Problem 88 - A Graphical Approach to Precalculus with Limits

Question

Show that \(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\) is a square root of \(i\)

Step-by-Step Solution

Answer

The number is a square root of \( i \) since squaring it gives \( i \).

Step 1: Understand the Problem

We are asked to show that the complex number \( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i \) is a square root of \( i \). This means that when squared, it should equal \( i \), or \((\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i)^2 = i\).

Step 2: Multiply the Complex Number by Itself

Compute the square of \( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i \). Use the formula \((a+b)^2 = a^2 + 2ab + b^2\) where \( a = \frac{\sqrt{2}}{2} \) and \( b = \frac{\sqrt{2}}{2}i \).\[\left(\frac{\sqrt{2}}{2}\right)^2 + 2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2}i + \left(\frac{\sqrt{2}}{2}i\right)^2\]

Step 3: Calculate the Individual Terms

Simplify each term separately:1. \( \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2} \)2. \( 2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2}i = \frac{2}{2}i = i \)3. \( \left(\frac{\sqrt{2}}{2}i\right)^2 = \frac{2}{4} \, i^2 = \frac{1}{2}(-1) = -\frac{1}{2} \)

Step 4: Add the Terms Together

Combine the results from Step 3:\[ \frac{1}{2} + i - \frac{1}{2} = i \]Thus, \( \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\right)^2 = i \), affirming that \( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i \) is indeed a square root of \( i \).

Key Concepts

Square Roots

In the realm of complex numbers, finding a square root involves discovering a number which, when squared, will yield the original number. For complex numbers, this process works similarly to real numbers but requires additional steps due to the presence of the imaginary unit.

In our problem, we're tasked with showing that the complex number \( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i \) is a square root of \( i \). This means when the expression is squared, the result should be \( i \).

• First step: Represent the complex number as \((a + bi)\).
• Square it: \((a + bi)^2 = a^2 + 2abi + b^2i^2\).
• Calculate each part: substitute \(a = \frac{\sqrt{2}}{2}\) and \(b = \frac{\sqrt{2}}{2}\), keeping \(i^2 = -1\) in mind.

Each component must strictly adhere to both real and imaginary rules, allowing us to rightly determine \( i \) as the final result, proving our initial square root condition.

Complex Conjugate

While the concept of complex conjugates isn't directly utilized in proving that a complex number is a square root, it's fundamental to understanding operations involving complex numbers.

The complex conjugate of a number \( a + bi \) is \( a - bi \). Conjugates are critical when dividing complex numbers, as they help to eliminate the imaginary part from the denominator.

• To find a complex conjugate, simply change the sign of the imaginary part.
• A useful property is that multiplying a complex number by its conjugate results in a real number: \((a + bi)(a - bi) = a^2 + b^2\).

Understanding conjugates makes many complex number operations more manageable and emphasizes symmetry present in the complex field. It ensures deepened comprehension of how different algebraic manipulations might inverse or balance each part of a complex number.

Multiplication of Complex Numbers

Multiplying complex numbers involves applying standard arithmetic distribution, specifically the distributive property, which handles both real and imaginary components. This operation becomes slightly more intricate than regular multiplication due to the inclusion of the imaginary unit \( i \), which satisfies \( i^2 = -1 \).

Let's explore it in the context of our exercise.

• Write both components as \( a+bi \), then distribute: \((a+bi)(c+di) = ac + adi + bci + bdi^2\).
• Simplify the expression: remember \( i^2 = -1 \), meaning real parts are combined separately, and imaginary parts get added up with considerations to \( i \).

For the given exercise, after applying multiplication, terms should simplify properly to show that the resultant expression aligns with the number claimed. In this example, we verified that squaring \( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i \) indeed matches the original number \( i \).
Mastery of multiplication aids in various algebraic manipulations necessary in complex number theory, paving the path for more advanced operations and proofs.