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Chapter 1: Problem 68

Let \(z=2-5 i\) and \(w=7+3 i\). Find: (a) \(v+w\); (b) \(z w\); (c) \(z / w\) (d) \(\bar{z}, \bar{w}\) (e) \(|z|,|w|\).

Short Answer

Expert verified
(a) \(z + w = 9 - 2i\) (b) \(zw = 29 - 29i\) (c) \(z / w = -\frac{1}{58} - \frac{29}{58}i\) (d) \(\bar{z} = 2 + 5i\), \(\bar{w} = 7 - 3i\) (e) \(|z| = \sqrt{29}\), \(|w| = \sqrt{58}\)

Step by step solution

01

(a) Calculating \(z + w\)

To add two complex numbers, simply add their real and imaginary parts separately. So, if \(z = a + bi\) and \(w = c + di\), then \(z + w = (a + c) + (b + d)i\). Given \(z = 2 - 5i\) and \(w = 7 + 3i\), we have: \(z + w = (2 + 7) + (-5 + 3)i = 9 - 2i\)
02

(b) Calculating \(zw\)

To multiply two complex numbers, use the distributive property and simplify as needed. So, if \(z = a + bi\) and \(w = c + di\), then \(zw = (a + bi)(c + di) = ac + adi + bci + bdi^2\). Remember that \(i^2 = -1\). Given \(z = 2 - 5i\) and \(w = 7 + 3i\), we have: \(zw = (2)(7) + (2)(3i) - (5i)(7) - (5i)(3i) = 14 + 6i - 35i - 15(-1) = 29 - 29i\)
03

(c) Calculating \(\frac{z}{w}\)

To divide two complex numbers, use the fact that \(\frac{z}{w} = z \cdot \frac{1}{w}\), and expressing the denominator in polar form allows us to compute the reciprocal easily. So, if \(w = c + di\), then \(\frac{1}{w} = \frac{c - di}{c^2 + d^2}\). Then, calculate \(z \cdot \frac{1}{w}\) using the methods from part (b). Given \(z = 2 - 5i\) and \(w = 7 + 3i\), we have: \(\frac{1}{w} = \frac{7 - 3i}{7^2 + 3^2} = \frac{7 - 3i}{58}\), and \(\frac{z}{w} = \frac{2 - 5i}{7 + 3i} = (2 - 5i) \cdot \frac{7 - 3i}{58} = \frac{(-1 - 29i)}{58} = -\frac{1}{58} - \frac{29}{58}i\)
04

(d) Finding \(\bar{z}\) and \(\bar{w}\)

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. So, if \(z = a + bi\), then \(\bar{z} = a - bi\). Given \(z = 2 - 5i\) and \(w = 7 + 3i\), we have: \(\bar{z} = 2 + 5i\), and \(\bar{w} = 7 - 3i\)
05

(e) Calculating \(|z|\) and \(|w|\)

The modulus (magnitude) of a complex number is given by the square root of the sum of the squares of its real and imaginary parts. So, if \(z = a + bi\), then \(|z| = \sqrt{a^2 + b^2}\). Given \(z = 2 - 5i\) and \(w = 7 + 3i\), we have: \(|z| = \sqrt{2^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}\), and \(|w| = \sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58}\)

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Key Concepts

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

Complex Addition
Complex addition is a straightforward and essential operation when working with complex numbers. Simply put, you add the real parts together and the imaginary parts together. Consider two complex numbers:
  • If you have a complex number \(z = a + bi\) where \(a\) is the real part and \(b\) is the imaginary part.
  • Another complex number \(w = c + di\), with \(c\) and \(d\) as the real and imaginary components, respectively.

The sum is expressed as:
  • \(z + w = (a+c) + (b+d)i\)
  • This results in a new complex number combining both.
For example, if \(z = 2 - 5i\) and \(w = 7 + 3i\), the addition is:
  • \((2+7) + (-5+3)i = 9 - 2i\)
Complex Multiplication
Complex multiplication requires a slightly deeper understanding as it relies on understanding distributive property and special rules for imaginary numbers. Let's assume:
  • \(z = a + bi\)
  • \(w = c + di\)

Whenever multiplying complex numbers:
  • Use the expression: \((a+bi)(c+di) = ac + adi + bci + bdi^2\)
  • Remember \(i^2 = -1\) which simplifies the expression effectively

For example, if \(z = 2 - 5i\) and \(w = 7 + 3i\), the calculation is:
  • \(zw = 14 + 6i - 35i - 15 = 29 - 29i\)
Complex Division
Division with complex numbers involves the use of complex conjugates. The idea is to make the denominator real by its complex conjugate. Let鈥檚 denote:
  • \(z = a + bi\)
  • \(w = c + di\)
The division \(\frac{z}{w}\) follows these steps:
  • Find the reciprocal: \(\frac{1}{w} = \frac{c-di}{c^2+d^2}\)
  • Then compute: \(z \cdot \frac{1}{w}\)

For instance, divide \(z = 2 - 5i\) by \(w = 7 + 3i\):
  • Compute \(\frac{1}{w} = \frac{7-3i}{58}\)
  • Then multiply: \((2 - 5i)\cdot \frac{7 - 3i}{58} = -\frac{1}{58} - \frac{29}{58}i\)
Complex Conjugate
The complex conjugate is key when managing operations such as division. Form the conjugate by altering the sign of the imaginary component. For any complex number:
  • \(z = a + bi\)
The conjugate is:
  • \(\bar{z} = a - bi\)

When applied to our example, for \(z = 2 - 5i\) and \(w = 7 + 3i\):
  • \(\bar{z} = 2 + 5i\)
  • \(\bar{w} = 7 - 3i\)
Complex Modulus
The modulus or magnitude is often used to understand the size of a complex number. It's computed using the Pythagorean Theorem as the distance in the complex plane:
  • For \(z = a + bi\), the modulus is \(|z| = \sqrt{a^2 + b^2}\)
Calculating for \(z = 2 - 5i\) and \(w = 7 + 3i\) yields:
  • \(|z| = \sqrt{2^2 + (-5)^2} = \sqrt{29}\)
  • \(|w| = \sqrt{7^2 + 3^2} = \sqrt{58}\)
This measurement helps in evaluating and comparing complex numbers in terms of their size.

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