Problem 35 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ \frac{1}{i} $$

Step-by-Step Solution

Answer

The result is \(0 - i\).

Step 1: Understand the expression

The expression given is \( \frac{1}{i} \). Our goal is to simplify this expression and express it in the complex form \( a + bi \), where \( a \) and \( b \) are real numbers.

Step 2: Multiply by the conjugate

To simplify \( \frac{1}{i} \), we multiply both the numerator and the denominator by the conjugate of \( i \), which is \( -i \). This helps in removing the imaginary number from the denominator. So we have:\[\frac{1}{i} \times \frac{-i}{-i} = \frac{-i}{i \cdot (-i)}\]

Step 3: Simplify the expression

Now simplify the expression by calculating the denominator:\[i \cdot (-i) = -i^2\]Since \(i^2 = -1\), we get\[-i^2 = 1\]Thus, the expression becomes:\[\frac{-i}{1} = -i\]

Step 4: Express in a + bi form

The simplified form \( -i \) can be written as \( 0 - 1i \), which is in the form \( a + bi \) with \( a = 0 \) and \( b = -1 \).

Key Concepts

Imaginary Unit

In the world of complex numbers, the imaginary unit is a fundamental building block. It is denoted by the letter \( i \) and is defined by the property \( i^2 = -1 \). This curious mathematical entity allows us to extend the real number system to include solutions to equations that do not have solutions in the real numbers alone, like \( x^2 + 1 = 0 \).

Understanding \( i \) as an imaginary number can initially be challenging. However, it can be visualized as a number that "rotates" the real number line by 90 degrees in the complex plane. This imaginary number is crucial in calculations involving complex expressions, as it often appears in the denominator or in other critical parts of an expression.

Whenever we encounter \( i \) in a denominator, like in \( \frac{1}{i} \), our goal is to address it so that our final result is presented in the form \( a + bi \), where \( a \) and \( b \) are real numbers. This involves techniques such as using the conjugate, which helps us simplify and properly express complex numbers.

Complex Conjugate

The complex conjugate of a complex number is a tool that's extremely helpful in simplifying expressions. For any complex number \( a + bi \), its complex conjugate would be \( a - bi \). This simple swap of the sign in front of the imaginary part makes it possible to remove the imaginary unit from a denominator.

In our exercise, we use the conjugate of \( i \), which is \( -i \), to simplify \( \frac{1}{i} \). Here's how it helps:

• By multiplying the numerator and the denominator of \( \frac{1}{i} \) by \( -i \), the denominator becomes \( i \times (-i) = -i^2 \).
• Since we know \( i^2 = -1 \), we simplify \( -i^2 \) to just \( 1 \).
• The expression \( \frac{-i}{1} \) results, leading us to the simplified form of \( -i \).

The idea here is to "cancel out" the imaginary unit in the denominator, letting us express our result purely in terms of real numbers and \( i \) within the proper format.

Simplifying Complex Expressions

When dealing with complex expressions, like \( \frac{1}{i} \), simplifying them requires certain techniques. The goal is to express the final result in the form \( a + bi \).

Here are the core steps you can generally follow to simplify complex expressions:

• Identify the expression: It's essential to understand what you're dealing with, be it a division, addition, or multiplication of complex numbers.
• Use conjugates wisely: If there's an imaginary unit in the denominator, multiply both the numerator and denominator by its complex conjugate.
• Simplify the expression: Calculate any operations needed, such as multiplication or addition, and use properties like \( i^2 = -1 \) to simplify terms.

For \( \frac{1}{i} \), these steps led us to find that multiplying by \( -i \) results in a denominator that is a real number. Then, the expression simplifies to \( -i \), which can be written in the standard form \( 0 - 1i \) or simply \( -i \).

Simplifying complex expressions often involves recognizing and using these fundamental steps effectively, making complex numbers more manageable and understandable.