Problem 30 - Elementary and Intermediate Algebra

Question

Write each number in the form a + bi. See Example 2. a. \(-43\) b. \(\sqrt{-169}\)

Step-by-Step Solution

Answer

a. \(-43 = -43 + 0i\); b. \(\sqrt{-169} = 0 + 13i\).

Step 1: Express Real Number in Complex Form (Part a)

To express \(-43\) in the form \(a + bi\), we recognize that this number is purely real, with no imaginary part. Thus, \(-43\) can be written as \(-43 + 0i\).

Step 2: Express Imaginary Number (Square Root) in Complex Form (Part b)

Now, consider the square root of a negative number, \(\sqrt{-169}\). Since square roots of negative numbers involve imaginary numbers, we rewrite \(\sqrt{-169}\) as \(\sqrt{169} \times \sqrt{-1}\). This simplifies to \(13i\), since \(\sqrt{169} = 13\) and \(\sqrt{-1} = i\). Therefore, \(\sqrt{-169} = 0 + 13i\).

Key Concepts

Real Numbers

Real numbers are the numbers that we use in everyday life. They include whole numbers like 0, 1, 2, negative numbers like -1, -2, and fractions or decimals like 1.5 or 3/4.
Real numbers can be plotted on a number line, making it easy to visualize their position relative to each other.
One unique aspect of real numbers is that they do not include imaginary units such as 'i'. When we express any real number, whether positive or negative, in complex form, it looks like this: \(-43 = -43 + 0i\).
This shows that its imaginary part is zero, hence it lies entirely on the real number line.

• Composed of both rational numbers (fractions) and irrational numbers (numbers that cannot be expressed as simple fractions, such as π or √2).
• No imaginary component; only the 'a' part in 'a + bi' form is present.
• Essentially just like the numbers you grew up using with a small twist to include a potential, but zero, imaginary component.

Imaginary Numbers

Imaginary numbers introduce an exciting twist to our otherwise straightforward number system.
These are numbers that, when squared, produce a negative result—including numbers like \(i\), the widely-used imaginary unit.The imaginary unit \(i\) is defined by the property \(i^2 = -1\).
This means that \(i\) is the square root of \(-1\).
Thus, when dealing with square roots of negative numbers, as we see in equations like \(\sqrt{-169}\), imaginary numbers naturally arise.A number purely imaginary will look like \(bi\) in complex form, where b is nonzero and a is zero.

• Used to extend the real number system to include solutions to equations that have no real solutions.
• Allow for the expression of negative square roots which otherwise wouldn’t make sense in real numbers alone.
• Introduces the concept of rotational vectors and more advanced mathematics beyond basic algebra.

Square Roots of Negative Numbers

Square roots of negative numbers can seem perplexing because no real number squared gives a negative result.
This is where the concept of imaginary numbers becomes useful.For example, \(\sqrt{-169}\) would ordinarily be unsolvable in terms of real numbers alone.
By using imaginary numbers, however, we reframe the problem: rewrite \(\sqrt{-169}\) as \(\sqrt{169} \times \sqrt{-1}\).
As such, it simplifies to \(13i\), with \(13\) from the \(\sqrt{169}\) and \(i\) because \(\sqrt{-1}\) equals \(i\).This solution demonstrates how every square root of a negative number can be expressed in terms of an imaginary unit.

• These roots reveal the power of the imaginary unit 'i' as it transforms impossible problems in the real number world into solvable ones in the complex number system.
• Vital in electrical engineering, quantum physics, and various fields leveraging complex numbers.
• Allows the unique expression of complex solutions in a mathematically consistent manner.