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Chapter 6: Problem 49

Write each quotient in the form \(a+\) bi. $$ \frac{-2+6 i}{2} $$

Short Answer

Expert verified
-1 + 3i

Step by step solution

01

Write the given expression

Write the given quotient \ \ \ \( \frac{-2+6i}{2} \)
02

Separate the numerator

Separate the real and imaginary parts of the numerator over the denominator: \ \ \( \frac{-2}{2} + \frac{6i}{2} \)
03

Simplify each part

Simplify each part separately: \ \ \( -1 + 3i \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quotient of Complex Numbers
The quotient of complex numbers involves dividing one complex number by another. To illustrate, consider our problem: \( \frac{-2+6i}{2} \). Complex numbers are usually written in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined as \( i^2 = -1 \). For division:
- Complex numbers can't be divided in the same way as regular fractions.
- We need to separate the real and imaginary parts and simplify each part individually.
In our example:
- The numerator is \( -2 + 6i \) and the denominator is \( 2 \).
By separating the fraction into two parts, we get:
\( \frac{-2}{2} + \frac{6i}{2} \).
Simplifying Complex Numbers
Once you've separated the numerator over the denominator, the next step is simplifying each part. Let's break it down further using our example. We had:
\( \frac{-2}{2} + \frac{6i}{2} \)
- Simplify the real part: \( \frac{-2}{2} = -1 \).
- Simplify the imaginary part: \( \frac{6i}{2} = 3i \).
This means that our complex number simplifies to:
\( -1 + 3i \)
Each term in the complex number is now in its simplest form. This simplification process is crucial as it gives us a clearer understanding of the real and imaginary parts.
Imaginary and Real Parts
Complex numbers consist of two parts: the real part and the imaginary part. In our final simplified form:
\( -1 + 3i \)
- The real part is \( -1 \)
- The imaginary part is \( 3i \) (with \( 3 \) being the coefficient of the imaginary unit \( i \))
Understanding this split helps in many complex number operations, including addition, subtraction, and especially when visualizing them on a complex plane where the x-axis represents the real part and the y-axis represents the imaginary part.
Knowing how to separate and simplify these parts is essential for mastering complex number arithmetic.

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