Problem 61 - Algebra 2

Question

Simplify. $$ \frac{2-i}{3-4 i} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{2}{5} + \frac{i}{5}\).

Step 1: Identify the complex fractions

The given fraction is \( \frac{2-i}{3-4i} \). In order to simplify a fraction with complex numbers, we need to eliminate the imaginary part from the denominator by multiplying both numerator and denominator by the conjugate of the denominator.

Step 2: Conjugate of the denominator

The denominator is \(3 - 4i\). The conjugate of \(3 - 4i\) is \(3 + 4i\). We will multiply both the numerator and the denominator of the fraction by this conjugate.

Step 3: Multiply by the conjugate

Multiply the numerator and the denominator by \(3 + 4i\): \[\frac{(2 - i)(3 + 4i)}{(3 - 4i)(3 + 4i)}\]

Step 4: Simplify the denominator

The denominator becomes \((3 - 4i)(3 + 4i) = 3^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25\). Now, the denominator is 25.

Step 5: Expand the numerator

Expand the numerator using distributive property:\((2 - i)(3 + 4i) = 2(3) + 2(4i) - i(3) - i(4i)\). Simplifying this gives:\[6 + 8i - 3i - 4i^2\]. Note that \(i^2 = -1\), so \(-4i^2 = 4\).

Step 6: Combine like terms

Now, combine like terms in the expanded form:\[6 + 4 + 5i = 10 + 5i\].

Step 7: Write the simplified fraction

Substitute back the simplified parts into the fraction:\[\frac{10 + 5i}{25}\]

Step 8: Simplify the fraction further

Divide both real and imaginary components by 25:\[\frac{10}{25} + \frac{5i}{25} = \frac{2}{5} + \frac{i}{5}\]

Step 9: Final Simplified Form

The simplified form of the original complex fraction is:\[\frac{2}{5} + \frac{i}{5}\]

Key Concepts

Imaginary Numbers

Imaginary numbers might seem mysterious at first, but they are simply numbers that give a negative product when squared. To understand them, let's start with a typical number line, which includes zero as its midpoint and stretches infinitely in both positive and negative directions. Imaginary numbers extend this idea by providing a perpendicular number line.

At the heart of imaginary numbers is the symbol \( i \), defined as the square root of \(-1\). This means that \( i^2 = -1 \). When you multiply \( i \) by any real number, the result is purely imaginary. For example:

• \( 3i \) means 3 times \( i \)
• \( -2i \) means -2 times \( i \)

Imaginary numbers are crucial in engineering and physics, where they help model real-world phenomena that regular numbers cannot easily express.

In our original problem, the imaginary unit \( i \) appears, which is addressed by using algebraic techniques to make calculations with complex numbers feasible.

Complex Conjugate

A complex conjugate is a significant concept when working with complex numbers. By definition, the complex conjugate of a complex number is obtained by changing the sign of the imaginary component. For instance, the conjugate of \( a + bi \) is \( a - bi \).

This operation is particularly useful for simplifying complex fractions. The multiplication of a complex number by its conjugate results in a real number because the imaginary parts cancel each other out. For example:

• \((a + bi)(a - bi) = a^2 - (bi)^2\)
• Since \( i^2 = -1 \), this simplifies to: \( a^2 + b^2 \)

In the original exercise, the denominator is \(3 - 4i\), and its conjugate is \(3 + 4i\). Multiplying by the conjugate simplifies the denominator to a real number, thus easing the simplification process.

Algebraic Simplification

Algebraic simplification is a crucial step when working with complex fractions. It involves breaking down expressions into simpler forms for more manageable calculations.

In complex fractions, both the numerator and denominator can be complex expressions. Simplifying them requires a good grasp of basic algebraic operations such as expansion, combination of like terms, and the use of properties of imaginary numbers, particularly noting that \( i^2 = -1 \):

• Expand expressions using distributive property
• Combine like terms, both real and imaginary

For instance, in our original problem, multiplying the numerator and the denominator by the conjugate leads to a simplified expression by clearing the imaginary part in the denominator. This was achieved by calculating the products and simplifying them according to algebraic rules.

Complex Numbers

Complex numbers are extensions of the real number line into a plane system, involving both real and imaginary parts. Any complex number can be expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

These numbers are incredibly versatile and are used in various fields, including engineering, physics, and applied mathematics. They allow for the solution of equations that have no real solutions, such as those involving roots of negative numbers.

In the exercise presented, we deal with the expression \( \frac{2-i}{3-4i} \), which involves both real and imaginary components. Simplifying such fractions means manipulating both components correctly to derive a fully simplified form. The final simplified form, \( \frac{2}{5} + \frac{i}{5} \), exemplifies how real and imaginary numbers can be combined and simplified in useful ways, encapsulating how complex numbers make calculations involving imaginary units feasible.