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Chapter 2: Problem 8

Find real numbers \(a\) and \(b\) such that the equation is true. $$a+b i=13+4 i$$

Short Answer

Expert verified
Solving the given equation, we find that the value of \(a\) is 13 and the value of \(b\) is 4.

Step by step solution

01

Identifying the Real and Imaginary Parts

Identify the real and imaginary parts from both sides of the equation. The real part is the number not associated with the imaginary unit \(i\), and the imaginary part is the number associated with \(i\). On the left hand side (LHS), \(a\) is the real part and \(b\) is the imaginary part. On the right hand side (RHS), 13 is the real part and 4 is the imaginary part.
02

Equating the Real Parts

Equating the real parts from both sides as the equation needs to hold true for both the real and imaginary parts, we have \(a = 13\).
03

Equating the Imaginary Parts

Now, equate the imaginary parts from both sides, which gives us \(b = 4\).

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Key Concepts

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

Real Numbers
Real numbers are fundamental in both algebra and calculus. They include all the numbers that can be found on the number line, encompassing both positive and negative integers, fractions, and irrational numbers. Real numbers are all the "normal" numbers that we day-to-day use. They can be thought of as numbers without any imaginary component.

In the context of complex numbers, real numbers constitute the part that is not multiplied by the imaginary unit (often denoted as \(i\)). For example, in the complex number \(a + bi\), \(a\) is the real part. In our problem, the real number we are seeking is \(13\). It serves as the benchmark for comparison when we equate parts later.
Imaginary Unit
The imaginary unit is a mathematical construct that allows us to extend the real number line into a two-dimensional plane, creating what we know as complex numbers. It is denoted most commonly by the letter \(i\), and it is defined by the property \(i^2 = -1\). This peculiar property is what distinguishes imaginary numbers from real numbers.

When expressed in a complex number format, like \(a + bi\), the term \(bi\) represents the imaginary part, where \(b\) is a real number coefficient and \(i\) is the imaginary unit. This combination allows for the analysis and computation of numbers that cannot be expressed as real numbers alone, as is necessary in fields such as engineering and physics. In the exercise given, the imaginary term \(bi\) equates to \(4i\) on the right hand side.
Equating Parts
When dealing with equations involving complex numbers, equating parts is a method used to find solutions for unknowns separately in the real and imaginary components.

In our context, the equation \(a + bi = 13 + 4i\), requires us to equate the real parts and the imaginary parts independently. This is because, for two complex numbers to be equal, both their real and imaginary components must match.
  • First, equate the real parts: set \(a = 13\).
  • Then, equate the imaginary parts: set \(b = 4\).
This approach allows us to solve for both \(a\) and \(b\), providing the complete solution to the problem.

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Most popular questions from this chapter

Write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point. Vertex: (1,-2)\(;\) point: (-1,14)

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