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Chapter 12: Problem 54

$$\text {solve the given problems.}$$ For \(\frac{3}{5}+\frac{4}{5} j,\) find: \((a)\) the conjugate; \((b)\) the reciprocal.

Short Answer

Expert verified
(a) The conjugate is \( \frac{3}{5} - \frac{4}{5}j \). (b) The reciprocal is \( \frac{3}{5} - \frac{4}{5}j \).

Step by step solution

01

Identify the Complex Number

The given complex number is \( \frac{3}{5} + \frac{4}{5}j \). This can be expressed as \( a + bj \) where \( a = \frac{3}{5} \) and \( b = \frac{4}{5} \).
02

Find the Conjugate

The conjugate of a complex number \( a + bj \) is \( a - bj \). Therefore, the conjugate of \( \frac{3}{5} + \frac{4}{5}j \) is \( \frac{3}{5} - \frac{4}{5}j \).
03

Find the Magnitude of the Complex Number

The magnitude of \( a + bj \) is given by \( \sqrt{a^2 + b^2} \). Substitute \( a = \frac{3}{5} \) and \( b = \frac{4}{5} \):\[ \sqrt{\left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2} = \sqrt{\frac{9}{25} + \frac{16}{25}} = \sqrt{\frac{25}{25}} = 1. \]
04

Find the Reciprocal

The reciprocal of a complex number \( a + bj \) is \( \frac{a - bj}{a^2 + b^2} \). Since the magnitude is 1, the reciprocal simplifies to \( a - bj \). For our number, the reciprocal is \( \frac{3}{5} - \frac{4}{5}j \), which matches the conjugate due to the magnitude being 1.

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Key Concepts

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

Conjugate of a Complex Number
A complex number is composed of a real part and an imaginary part, commonly written as \( a + bj \). To find the conjugate, you simply change the sign of the imaginary part. This means if you have a complex number like \( \frac{3}{5} + \frac{4}{5}j \), its conjugate will be \( \frac{3}{5} - \frac{4}{5}j \). It鈥檚 like an identical twin, but with a subtle and crucial difference鈥攐ne is positive and the other is negative in their imaginary part.

Why is the conjugate useful? Taking the conjugate of a complex number helps in tasks like simplifying division by what feels like an imaginary hurdle, literally! It鈥檚 a handy tool in many mathematical computations.
Reciprocal of a Complex Number
Just like with real numbers, finding the reciprocal of a complex number means flipping it around. However, with complex numbers, there's a bit more to it. The reciprocal of a complex number \( a + bj \) is given by \( \frac{a - bj}{a^2 + b^2} \). This formula might look familiar because it involves using the conjugate again!

For our specific example \( \frac{3}{5} + \frac{4}{5}j \), the calculation turns out especially simple because of its magnitude. Since the magnitude is 1, the reciprocal is \( \frac{3}{5} - \frac{4}{5}j \), precisely the conjugate of the number. Hence, the reciprocal and conjugate can indeed be the same when the magnitude is 1.
Magnitude of a Complex Number
The magnitude of a complex number gives you an idea of its size or distance from the origin in the complex plane. Imagine plotting a point in this plane; the magnitude is like measuring the straight-line distance from the origin to this point.

To calculate it, use the formula \( \sqrt{a^2 + b^2} \) for a complex number \( a + bj \). Here, with \( \frac{3}{5} + \frac{4}{5}j \), you'd compute \( \sqrt{\left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2} = 1 \). This result, 1, indicates that this complex number is just one unit away from the origin, giving it a special simplicity in calculations due to its friendly magnitude.
Step by Step Solution for Complex Number Problems
Breaking down a problem into clear, manageable steps can turn complexity into simplicity:
  • **Identify the Complex Number:** Recognize and write it plainly as \( a + bj \).
  • **Find the Conjugate:** Flip the sign of the imaginary part. This is crucial for many operations.
  • **Measure the Magnitude:** Use \( \sqrt{a^2 + b^2} \) to know how far your number is from zero.
  • **Determine the Reciprocal:** Engage the conjugate and magnitude to unravel this component effortlessly.
These steps offer a structured way to approach complex numbers. Whether preparing for exams or solving homework, keeping this sequence and its underlying principles in mind can greatly ease your numbers journey.

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Most popular questions from this chapter

$$\text {solve the given problems.}$$ Multiply \(-3+j\) by its conjugate.

Answer the given questions. Is it possible that a given complex number and the negative of its conjugate are equal? Explain.

Answer the given questions. How is a number changed if it is multiplied by \((a) j^{4} ;(b) j^{2} ?\)

Express the given numbers in exponential form. $$5.90+2.40 j$$

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