Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 29
Evaluate the expression and write the result in the form \(a+b i .\) $$ \frac{2-3 i}{1-2 i} $$
Short Answer
Expert verified
The result is \(\frac{8}{5} + \frac{1}{5} i\).
Step by step solution
01
Understand the Problem
The task is to evaluate the expression \( \frac{2-3i}{1-2i} \) and express it in the form \( a + bi \), which is the standard form of a complex number.
02
Multiply by the Conjugate
To simplify the expression, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 1-2i \) is \( 1+2i \). Thus, multiply:\[\frac{2-3i}{1-2i} \times \frac{1+2i}{1+2i}\]
03
Apply Distribution (Numerator)
Distribute \( (2-3i) (1+2i) \):\[(2-3i)(1+2i) = 2 \times 1 + 2 \times 2i + (-3i) \times 1 + (-3i) \times 2i\]\[= 2 + 4i - 3i - 6i^2\]Since \(i^2 = -1\), replacing \(-6i^2\) gives:\[= 2 + 4i - 3i + 6\]Simplifying gives:\[= 8 + i\]
04
Apply Distribution (Denominator)
Distribute \( (1-2i)(1+2i) \):\[(1-2i)(1+2i) = 1^2 + 1 \times 2i - 2i \times 1 - 2i \times 2i\]\[= 1 + 2i - 2i - 4i^2\]Since \(i^2 = -1\), replacing \(-4i^2\) gives:\[= 1 + 4\]Thus, the denominator simplifies to:\[= 5\]
05
Simplify the Expression
Now, place the results from Steps 3 and 4 into the fraction:\[\frac{8+i}{5}\]Divide both terms in the numerator by 5:\[= \frac{8}{5} + \frac{1}{5} i\]
06
Write the Result in Standard Form
The expression is now in the form \( a + bi \), where \( a = \frac{8}{5} \) and \( b = \frac{1}{5} \). Thus, the result in the form \( a + bi \) is:\[\frac{8}{5} + \frac{1}{5} i\]
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.
Complex Number Simplification
In mathematics, simplifying complex numbers is a crucial step to make expressions easier to handle. Complex numbers are in the form of \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined by \( i^2 = -1 \).
When given an expression of complex numbers in fraction form, such as \( \frac{2-3i}{1-2i} \), the goal is to remove the imaginary part from the denominator. This process enables easier evaluation of the expression.
Here's a step-by-step breakdown:
When given an expression of complex numbers in fraction form, such as \( \frac{2-3i}{1-2i} \), the goal is to remove the imaginary part from the denominator. This process enables easier evaluation of the expression.
Here's a step-by-step breakdown:
- Multiply both numerator and denominator by the conjugate of the denominator to eliminate the imaginary part.
- Distribute and simplify each multiplication, using \( i^2 = -1 \) to replace any squares of \( i \).
- Combine like terms and finalize the expression in standard form by dividing each term by the new denominator.
Conjugate of Complex Numbers
The conjugate of a complex number is a key concept when it comes to simplifying fractions involving complex numbers. If you have a complex number \( a + bi \), its conjugate is \( a - bi \).
The conjugate is used to rationalize the denominator of a complex fraction. In the exercise \( \frac{2-3i}{1-2i} \), the conjugate of the denominator \( 1-2i \) is \( 1+2i \). By multiplying both the numerator and the denominator by the conjugate, the denominator becomes a real number.
The conjugate is used to rationalize the denominator of a complex fraction. In the exercise \( \frac{2-3i}{1-2i} \), the conjugate of the denominator \( 1-2i \) is \( 1+2i \). By multiplying both the numerator and the denominator by the conjugate, the denominator becomes a real number.
- Multiplication of a complex number by its conjugate results in a real number because \( (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 \).
- With a real denominator, the fraction becomes much easier to deal with.
Standard Form of Complex Numbers
The standard form of complex numbers is \( a + bi \), where \( a \) represents the real part, and \( b \) represents the imaginary part, with \( i \) being the imaginary unit. Expressing a complex number in this form is essential for clarity and ease of understanding.
In our original exercise, after simplifying the fraction \( \frac{2-3i}{1-2i} \), we obtained a simplified expression \( \frac{8}{5} + \frac{1}{5}i \), which is indeed in the standard form \( a + bi \).
Here are some key points:
In our original exercise, after simplifying the fraction \( \frac{2-3i}{1-2i} \), we obtained a simplified expression \( \frac{8}{5} + \frac{1}{5}i \), which is indeed in the standard form \( a + bi \).
Here are some key points:
- This standard form clearly separates the real and imaginary components of the complex number.
- It helps in performing further calculations, like addition or subtraction of complex numbers, more straightforwardly.
- When expressing solutions, ensuring the result is in this format aids in understanding and interpreting the result accurately.
