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Magazine Chapter 6: Problem 98
Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-3+8 i}{-2+i} $$
Short Answer
Expert verified
The quotient is \(\frac{14}{5} - \frac{13}{5}i\).
Step by step solution
01
Write the problem
To solve the problem, we have the complex division \( \frac{-3+8i}{-2+i} \). We need to find the quotient and express it in the form \( a + bi \).
02
Multiply numerator and denominator by the conjugate of the denominator
We multiply both the numerator and the denominator by the conjugate of the denominator, which is \(-2-i\). So we have: \[\left(\frac{-3+8i}{-2+i}\right) \cdot \left(\frac{-2-i}{-2-i}\right)\].
03
Simplify the denominator
Now calculate the denominator by using the difference of squares: \((-2 + i)(-2-i) = (-2)^2 - (i)^2 = 4 - (-1) = 5\).
04
Simplify the numerator
Now calculate the numerator: \((-3+8i)(-2-i) = (-3)(-2) - (-3)(i) + (8i)(-2) - (8i)(i)\) which equals \[6 + 3i - 16i - 8i^2\].
05
Substitute values back and simplify
Note that \(i^2 = -1\), so the expression becomes \[6 + 3i - 16i + 8 = 14 - 13i\].The final division is: \[\frac{14 - 13i}{5} = \frac{14}{5} - \frac{13}{5}i\].
06
Write the solution in standard form
The quotient expressed in standard complex form is \(\frac{14}{5} - \frac{13}{5}i\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Division
In mathematics, division involving complex numbers is a common task, and it requires us to simplify expressions to a more manageable form. Complex division is the process of dividing one complex number by another. Complex numbers are of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\).
To perform complex division, we follow these steps:
To perform complex division, we follow these steps:
- Identify the complex numbers in the division, usually expressed as \(\frac{a+bi}{c+di}\).
- To simplify the expression, we need to eliminate the imaginary unit \(i\) from the denominator.
- This is done by utilizing the 'conjugate of the denominator'.
Conjugate of a Denominator
The conjugate of a complex number \(c+di\) is \(c-di\). This concept is crucial when dividing complex numbers because it helps eliminate the imaginary part in the denominator.
Here's how it works:
Here's how it works:
- For the complex number \(-2 + i\), its conjugate is \(-2 - i\).
- When we multiply a complex number by its conjugate, we create a real number, therefore simplifying our division task.
- Specifically, to divide \(\frac{-3+8i}{-2+i}\), we multiply both numerator and denominator by \(-2-i\).
Difference of Squares
The difference of squares is an algebraic identity useful in simplifying expressions, especially when dealing with complex numbers. It states that \((a+b)(a-b) = a^2 - b^2\).
For complex numbers:
For complex numbers:
- When we multiply a complex number by its conjugate, such as \((-2+i)(-2-i)\), we are applying the difference of squares.
- The result is \((-2)^2 - (i)^2 = 4 - (-1) = 5\).
- This transforms the denominator into a non-imaginary real number, significantly simplifying the division process.
Standard Form of Complex Numbers
The standard form of a complex number is expressed as \(a+bi\), where \(a\) represents the real part and \(b\) represents the imaginary part. It's the conventional way to represent complex numbers.
To express your complex quotient in standard form:
To express your complex quotient in standard form:
- Simplify your expression so that you separate the real and imaginary components.
- For instance, the division result \(\frac{14 - 13i}{5}\) simplifies to \(\frac{14}{5} - \frac{13}{5}i\).
- This formatting allows clearer communication and further mathematical manipulation or computation.
