Simplifying Complex Fractions: A Step-by-Step Guide
When dealing with complex numbers, it's crucial to know how to simplify expressions involving complex fractions. This article will walk you through the process of simplifying the fraction 17i/13i and explain the mathematical steps involved. We will use various techniques such as rationalizing the denominator and working with conjugates. Let's dive in!
Introduction to Simplifying Complex Fractions
A complex fraction is a fraction where the numerator, denominator, or both contain complex numbers. In this case, we will focus on simplifying the fraction 17i/13i. To simplify this fraction, we need to rationalize the denominator. Rationalization is the process of removing the imaginary unit i from the denominator.
Step-by-Step Simplification
To simplify 17i/13i, the first step is to multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 13i is 1-3i.
Multiplying by the Conjugate
The first step involves multiplying both the numerator and the denominator by 1-3i: [frac{17i}{13i} frac{17i(1-3i)}{13i(1-3i)}]
Now, simplify the expression in the numerator and the denominator:
[begin{align*}frac{17i(1-3i)}{13i(1-3i)} frac{17i - 51i^2}{13i - 39i^2} frac{17i - 51(-1)}{13i - 39(-1)} frac{17i 51}{13i 39} frac{68 17i}{52 13i}end{align*}]Since we have a common factor of 13i in both the numerator and the denominator, we can simplify further:
[begin{align*}frac{68 17i}{52 13i} frac{17(4 i)}{13(4 i)} frac{17}{13} cdot frac{4 i}{4 i} frac{17}{13} cdot 1 frac{17}{13}end{align*}]Final Simplified Form
The simplified form of the complex fraction 17i/13i is:
[frac{17i}{13i} frac{11}{5} frac{2}{5}i]Additional Scenarios
Let's consider some additional scenarios to further understand the process of simplifying complex fractions.
Scenario 1: 17i/13i * 13i/13i
Multiplying both the numerator and the denominator by 13i yields:
[frac{17i}{13i} cdot frac{13i}{13i} frac{224i}{10} frac{112i}{5} frac{11}{5} cdot frac{2}{5}i]Scenario 2: 1 7i/1 3i
To rationalize this fraction, multiply both the numerator and the denominator by the conjugate of the denominator, 1-3i: [frac{1 7i}{1 3i} cdot frac{1-3i}{1-3i} frac{(1 7i)(1 - 3i)}{(1 3i)(1 - 3i)}]
Expanding the expressions in the numerator and denominator gives:
[frac{1 - 3i 7i - 21i^2}{1 - 9i^2}]Since i^2 -1, this simplifies to:
[frac{1 4i 21}{1 9} frac{22 4i}{10} frac{11}{5} frac{2}{5}i]Conclusion
Understanding how to simplify complex fractions is an essential skill in working with complex numbers. By using methods such as rationalizing the denominator and employing conjugates, we can transform complex numbers into more manageable forms.