Division Of Complex Numbers Worksheet: Practice & Solve
Table of Contents :
Complex numbers are an essential part of mathematics, particularly in fields such as engineering, physics, and applied sciences. They allow us to work with equations that have no real solutions and provide a framework for understanding phenomena such as electrical circuits and waveforms. In this article, we'll dive into the division of complex numbers, providing you with practice problems, solutions, and tips to master this crucial concept. Let's get started! 📚✨
Understanding Complex Numbers
Before we delve into division, let's briefly review what complex numbers are. A complex number is expressed in the form:
[ z = a + bi ]
Where:
- ( a ) is the real part
- ( b ) is the imaginary part
- ( i ) is the imaginary unit, defined as ( i = \sqrt{-1} )
Basic Operations with Complex Numbers
The fundamental operations with complex numbers include addition, subtraction, multiplication, and division. We will focus on division in this article.
Division of Complex Numbers
To divide complex numbers, we follow these steps:
-
Multiply the numerator and denominator by the conjugate of the denominator.
- The conjugate of a complex number ( b + ci ) is ( b - ci ).
-
Simplify the expression.
Formula for Division
If we want to divide two complex numbers, ( z_1 = a + bi ) and ( z_2 = c + di ), the formula is:
[ \frac{z_1}{z_2} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} ]
This results in:
[ \frac{z_1}{z_2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2} i ]
Practice Problems
Problem Set
Here’s a set of practice problems you can try to enhance your skills in dividing complex numbers:
| Problem Number | Problem |
|---|---|
| 1 | (\frac{3 + 4i}{1 + 2i}) |
| 2 | (\frac{5 - 2i}{3 + 3i}) |
| 3 | (\frac{-2 + 5i}{4 - i}) |
| 4 | (\frac{7 + 8i}{0 + 1i}) |
| 5 | (\frac{1 + i}{2 - 3i}) |
Important Notes
When dividing complex numbers, remember that the denominator should never equal zero, as this makes the expression undefined.
Solutions
Let’s go through the solutions step by step for each of the problems listed above.
Solution to Problem 1
Problem: (\frac{3 + 4i}{1 + 2i})
-
Multiply numerator and denominator by the conjugate of the denominator:
[ \frac{(3 + 4i)(1 - 2i)}{(1 + 2i)(1 - 2i)} = \frac{3 - 6i + 4i + 8}{1 + 4} = \frac{11 - 2i}{5} ]
-
Divide:
[ \frac{11}{5} - \frac{2}{5}i ]
Final Answer: (\frac{11}{5} - \frac{2}{5}i)
Solution to Problem 2
Problem: (\frac{5 - 2i}{3 + 3i})
-
Multiply by the conjugate:
[ \frac{(5 - 2i)(3 - 3i)}{(3 + 3i)(3 - 3i)} = \frac{15 - 15i - 6i + 6}{9 + 9} = \frac{21 - 21i}{18} ]
-
Simplify:
[ \frac{7}{6} - \frac{7}{6}i ]
Final Answer: (\frac{7}{6} - \frac{7}{6}i)
Solution to Problem 3
Problem: (\frac{-2 + 5i}{4 - i})
-
Multiply by the conjugate:
[ \frac{(-2 + 5i)(4 + i)}{(4 - i)(4 + i)} = \frac{-8 - 2i + 20i + 5}{16 + 1} = \frac{-3 + 18i}{17} ]
-
Simplify:
[ -\frac{3}{17} + \frac{18}{17}i ]
Final Answer: (-\frac{3}{17} + \frac{18}{17}i)
Solution to Problem 4
Problem: (\frac{7 + 8i}{0 + 1i})
-
Multiply by the conjugate:
[ \frac{(7 + 8i)(0 - i)}{i(0 - i)} = \frac{-7i - 8}{-1} = 8 + 7i ]
Final Answer: (8 + 7i)
Solution to Problem 5
Problem: (\frac{1 + i}{2 - 3i})
-
Multiply by the conjugate:
[ \frac{(1 + i)(2 + 3i)}{(2 - 3i)(2 + 3i)} = \frac{2 + 3i + 2i - 3}{4 + 9} = \frac{-1 + 5i}{13} ]
-
Simplify:
[ -\frac{1}{13} + \frac{5}{13}i ]
Final Answer: (-\frac{1}{13} + \frac{5}{13}i)
Conclusion
Dividing complex numbers can be straightforward once you understand the steps involved. Practice regularly with different problems to sharpen your skills! With time and practice, you will find that this fundamental concept becomes second nature. Don’t hesitate to use the examples and solutions provided here as a resource in your mathematical journey! Happy solving! 🧮🌟