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Magazine Chapter 8: Problem 49
Find the quotient \(z_{1} / z_{2}\) in standard form. Then write \(z_{1}\) and \(z_{2}\) in trigonometric form and find their quotient again. Finally, convert the answer that is in trigonometric form to standard form to show that the two auotients are equal. $$ z_{1}=2+2 i, z_{2}=1+i $$
Short Answer
Expert verified
The quotient is 2, both in standard and trigonometric forms.
Step by step solution
01
Find the quotient in standard form
Given two complex numbers, \(z_1 = 2 + 2i\) and \(z_2 = 1 + i\), the quotient \(z_1 / z_2\) in standard form is calculated by multiplying the numerator and denominator by the conjugate of the denominator. \(z_2\) has a conjugate \(1 - i\). Thus, the operation is:\[\frac{2 + 2i}{1 + i} \times \frac{1 - i}{1 - i} = \frac{(2 + 2i)(1 - i)}{(1 + i)(1 - i)}\]The denominator simplifies as: \[(1 + i)(1 - i) = 1^2 - i^2 = 1 + 1 = 2\]The numerator is:\[(2 + 2i)(1 - i) = 2 \cdot 1 - 2 \cdot i + 2i \cdot 1 - 2i \cdot i = 2 + 2i - 2i - 2i^2\]Since \(i^2 = -1\), it becomes:\(2 + 2 = 4\).The resulting standard form is:\[\frac{4}{2} = 2\].
02
Convert to trigonometric form
Convert \(z_1 = 2 + 2i\) and \(z_2 = 1 + i\) into trigonometric form. The magnitude (modulus) and argument (angle) must be calculated.For \(z_1 = 2 + 2i\):Magnitude \(|z_1| = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}\)Argument \(\theta_1 = \tan^{-1}\left(\frac{2}{2}\right) = \frac{\pi}{4}\)Trigonometric form: \(z_1 = 2\sqrt{2} \left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)\).For \(z_2 = 1 + i\):Magnitude \(|z_2| = \sqrt{1^2 + 1^2} = \sqrt{2}\)Argument \(\theta_2 = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}\)Trigonometric form: \(z_2 = \sqrt{2} \left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)\).
03
Find the quotient in trigonometric form
Using the trigonometric forms of \(z_1\) and \(z_2\), the quotient \(\frac{z_1}{z_2}\) in trigonometric form is:Magnitude:\[\frac{|z_1|}{|z_2|} = \frac{2\sqrt{2}}{\sqrt{2}} = 2\]Argument:\[\theta_{\text{quotient}} = \theta_1 - \theta_2 = \frac{\pi}{4} - \frac{\pi}{4} = 0\]Thus, in trigonometric form:\[\frac{z_1}{z_2} = 2 \left(\cos 0 + i \sin 0 \right) = 2\]
04
Convert trigonometric form to standard form
Convert the trigonometric form back to standard form to verify equality with the original standard form result. The expression \(2 (\cos 0 + i \sin 0)\) simplifies to:\[2 (1 + 0i) = 2\].So, the standard form obtained here is \(2\), which matches the result from Step 1.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Standard Form
In the world of complex numbers, the standard form refers to expressing a complex number as a sum of a real part and an imaginary part. For example, a complex number is generally written as \(a + bi\), where \(a\) is the real part, and \(bi\) represents the imaginary part with \(i\) being the imaginary unit. The imaginary unit \(i\) is defined as \(\sqrt{-1}\). This concept helps in performing arithmetic operations such as addition, subtraction, multiplication, and division between complex numbers.
- Standard form is straightforward for calculations.
- It allows easy comparison of complex numbers.
- Facilitates the use of algebraic operations.
Trigonometric Form
Trigonometric form of a complex number is an alternative way to represent a complex number. It expresses the number in terms of its magnitude (or modulus) and an angle (or argument) on the complex plane. It is generally written as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude and \(\theta\) is the argument.
- This form is helpful in simplifying multiplication and division of complex numbers.
- Connecting complex numbers to geometric ideas makes visualization easier.
- Translates well to polar coordinates.
Complex Conjugate
The concept of a complex conjugate is pivotal in complex number calculations, especially when dividing them. The conjugate of a complex number \(a + bi\) is \(a - bi\). It is achieved by simply changing the sign of the imaginary part.
- It makes dividing complex numbers more manageable.
- When a number is multiplied by its conjugate, the imaginary part cancels out, resulting in a real number (without imaginary part).
- Conjugates are particularly useful in rationalizing denominators.
Magnitude
Magnitude or modulus of a complex number is a measure of its size or length in the complex plane. It is computed using the Pythagorean theorem and is always a non-negative real number. For a complex number \(a + bi\), the magnitude is \(|a + bi| = \sqrt{a^2 + b^2}\).
- Useful in finding the distance of points in the complex plane from the origin.
- Informs the size or absolute scale of the complex number.
- Helps in converting complex numbers to polar form.
Argument of a Complex Number
The argument of a complex number refers to the angle formed with the positive real axis in the complex plane. Calculated with the inverse tangent function, \(\tan^{-1}\), it provides a direction for the complex number in polar coordinates.
- Offers a geometric interpretation of complex numbers.
- Essential for converting to trigonometric forms.
- Influences the phase or angle in complex analysis.
