Problem 29 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ \frac{2-3 i}{1-2 i} $$

Step-by-Step Solution

Answer

The result is \(\frac{8}{5} + \frac{1}{5} i\).

Step 1: Understand the Problem

The task is to evaluate the expression \( \frac{2-3i}{1-2i} \) and express it in the form \( a + bi \), which is the standard form of a complex number.

Step 2: Multiply by the Conjugate

To simplify the expression, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 1-2i \) is \( 1+2i \). Thus, multiply:\[\frac{2-3i}{1-2i} \times \frac{1+2i}{1+2i}\]

Step 3: Apply Distribution (Numerator)

Distribute \( (2-3i) (1+2i) \):\[(2-3i)(1+2i) = 2 \times 1 + 2 \times 2i + (-3i) \times 1 + (-3i) \times 2i\]\[= 2 + 4i - 3i - 6i^2\]Since \(i^2 = -1\), replacing \(-6i^2\) gives:\[= 2 + 4i - 3i + 6\]Simplifying gives:\[= 8 + i\]

Step 4: Apply Distribution (Denominator)

Distribute \( (1-2i)(1+2i) \):\[(1-2i)(1+2i) = 1^2 + 1 \times 2i - 2i \times 1 - 2i \times 2i\]\[= 1 + 2i - 2i - 4i^2\]Since \(i^2 = -1\), replacing \(-4i^2\) gives:\[= 1 + 4\]Thus, the denominator simplifies to:\[= 5\]

Step 5: Simplify the Expression

Now, place the results from Steps 3 and 4 into the fraction:\[\frac{8+i}{5}\]Divide both terms in the numerator by 5:\[= \frac{8}{5} + \frac{1}{5} i\]

Step 6: Write the Result in Standard Form

The expression is now in the form \( a + bi \), where \( a = \frac{8}{5} \) and \( b = \frac{1}{5} \). Thus, the result in the form \( a + bi \) is:\[\frac{8}{5} + \frac{1}{5} i\]

Key Concepts

Complex Number Simplification

In mathematics, simplifying complex numbers is a crucial step to make expressions easier to handle. Complex numbers are in the form of \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined by \( i^2 = -1 \).
When given an expression of complex numbers in fraction form, such as \( \frac{2-3i}{1-2i} \), the goal is to remove the imaginary part from the denominator. This process enables easier evaluation of the expression.
Here's a step-by-step breakdown:

• Multiply both numerator and denominator by the conjugate of the denominator to eliminate the imaginary part.
• Distribute and simplify each multiplication, using \( i^2 = -1 \) to replace any squares of \( i \).
• Combine like terms and finalize the expression in standard form by dividing each term by the new denominator.

This process results in a simplified and more manageable expression.

Conjugate of Complex Numbers

The conjugate of a complex number is a key concept when it comes to simplifying fractions involving complex numbers. If you have a complex number \( a + bi \), its conjugate is \( a - bi \).
The conjugate is used to rationalize the denominator of a complex fraction. In the exercise \( \frac{2-3i}{1-2i} \), the conjugate of the denominator \( 1-2i \) is \( 1+2i \). By multiplying both the numerator and the denominator by the conjugate, the denominator becomes a real number.

• Multiplication of a complex number by its conjugate results in a real number because \( (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 \).
• With a real denominator, the fraction becomes much easier to deal with.

This technique is essential for transforming complex numbers into their simplest forms.

Standard Form of Complex Numbers

The standard form of complex numbers is \( a + bi \), where \( a \) represents the real part, and \( b \) represents the imaginary part, with \( i \) being the imaginary unit. Expressing a complex number in this form is essential for clarity and ease of understanding.
In our original exercise, after simplifying the fraction \( \frac{2-3i}{1-2i} \), we obtained a simplified expression \( \frac{8}{5} + \frac{1}{5}i \), which is indeed in the standard form \( a + bi \).
Here are some key points:

• This standard form clearly separates the real and imaginary components of the complex number.
• It helps in performing further calculations, like addition or subtraction of complex numbers, more straightforwardly.
• When expressing solutions, ensuring the result is in this format aids in understanding and interpreting the result accurately.

Maintaining this form is a fundamental practice in complex number calculations.