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Chapter 2: Problem 9

Find real numbers \(a\) and \(b\) such that the equation is true. $$(a-1)+(b+3) i=5+8 i$$

Short Answer

Expert verified
The real numbers \(a\) and \(b\) that make the equation hold true are \(a=6\) and \(b=5\).

Step by step solution

01

Identify real and imaginary parts

Rewrite the complex number in standard form \(z=a+bi\), where \(a\) is the real part and \(b\) is the imaginary part. In this case, the given equation is \((a-1) +(b+3)i = 5+8i.\) So, the real parts are \(a-1\) and \(5\) and the imaginary parts are \(b+3\) and \(8\).
02

Equate real parts

From step 1, equate the real parts \(a-1 = 5\). Solving for \(a\) gives \(a=5+1=6\).
03

Equate imaginary parts

From step 1, equate the imaginary parts \(b+3 = 8\). Solving for \(b\) gives \(b=8-3=5\).

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Key Concepts

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

Understanding the Real Part
The real part of a complex number is the component that does not involve the imaginary unit, \(i\). In other words, it can be thought of as the "non-imaginary" component of a complex number. For example, in the complex number \((a-1) + (b+3)i\), the real part is \(a-1\). Similarly, in the complex number \(5 + 8i\), the real part is \(5\).
To find the real part of a complex number in an equation, identify and isolate the term without the \(i\). Complex numbers are often expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
  • Real parts are like regular numbers you use every day and don't involve \(i\).
  • To solve for the real part, equate the non-\(i\) terms on both sides of the equation.
  • In the exercise example, we have \(a-1 = 5\).
Exploring the Imaginary Part
The imaginary part of a complex number is associated with the imaginary unit \(i\). This part is multiplied by \(i\), representing a dimension orthogonal to the real numbers on the complex plane. In the expression \((a-1) + (b+3)i\), the imaginary part is \(b+3\), while for \(5+8i\), it's \(8\).
When working with complex equations, like in our exercise, separate the imaginary terms. These are the terms that have \(i\) attached to them. Imaginary parts allow for a broader range of solutions, often handling phenomena like electrical currents and waveforms.
  • The imaginary part has a factor of \(i\), which signifies its nature in the complex plane.
  • Equate the imaginary terms from each side of the equation to solve for the unknown.
  • In this case, this means solving \(b+3 = 8\).
Equation Solving with Complex Numbers
Solving equations with complex numbers involves handling both real and imaginary components separately. Each part is treated like an independent equation. This separation makes it easier to find the unknown values.
In the original exercise context, the equation \((a-1) + (b+3)i = 5+8i\) provides separate paths for determining \(a\) and \(b\).
  • First, identify the real components of the equation and solve them independently. For example, solve \(a - 1 = 5\) to find \(a\).
  • Next, look at the imaginary components, such as \(b + 3 = 8\), and solve for \(b\).
  • The answers can then be combined to form the complete solution for the complex number.
Complex equations require adjusting both parts but don't overlook the straightforward step of separation. This allows calculations to remain simple and organized.

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