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Chapter 10: Problem 1

Express each of the following numbers in the form \(a+b i\). $$5+\sqrt{-4}$$

Short Answer

Expert verified
5 + 2i

Step by step solution

01

Understanding the Problem

We need to express the given number, \(5 + \sqrt{-4}\), in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with \(i^2 = -1\).
02

Identify the Imaginary Component

Recognize that \(\sqrt{-4}\) can be rewritten by using the identity \(\sqrt{-1} = i\). Thus, \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2 \cdot i\).
03

Rewrite in Standard Form

Substitute the expression for \(\sqrt{-4}\) back into the original expression to get \(5 + 2i\). Now, \(5 + 2i\) is in the form \(a + bi\).
04

Simplify if Necessary

In this particular problem, there are no further simplifications needed. Thus, the expression \(5 + 2i\) already represents the answer in the required form \(a + bi\).

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Key Concepts

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

Imaginary Unit
When we deal with square roots of negative numbers, we encounter a fascinating concept called the imaginary unit. Denoted by the letter \(i\), this unit helps us manage the square roots of negative numbers. The imaginary unit is defined by the equation \(i^2 = -1\). This means when we square \(i\), we get \(-1\). Having the imaginary unit lets us calculate and express complex numbers.
For example:
  • \(\sqrt{-1} = i\)
  • \(\sqrt{-4} = 2i\)
Whenever you see \(i\) in mathematics, it indicates the involvement of an imaginary component. Remember, \(i\) isn't a real number, but it is immensely useful for handling square roots of negative values.
Real Numbers
Real numbers constitute the major part of numbers we use daily. These include all positive numbers, negative numbers, fractions, and irrational numbers like \(\pi\) or \(\sqrt{2}\). Real numbers are tangible and can plot on a number line.
In the expression \(a + bi\), the part \(a\) is the real number, while \(bi\) involves the imaginary unit.
Here's how you can identify real numbers:
  • They have a specific location on the number line.
  • They include both rational and irrational numbers.
  • A real number does not involve \(i\).
In the given expression \(5 + 2i\), \(5\) is the real part, making the number complex since it also incorporates an imaginary part \(2i\).
Square Roots of Negative Numbers
Finding the square root of a negative number was once considered impossible because no real number multiplied by itself results in a negative. This challenge led to the introduction of imaginary numbers that make square roots of negative numbers manageable.
Here's how it works:
  • Observe the negative number. For example, \(-4\).
  • Break it down using the identity \(\sqrt{-1} = i\) and the square root of the positive part, in this case, \(\sqrt{4}\).
  • So, \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i\).
Using this method, we combine the real part (from the positive square root) with the imaginary unit \(i\). This technique is vital in complex analysis and helps express complex numbers in forms like \(a + bi\).

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Most popular questions from this chapter

Perform the following operations and express your answer in the form \(a+b i\). $$(1+i)(2-3 i)$$

Write each complex number in exponential form. $$-3 \sqrt{3}-3 i$$

Perform the following operations and express your answer in the form \(a+b i\). $$\frac{13}{5-12 i}$$

Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the reciprocals \(\frac{1}{z_{1}}, \frac{1}{z_{2}^{\prime}}\) the product \(z_{1} z_{2}\) and the quotient \(\frac{z_{1}}{z_{2}}(-\pi<\theta<\pi)\). $$z_{1}=\sqrt{5}+i \sqrt{5} \text { and } z_{2}=2 i \sqrt{2}$$

Perform the following operations and express your answer in the form \(a+b i\). $$\frac{5-\sqrt{-144}}{3+\sqrt{-16}}$$
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