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Chapter 3: Problem 6

Find the real and imaginary parts of the complex number. $$-6+4 i$$

Short Answer

Expert verified
Real: -6, Imaginary: 4

Step by step solution

01

Identify the Real Part

A complex number is generally expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. In the complex number \(-6 + 4i\), the real part is the number without the \(i\), which is \(-6\).
02

Identify the Imaginary Part

The imaginary part of a complex number \(a + bi\) is the coefficient of \(i\). In the complex number \(-6 + 4i\), the coefficient of \(i\) is \(4\), so the imaginary part is \(4\).

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Key Concepts

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

Real Part
In mathematics, the real part of a complex number refers to the component of the complex number that does not involve the imaginary unit \(i\). A standard complex number is written in the form \(a + bi\), where \(a\) represents the real part. This part is a regular real number that you are familiar with from everyday arithmetic.
For example, in the complex number \(-6 + 4i\), \(-6\) is the real part.
  • The real part is not affected by the presence of the imaginary unit \(i\).
  • It is the number that can be plotted on the real axis of the complex plane.
By understanding the real part, you'll be better equipped to handle the arithmetic of complex numbers.
Imaginary Part
The imaginary part of a complex number is the portion that includes the imaginary unit \(i\). This part of a complex number is equally crucial as it extends our numeric system to accommodate roots of negative numbers.
For a complex number written as \(a + bi\), the imaginary part corresponds to \(b\). This "\(b\)" is not the \(i\) itself, but the coefficient in front of \(i\).
In the complex number \(-6 + 4i\), the imaginary part is \(4\).
  • It shows how much of the imaginary unit \(i\) is included.
  • The imaginary part can also be plotted, but on the imaginary (vertical) axis of the complex plane.
Understanding the imaginary part is key for working with complex functions and equations.
Imaginary Unit
The imaginary unit, denoted by \(i\), plays a fundamental role in complex numbers. By definition, \(i\) is the square root of \(-1\). This means \(i^2 = -1\), a property that is used extensively in computations with complex numbers.
The introduction of \(i\) allows us to expand our number system beyond real numbers to solve equations that wouldn't otherwise have solutions, like \(x^2 + 1 = 0\).
  • \(i\) helps in expressing square roots of negative numbers.
  • It forms the imaginary axis in the complex plane, orthogonal to the real axis.
By integrating \(i\), mathematics captures a broader range of phenomena and problems. Remember, \(i\) itself is not a number but a concept that extends how we understand numbers.

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Most popular questions from this chapter

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$t(x)=\frac{x-2}{x^{2}-4 x}$$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{2}+5 x+4}{x-3}$$

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. $$2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0 ; \quad[-2,5] \text { by }[-40,40]$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{2 x(x+2)}{(x-1)(x-4)}$$

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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