \(a + bi\), simplifying involves treating the real and imaginary parts separately. When faced with an expression like \((3 + 2i) - (6 + i)\), start by tackling each component independently.
  • First, distribute any signs, especially negatives, across the expression. This will help in organizing terms without confusion.
  • Next, combine like terms鈥攔eal parts together, imaginary parts together. The real numbers \(3\) and \(-6\) combine to form \(-3\), while \(2i\) and \(-i\) simplify to \(i\).
After these steps, you can see complex numbers as just an extension of algebraic expressions, just with an added dimension.
Addition of Complex Numbers
Once you have simplified a complex expression, you might need to add it to another complex number. The key to achieving this easily is treating the addition as two separate tasks: one for real parts and the other for imaginary parts. Consider the sum \((-3 + i) + (5 + i)\).
  • For the real parts, add \(-3\) and \(5\). This gives a result of \(2\).
  • For the imaginary parts, combine \(i\) and \(i\) to get \(2i\).
Thus, the combined complex number is \(2 + 2i\). By separating the real from the imaginary parts, you can easily manage operations with complex numbers without confusion.
Real and Imaginary Parts
Understanding complex numbers requires recognizing the two main components of these numbers, the real and imaginary parts. In \(a + bi\), \(a\) is the real part and \(bi\) is the imaginary part.
  • The real part, \(a\), behaves just like regular numbers in operations like addition and subtraction.
  • The imaginary part, \(bi\), isn't something from science fiction鈥攊t's simply based on the imaginary unit \(i\), where \(i = \sqrt{-1}\).
When working with complex numbers, keep in mind that these parts are entirely separate and should be treated differently during mathematical operations. By understanding their distinct roles, you can better navigate problems involving complex numbers and make more sense of their unique behavior in equations.

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Most popular questions from this chapter

Write each complex number in standard form.\(2\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\)

Graph each complex number. In each case, give the absolute value of the number.\(-4-3 i\)

Graph each complex number along with its opposite and conjugate.\(.5\)

Let \(z_{1}=2+3 i, z_{2}=2-3 i\), and \(z_{3}=4+5 i\), and find \(z_{1} z_{2}\)

\(r=6 \cos \theta\)
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