Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Find the modulus \(r\) of the number. Do not use a calculator. $$15 i$$

Step-by-Step Solution

Answer

The modulus is 15.

Step 1: Understand the problem

We are tasked with finding the modulus \(r\) of the complex number \(15i\).

Step 2: Recall the formula for modulus

Given a complex number \(a + bi\), its modulus is \(r = \sqrt{a^2 + b^2}\). Here, \(a = 0\) and \(b = 15\) because \(15i\) is equivalent to \(0 + 15i\).

Step 3: Apply the formula

Substitute \(a = 0\) and \(b = 15\) into the modulus formula: \[ r = \sqrt{0^2 + 15^2} \]

Step 4: Simplify

Calculate the expression inside the square root: \[ r = \sqrt{0 + 225} = \sqrt{225} \]

Step 5: Find the square root

Calculate \(\sqrt{225}\), which is \(15\), since \(15^2 = 225\).

Key Concepts

Modulus of Complex Numbers

The modulus of a complex number is a measure of its "size" or absolute value.
This concept is akin to finding the length from the origin to the point representing the complex number in the complex plane.
For complex numbers, represented as \(a + bi\), the modulus \(r\) is calculated by the formula:

• \(r = \sqrt{a^2 + b^2}\)

Here, \(a\) is the real part, and \(b\) is the imaginary part.
In this case, we have the complex number \(15i\), which can be seen as \(0 + 15i\).
This implies:

• \(a = 0\)
• \(b = 15\)

To find the modulus, substitute these values into the formula:

• \(r = \sqrt{0^2 + 15^2}\)
• \(r = \sqrt{0 + 225}\)
• \(r = \sqrt{225}\)
• \(r = 15\)

Thus, the modulus of the complex number \(15i\) is \(15\). The numerical result, in this case, provides the magnitude of the imaginary number, leading us to our next topic.

Imaginary Numbers

Imaginary numbers are an intriguing concept in mathematics.
They arise when we deal with square roots of negative numbers.
The fundamental imaginary unit is denoted by \(i\), which is defined as \(i = \sqrt{-1}\).

• Multiplying \(i\) by itself results in \(i^2 = -1\).
• This means imaginary numbers can express quantities that are otherwise not possible in the realm of real numbers.

Complex numbers are expressed as \(a + bi\), where \(bi\) represents the imaginary part.
These numbers allow solutions to equations that do not have real solutions.
For example, the imaginary number \(15i\) has no real part (because \(a = 0\)) but represents a unit in the direction of the complex plane.
This visualization helps understand how imaginary numbers extend real numbers to solve new and broader equations.

Square Root Calculation

Square root calculation is fundamental in finding the modulus of a complex number and in many other areas of math.
The square root of a number is a value that, when multiplied by itself, gives the original number.
Written as \(\sqrt{x}\), it represents one of the two equal factors of \(x\).

• For example, \(\sqrt{225} = 15\) because \(15 \times 15 = 225\).

When dealing with complex numbers, mastery of square root calculations is crucial.
This expertise helps in solving the modulus, among various other problems.
Knowing which squares yield which roots is valuable, especially when calculators aren't an option.
Remembering squares and their roots, like \(15^2 = 225\), provides quick solutions without additional computational tools.
It also deepens understanding of number relationships in mathematical schemas.