Problem 50 $$\text {solve the given problem... [FREE SOLUTION]

Chapter 12: Problem 50

$$\text {solve the given problems.}$$ Write the reciprocal of $2+5 j$ in rectangular form.

Short Answer

Expert verified

The reciprocal of $2+5j$ in rectangular form is $ \frac{2}{29} - \frac{5}{29}j $.

Step by step solution

Identify the Given Number

The complex number given is $ 2 + 5j $. It is already in rectangular form, where $ 2 $ is the real part and $ 5j $ is the imaginary part.

Constructing the Reciprocal Formula

To find the reciprocal of a complex number $ z = a + bj $, we use the formula: \[ \text{Reciprocal of } z = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}j \]

Calculate Components of the Reciprocal

For $ z = 2 + 5j $, we identify $ a = 2 $ and $ b = 5 $. Calculate $ a^2 + b^2 = 2^2 + 5^2 = 4 + 25 = 29 $.

Apply the Reciprocal Formula

Using the reciprocal formula, substitute the values: \[ \frac{2}{29} - \frac{5}{29}j \]. This gives us the reciprocal of $ 2 + 5j $ as $ \frac{2}{29} - \frac{5}{29}j $ in rectangular form.

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Key Concepts

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

Rectangular Form

When dealing with complex numbers, we often use the rectangular form. It's a way to express these numbers in the form $ a + bj $, where $ a $ is the real part, and $ bj $ is the imaginary part. In our exercise, the complex number $ 2 + 5j $ is already given in this format.

Real Part: This is the part without the "j," in this case, it's $ 2 $.
Imaginary Part: This refers to the part that includes "j," which is $ 5j $ here.

This form is helpful because it's straightforward. It allows us to easily identify and work with each component, especially when performing operations like addition, subtraction, or finding reciprocals.

Reciprocal of Complex Numbers

Finding the reciprocal of a complex number involves using a specific mathematical formula. A reciprocal, in simple terms, is what you'd multiply a number by to get 1. For a complex number $ z = a + bj $, its reciprocal is given by the formula:\[\text{Reciprocal of } z = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}j\]The rationale is to eliminate the imaginary component by using the denominator, $ a^2 + b^2 $, which is a sum of squares of the real and imaginary parts.
For example, given $ 2 + 5j $:

Identify $ a $ and $ b $: here $ a = 2 $ and $ b = 5 $.
Calculate $ a^2 + b^2 $: $ 4 + 25 = 29 $.
Apply the formula to find: $ \frac{2}{29} - \frac{5}{29}j $.

This gives us the reciprocal $ \frac{2}{29} - \frac{5}{29}j $, still in rectangular form.

Imaginary Numbers

Imaginary numbers are an important concept in complex numbers. They are defined as multiples of the imaginary unit $ j $, where $ j^2 = -1 $. This might seem puzzling at first because these numbers don't exist on the traditional number line.

Understanding $ j $: It symbolically represents the square root of $ -1 $.
Role in Complex Numbers: Imaginary numbers combine with real numbers to form complex numbers, like $ 2 + 5j $.

Imaginary numbers are crucial in many fields, including electrical engineering, quantum physics, and applied mathematics. They offer solutions to equations that have no real solutions, expanding the scope of mathematical problem-solving. The quality "imaginary" doesn't mean they are less real; it's simply a label for this useful concept in mathematics.

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91影视

$$\text {solve the given problems.}$$ Write the reciprocal of \(2+5 j\) in rectangular form.

Short Answer

Step by step solution

Identify the Given Number

Constructing the Reciprocal Formula

Calculate Components of the Reciprocal

Apply the Reciprocal Formula

Key Concepts

Rectangular Form

Reciprocal of Complex Numbers

Imaginary Numbers

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