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Chapter 2: Problem 15

Write the complex number in standard form. $$-8 i-i^{2}$$

Short Answer

Expert verified
The standard form of the given complex number is 1 - 8i.

Step by step solution

01

Recognize and Use Imaginary Unit

In complex numbers, 'i' is the square root of -1. Thus, the square of 'i' i.e \(i^2\) equals -1.
02

Substitute Value of \(i^2\)

Now, substitute \(i^2\) with -1 in the given expression. So, -8i -\(i^2\) becomes -8i - (-1), which simplifies to -8i + 1.
03

Reorder terms

The given expression is now a combination of a real number and a complex number. However, the standard form of a complex number is a + bi. So, reorder the simplified expression as 1 - 8i.

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Key Concepts

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

Imaginary Unit
The imaginary unit is a fundamental concept in complex numbers, represented by the symbol 'i'. The defining property of 'i' is that it satisfies the equation \(i^2 = -1\). This means that 'i' is the principal square root of -1. Imaginary numbers enable us to work with square roots of negative numbers, which are not possible in the set of real numbers. This concept opens up a new dimension of numbers, known as complex numbers, providing solutions to equations that have no real solutions.
Standard Form
Complex numbers are typically expressed in a standard form, which is \(a + bi\). Here, 'a' is the real part, and 'b' is the imaginary part. Writing complex numbers in this format makes them easier to standardize and compare, as well as perform operations like addition, subtraction, and multiplication. For example, in the expression \(1 - 8i\), '1' is the real part, and '-8i' is the imaginary part. Standard form allows us to easily visualize the components of a complex number and is essential for analyzing and solving complex-number problems.
Real Part
The real part of a complex number is the component that does not involve the imaginary unit 'i'. In the standard form \(a + bi\), the real part is 'a'. This part of the complex number is essentially a real number, just as we encounter in everyday arithmetic. For the expression \(1 - 8i\), the real part is 1. The real part becomes particularly important when plotting complex numbers on a complex plane, as it helps determine the number's horizontal position.
Imaginary Part
The imaginary part of a complex number involves the imaginary unit 'i'. In the standard form \(a + bi\), it corresponds to 'b' in 'bi'. Although it is called the imaginary part, it represents a real number multiplied by 'i'. In the expression \(1 - 8i\), the imaginary part is '-8i', with '-8' being the coefficient of 'i'. This component is crucial for various mathematical applications, allowing us to extend real number concepts into the complex plane. It helps determine the vertical position when graphing complex numbers.

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Most popular questions from this chapter

Match the equation with a method you would use to solve it. Explain your reasoning. (Use each method once and do not solve the equations.) (a) \(3 x^{2}+5 x-11=0 \quad\) (i) Factoring (b) \(x^{2}+10 x=3 \quad\) (ii) Extracting square roots (c) \(x^{2}-16 x+64=0 \quad\) (iii) Completing the square (d) \(x^{2}-15=0 \quad\) (iv) Quadratic Formula

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$2 x^{3}+3 x^{2}<11 x+6$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$x^{3}-3 x^{2}-x>-3$$

Two planes leave simultaneously from Chicago's O'Hare Airport, one flying due north and the other due east (see figure). The northbound plane is flying 50 miles per hour faster than the eastbound plane. After 3 hours, the planes are 2440 miles apart. Find the speed of each plane.

Without performing any calculations, match the inequality with its solution. Explain your reasoning. (a) \(2 x \leq-6\) (b) \(-2 x \leq 6\) (c) \(|x+2| \leq 6\) (d) \(|x+2| \geq 6\) (i) \(x \leq-8\) or \(x \geq 4\) (ii) \(x \geq-3\) (iii) \(-8 \leq x \leq 4\) (iv) \(x \leq-3\)
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