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Chapter 2: Problem 20

Write the complex number in standard form. \(\sqrt{-0.0049}\)

Short Answer

Expert verified
The standard form of the complex number \(\sqrt{-0.0049}\) is \(0.07i\).

Step by step solution

01

Identify the square root of a negative number

The square root of a negative number is an imaginary number. In this case, the square root of -0.0049 is \(i \sqrt{0.0049}\). Here, 'i' is the imaginary unit with the property \(i^2 = -1\).
02

Calculate the square root of the magnitude

Next, calculate the square root of 0.0049. \(\sqrt{0.0049} = 0.07\).
03

Complete the standard form

Finally, to write the complex number (which is an imaginary number in this case) in standard form, multiply the 'i' by the calculated square root. The standard form of our number then is \(0.07i\).

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Key Concepts

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

Imaginary Numbers
When dealing with complex numbers, imaginary numbers are key components to understand. Imaginary numbers are those numbers that give negative results when squared.
They are not found on the regular number line. Instead, they exist perpendicular to the real numbers on the complex plane.
The imaginary unit is denoted by 'i', which is defined as \(i = \sqrt{-1}\).
This fundamental concept allows us to work with square roots of negative numbers, which aren't possible within the realm of real numbers.
For example, the square root of \(-4\) would be \(2i\) because \(2i \times 2i = -4\).
  • Imaginary numbers enable us to expand our number system.
  • They are crucial in fields like engineering and physics.
  • Understanding them helps us solve equations beyond limitation of real numbers.
Mastering imaginary numbers allows you to write and manipulate complex numbers effectively.
Standard Form
Complex numbers, like the one in our exercise, are typically expressed in a specific format known as the standard form. This form is \(a + bi\) where 'a' represents the real part and 'bi' represents the imaginary part.
In this notation, both 'a' and 'b' are real numbers.
Even if a complex number has no real part (where 'a' is zero), it can still be represented in standard form, written as just 'bi'.
Our exercise resulted in just a pure imaginary number, \(0.07i\).
Sometimes complex numbers appear as just imaginary numbers with no real counterpart but still follow the standard format.
  • The letter 'i' indicates the imaginary component.
  • Standard form helps in algebraic operations such as addition, subtraction, and multiplication of complex numbers.
  • It is foundational for further complex analysis and applications.
Utilizing standard form eases calculation and comparison of complex numbers.
Square Roots of Negative Numbers
Square roots of negative numbers introduce the fascinating concept of imaginary numbers.
When we take the square root of a negative number, we step beyond the bounds of the real number system.
Basically, every negative number's square root will include the imaginary unit 'i'.
To break it down, let's say we're looking at \(\sqrt{-9}\). Normally, this would seem impossible as no real number multiplied by itself results in \(-9\).
By introducing the imaginary unit, we express this as \(3i\), where 'i' manages the negative sign.
  • Calculating \(\sqrt{-x}\) always results in \(i\sqrt{x}\).
  • This allows us to simplify expressions with negative radicals.
  • It also helps extend other mathematical operations into complex numbers.
Thus, understanding square roots of negative numbers becomes instrumental in working with imaginary and complex numbers.

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Most popular questions from this chapter

A rectangular page is designed to contain 64 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on each side are \(1 \frac{1}{2}\) inches wide. What should the dimensions of the page be so that the least amount of paper is used?

Use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. \(h(x)=\frac{12-2 x-x^{2}}{2(4+x)}\)

Solve the inequality. (Round your answers to two decimal places.) \(-0.5 x^{2}+12.5 x+1.6>0\)

Solve the inequality and graph the solution on the real number line. \(2 x^{3}+13 x^{2}-8 x-46 \geq 6\)

Solve the inequality and graph the solution on the real number line. \(-2 x^{2}+6 x+15 \leq 0\)
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