Problem 2 - Algebra 2 and Trigonometry

Question

Ethan said that the square of any pure imaginary number is a negative real number. Do you agree with Ethan? Justify your answer.

Step-by-Step Solution

Answer

Yes, Ethan is correct. The square of a pure imaginary number is a negative real number.

Step 1: Define a Pure Imaginary Number

A pure imaginary number can be represented as \( bi \), where \( b \) is a real number and \( i \) is the imaginary unit, defined by \( i^2 = -1 \).

Step 2: Calculate the Square of a Pure Imaginary Number

To find the square of \( bi \), square the imaginary number: \((bi)^2 = b^2i^2\). Since \( i^2 = -1 \), it follows that \( b^2i^2 = b^2(-1)\).

Step 3: Simplify the Expression

Simplify the expression to get \( -b^2 \). Since \( b^2 \) is a positive real number, \( -b^2 \) is a negative real number.

Step 4: Conclusion Based on Calculation

Based on our calculations, the square of any pure imaginary number \( bi \) results in \( -b^2 \), which is a negative real number.

Key Concepts

Pure Imaginary Numbers

In mathematics, pure imaginary numbers play a fundamental role in the realm of complex numbers. A pure imaginary number can be expressed in the form \( bi \), where \( b \) is a real number and \( i \) represents the imaginary unit. The characteristic feature of a pure imaginary number is that it solely has an imaginary part without any real component.

• The expression \( bi \) identifies it purely as an imaginary number because there is no added constant term, unlike a complex number which typically looks like \( a + bi \) where both \( a \) and \( b \) are real numbers.
• Pure imaginary numbers are fundamental in various fields including engineering and physics since they allow for the mathematical representation of quantities like alternating current, where the phase can be regarded as an imaginary component.

Understanding pure imaginary numbers is essential for broader studies of complex numbers and beyond.

Square of Imaginary Numbers

When you square a pure imaginary number, the result is quite intriguing. Consider a number \( bi \), where the square is computed as follows:

• The expression \( (bi)^2 = b^2i^2 \) emerges when you apply the formula for squaring a number.
• We substitute \( i^2 \) with \(-1\), a unique property of the imaginary unit \( i \).

This manipulation reveals the surprising result that \[ b^2i^2 = b^2(-1) = -b^2 \]The key takeaway here is:

• The square of a pure imaginary number yields a negative real number.
• This result emphasizes the peculiar nature of imaginary numbers, making them both fascinating and fundamental in mathematics.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a cornerstone of imaginary numbers and complex analysis. It is defined by the property that its square equals negative one: \( i^2 = -1 \).

• This defining trait of \( i \) is what enables the conversion of imaginary numbers into real outcomes upon squaring.
• Mathematically, \( i \) solves the equation \( x^2 + 1 = 0 \), providing a way to extend real numbers into the complex plane.

The significance of the imaginary unit extends beyond pure calculations:

• In electrical engineering, it represents phase shifts in alternating current circuits.
• It underlies Fourier transforms, used in signal processing, which fundamentally relies on imaginary numbers for converting signals from time to frequency domain.

Mastering the concept of the imaginary unit aids in unraveling the complexities of numerous advanced mathematical and engineering problems.