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Chapter 3: Problem 8

Find the real and imaginary parts of the complex number. $$i \sqrt{3}$$

Short Answer

Expert verified
Real part: 0, Imaginary part: \( \sqrt{3} \).

Step by step solution

01

Understanding the Problem

We are given the complex number \( i \sqrt{3} \). A complex number is generally expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Our task is to identify these parts.
02

Identifying the Real Part

In the expression \( i \sqrt{3} \), there is no separate real number added, i.e., it's equivalent to \( 0 + i \sqrt{3} \). Hence, the real part is \( a = 0 \).
03

Identifying the Imaginary Part

In the expression \( 0 + i \sqrt{3} \), the coefficient of \( i \) is \( \sqrt{3} \). Therefore, the imaginary part is \( b = \sqrt{3} \).
04

Conclusion

The real part of the complex number \( i \sqrt{3} \) is 0 and the imaginary part is \( \sqrt{3} \).

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Key Concepts

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

Real Part
In the realm of complex numbers, the **real part** is the component that does not involve the imaginary unit, which is represented by \( i \). A complex number has the form \( a + bi \), where \( a \) is the real part.
  • In the complex number \( i \sqrt{3} \), there isn't an explicit real number part included in the expression.
  • This means \( i \sqrt{3} \) is essentially equivalent to \( 0 + i \sqrt{3} \), where the "0" represents the real part.
Thus, it is crucial to recognize that the lack of a visible real component in the equation indicates that the real part is zero. Remember that this zero component is important, as it designates where the complex number stands on the real-imaginary plane. Without it, the number would simply be imaginary, residing on the vertical imaginary axis alone.
Imaginary Part
The **imaginary part** of a complex number is characterized by its association with the imaginary unit \( i \), which satisfies the equation \( i^2 = -1 \). In a complex expression, \( a + bi \), the term \( bi \) tells us about the imaginary part, with \( b \) being its coefficient.
  • In the expression \( i \sqrt{3} \), understanding the role of \( \sqrt{3} \) is essential.
  • Here, \( \sqrt{3} \) is the coefficient that multiplies the imaginary unit \( i \).
Therefore, the imaginary part of the complex number \( i \sqrt{3} \) is \( \sqrt{3} \). It's important to realize that this component determines the position of the number vertically on the complex plane, influencing the "height" or "position" along the imaginary axis. This positioning is vital for visualizing complex numbers.
Complex Number Representation
The **representation** of a complex number is a unique way to combine its real and imaginary components to visualize it on the complex plane. A generic complex number is expressed as \( a + bi \), blending both real and imaginary parts together.
  • This form helps in understanding and operating with numbers that have both dimensions.
  • For \( i \sqrt{3} \), while it appears to be solely imaginary, the format \( 0 + i \sqrt{3} \) speaks to its full complex representation.
This format clarifies its positioning: with a real part of 0 (on the horizontal axis) and an imaginary part of \( \sqrt{3} \) (on the vertical axis). Understanding this representation is key to performing complex-related calculations and visualizing how such numbers map onto a two-dimensional plane. By identifying both portions accurately, students can gain a deeper insight into how these numbers behave in mathematical applications.

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Most popular questions from this chapter

Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$P(x)=x^{8}-x^{5}+x^{4}-x^{3}+x^{2}-x+1$$

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{x^{3}-2 x^{2}+16}{x-2}, g(x)=x^{2}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{3 x^{2}+6}{x^{2}-2 x-3}$$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=x^{5}-7 x^{4}+9 x^{3}+23 x^{2}-50 x+24$$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{2}+2 x}{x-1}$$
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