Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{-2 i}{-5 i} $$

Step-by-Step Solution

Answer

\(\frac{2}{5} + 0i\)

Step 1: Simplify the fraction

The expression is \( \frac{-2i}{-5i} \). First, notice that both the numerator and denominator have an \( i \), which we can cancel out. This leaves us with \( \frac{-2}{-5} \).

Step 2: Evaluate the fraction

Now, simplify \( \frac{-2}{-5} \). The negative signs in the numerator and denominator cancel each other out, simplifying the fraction to \( \frac{2}{5} \).

Step 3: Write the number in standard form

The resulting number from simplification is \( \frac{2}{5} \), with no imaginary part. Thus, the expression in the form \( a + bi \) is \( \frac{2}{5} + 0i \).

Key Concepts

Simplifying Complex Expressions

When working with complex numbers, simplifying complex expressions is a fundamental skill. Complex expressions can look intimidating at first, but by following certain steps, we can simplify them to a more manageable form.
Take, for example, the expression \( \frac{-2i}{-5i} \). This initially daunting fraction can be simplified by recognizing that both the numerator and denominator have the imaginary unit \( i \).

When both parts of a fraction contain the same factor, in this case \( i \), we can cancel them out. Thus, \( \frac{-2i}{-5i} \) can be simplified to \( \frac{-2}{-5} \). It's similar to how you would simplify \( \frac{2x}{5x} \) to \( \frac{2}{5} \).

Further simplifying \( \frac{-2}{-5} \) results in \( \frac{2}{5} \), as negatives in both the numerator and the denominator negate each other. Always remember:

• Cancel common factors in numerator and denominator.
• Simplify negative fractions by canceling double negatives.

This reduces the expression into a standard and simpler form.

Imaginary Unit

The concept of the imaginary unit \( i \) is crucial in understanding complex numbers. The imaginary unit is defined as \( i = \sqrt{-1} \). It allows us to handle the square roots of negative numbers, which are impossible within the system of real numbers.
While dealing with expressions like the one in \( \frac{-2i}{-5i} \), recognizing that \( i \cdot i = -1 \) provides a way to simplify expressions more effectively. When an expression has \( i \) in both the numerator and denominator, these terms cancel each other out.
This key property helps simplify terms in complex numbers, allowing you to reduce them into their component parts easily. Remember

• \( i^2 = -1 \) is an essential identity.
• Canceling out \( i \) when it appears in both numerator and denominator.

Understanding \( i \) helps avoid misconceptions and simplifies complex expressions greatly.

Real and Imaginary Parts

In the expression \( \frac{-2i}{-5i} \), the end result is written as \( \frac{2}{5} + 0i \), highlighting the distinction between real and imaginary parts of a complex number.
This expression decomposes into:

• The real part is \( \frac{2}{5} \).
• The imaginary part is \( 0 \), since there's no remaining \( i \).

A complex number typically takes the form \( a + bi \), where \( a \) is the real part and \( b \) is the coefficient of the imaginary part.

The exercise results in \( a = \frac{2}{5} \) and \( b = 0 \). Clearly identifying these parts is crucial because it simplifies the handling of complex numbers in mathematical operations. The form \( a + bi \) helps in:

• Understanding the contribution of both the real and imaginary components.
• Applying algebraic properties and mathematical operations appropriately to complex numbers.

Mastering this breakdown leads to a deeper understanding of how to manipulate and simplify complex expressions efficiently.