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Chapter 3: Problem 8

Find the real and imaginary parts of the complex number. $$\frac{4+7 i}{2}$$

Short Answer

Expert verified
Real part is 2; imaginary part is \( \frac{7}{2} \).

Step by step solution

01

Identify the Complex Number Format

The given expression is \( \frac{4+7i}{2} \). This complex number is in the form \( a + bi \), where \( a = 4 \) is the real part, and \( bi = 7i \) is the imaginary part.
02

Simplify the Complex Number

Divide both the real part and the imaginary part by 2. This means you compute \( \frac{4}{2} + \frac{7i}{2} \).
03

Compute the Real Part

The real part of the complex number is \( \frac{4}{2} = 2 \).
04

Compute the Imaginary Part

The imaginary part of the complex number is \( \frac{7}{2}i \), or \( \frac{7}{2} \) when expressed as a multiplier of \( i \).

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Key Concepts

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

real part
When dealing with complex numbers, it's essential to understand their components. A complex number is usually written in the form \( a + bi \), where \( a \) and \( b \) are real numbers. The term \( a \) represents the "real part" of the complex number.
To find the real part of a complex number given in the expression \( \frac{4 + 7i}{2} \), we look at the numerator, which is \( 4 + 7i \). The part without the \( i \), the real part, is \( 4 \). However, since the entire expression \( \frac{4 + 7i}{2} \) is being divided by 2, we need to divide the real part \( 4 \) by 2 as well.
This computation leads us to:
  • The expression for real part: \( \frac{4}{2} \)
  • The real part is: 2
imaginary part
The imaginary part of a complex number is based on the coefficient of \( i \), a symbol signifying the square root of -1. In our expression \( \frac{4 + 7i}{2} \), the term \( 7i \) is where we find our imaginary part. The coefficient \( 7 \) is key, as the imaginary part centers on this number.
To extract the imaginary part from the expression \( \frac{4 + 7i}{2} \), we must likewise divide its coefficient \( 7 \) by 2. The solution breaks down as follows:
  • Imaginary part from \( 7i \): \( \frac{7}{2}i \)
  • Simplified imaginary part: \( \frac{7}{2} \) times \( i \)
Visualizing this helps make the concept clear: whenever you have a complex number, look to the \( bi \) part, and the coefficient of \( i \) tells you about the imaginary part.
simplifying complex numbers
Simplifying complex numbers can make them more digestible both for performing calculations and for proper understanding. The expression \( \frac{4 + 7i}{2} \) invites us to simplify it by distributing the division across the real and imaginary components separately.
Here's a quick step-by-step approach to simplify it:
  • Break down the original expression: \( \frac{4}{2} + \frac{7i}{2} \)
  • Simplify each part separately.
    • Real part simplified to 2
    • Imaginary part simplified to \( \frac{7}{2}i \)
The final result of these operations is \( 2 + \frac{7}{2}i \). By separately addressing each component, we render the complex number in a more manageable form. This method also helps to ensure accuracy in further mathematical operations with the complex number.

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Most popular questions from this chapter

So far, we have worked only with polynomials that have real coefficients. These exercises involve polynomials with real and imaginary coefficients. Find all solutions of the equation. (a) \(2 x+4 i=1\) (b) \(x^{2}-i x=0\) (c) \(x^{2}+2 i x-1=0\) (d) \(i x^{2}-2 x+i=0\)

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}-6 x+10=0$$

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}-4 x+5=0$$

The Conjugate Zeros Theorem says that the complex zeros of a polynomial with real coefficients occur in complex conjugate pairs. Explain how this fact proves that a polynomial with real coefficients and odd degree has at least one real zero.

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, comect to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{4}}{x^{2}-2}$$
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