Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 9
If \(c=2+i\) and \(d=4+3 i\) find \(c d\) and \(c / d\). Verify that the absolute value \(|c d|\) equals \(|c|\) times \(|d|,\) and \(|c / d|\) equals \(|c|\) divided by \(|d|\)
Short Answer
Expert verified
The product \(cd = 5 + 10i\); \(c/d = \frac{11}{25} - \frac{2}{25}i\). Absolute values verified.
Step by step solution
01
Calculate Product of Complex Numbers
To find the product of \(c\) and \(d\), use the formula for multiplying complex numbers: \((a+bi)(c+di) = (ac-bd) + (ad+bc)i\). Thus, \(cd = (2+i)(4+3i)\). Perform the multiplication: \(2 \cdot 4 + 2 \cdot 3i + i \cdot 4 + i \cdot 3i = 8 + 6i + 4i + 3i^2\). Since \(i^2 = -1\), simplify to get \(cd = 8 + 10i - 3 = 5 + 10i\).
02
Calculate Division of Complex Numbers
To find \(c/d\), multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(4 + 3i\) is \(4 - 3i\). Thus, \(\frac{c}{d} = \frac{(2+i)}{(4+3i)} \cdot \frac{(4-3i)}{(4-3i)} = \frac{(2+i)(4-3i)}{(4+3i)(4-3i)}\). Calculate the numerator: \((2+i)(4-3i) = 8 - 6i + 4i - 3i^2 = 8 - 2i + 3 = 11 - 2i\). Calculate the denominator: \((4+3i)(4-3i) = 16 - 9i^2 = 16 + 9 = 25\). Therefore, \(\frac{c}{d} = \frac{11-2i}{25} = \frac{11}{25} - \frac{2}{25}i\).
03
Calculate Absolute Values
Find \(|c|\), \(|d|\), and verify the properties. The absolute value of a complex number \(a + bi\) is \(|a + bi| = \sqrt{a^2 + b^2}\). For \(c = 2+i\), \(|c| = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}\). For \(d = 4+3i\), \(|d| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\).
04
Verify Absolute Value of Product
Check if \(|cd| = |c| \cdot |d|\). The product \(cd = 5 + 10i\) has an absolute value \(|5 + 10i| = \sqrt{5^2 + 10^2} = \sqrt{25 + 100} = \sqrt{125}\). Compare this to \(|c| \cdot |d| = \sqrt{5} \cdot 5 = \sqrt{125}\). Thus, \(|cd| = \sqrt{125}\), verifying that it equals \(|c| \cdot |d|\).
05
Verify Absolute Value of Quotient
Verify \(|\frac{c}{d}| = \frac{|c|}{|d|}\). The absolute value of the quotient \(\frac{11-2i}{25}\) is \(\left| \frac{11-2i}{25} \right| = \frac{|11-2i|}{25} = \frac{\sqrt{11^2 + (-2)^2}}{25} = \frac{\sqrt{121 + 4}}{25} = \frac{\sqrt{125}}{25}\). Compare with \(|c| / |d| = \sqrt{5}/5 = \sqrt{125}/25\). Thus, \(|\frac{c}{d}| = \frac{|c|}{|d|}\), confirming the relationship.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Multiplying Complex Numbers
When you're multiplying complex numbers like \( (2 + i) \) and \( (4 + 3i) \), the process involves using a formula similar to the distributive property in algebra. This formula can be expressed as \((a+bi)(c+di) = (ac-bd) + (ad+bc)i\).
The main steps are:
Multiplying complex numbers can initially seem tricky, but with practice, it becomes much like combining terms in algebra. Always remember to apply the rule \(i^2 = -1\) when dealing with the square of the imaginary part.
The main steps are:
- Multiply the real parts: \(2 \times 4 = 8\).
- Multiply the imaginary parts: \(2 \times 3i = 6i\) and \(i \times 4 = 4i\).
- The product of the imaginary units, \(i^2\), is \(-1\). So, \(i \times 3i = -3\).
- Add all these results together: \(8 + 6i + 4i - 3\).
- Combine like terms to get the final result: \(5 + 10i\).
Multiplying complex numbers can initially seem tricky, but with practice, it becomes much like combining terms in algebra. Always remember to apply the rule \(i^2 = -1\) when dealing with the square of the imaginary part.
Dividing Complex Numbers
Dividing complex numbers involves a special step called "rationalizing the denominator." This means you'll use the complex conjugate of the denominator to simplify the expression. For the complex numbers \(c = 2 + i\) and \(d = 4 + 3i\), we find:
The division then resolves to \(\frac{11}{25} - \frac{2}{25}i\). This method is crucial because it ensures that the denominator is a real number, simplifying the expression of the quotient.
- Determine the complex conjugate of \(4 + 3i\), which is \(4 - 3i\).
- Multiply both numerator and denominator by this conjugate: \(\frac{(2+i)(4-3i)}{(4+3i)(4-3i)}\).
- Carry out the multiplication separately for both parts:
- Numerator: \((2+i)(4-3i) = 11 - 2i\).
- Denominator: \((4+3i)(4-3i) = 25\) as it becomes a difference of squares, \(16 + 9\).
The division then resolves to \(\frac{11}{25} - \frac{2}{25}i\). This method is crucial because it ensures that the denominator is a real number, simplifying the expression of the quotient.
Absolute Value of Complex Numbers
The absolute value of a complex number \(a + bi\) represents its distance from the origin in the complex plane. You find it using the formula \(|a + bi| = \sqrt{a^2 + b^2}\). This concept is crucial for understanding the magnitude or size of complex numbers.
For example:
For example:
- For the complex number \(c = 2 + i\), \(|c| = \sqrt{2^2 + 1^2} = \sqrt{5}\).
- For \(d = 4 + 3i\), the absolute value is \(|d| = \sqrt{16 + 9} = 5\).
Complex Conjugate
The complex conjugate is a powerful tool when working with complex numbers, especially for simplifying division. The complex conjugate of a number \(a + bi\) is \(a - bi\). By multiplying by the conjugate, you turn the denominator into a real number due to the identity \((a + bi)(a - bi) = a^2 + b^2\).
This makes the process straightforward:
This makes the process straightforward:
- For a complex number like \(4 + 3i\), the conjugate is \(4 - 3i\).
- Using it in division, both numerator and denominator are multiplied, transforming the denominator into a real number: \((4 + 3i)(4 - 3i) = 25\).
