Stuck on the first chapter of Apostol:

• Gwyn
In summary, the conversation was about a difficult question involving a fraction with a denominator that could not be simplified using partial fractions. The solution was to rationalize the denominator by multiplying it by its conjugate, resulting in a difference of squares that could be simplified. The conversation also includes a note about the correct syntax for displaying mathematical equations.
Gwyn
I've been going over the book and everything was going well until I got to this question:

$$\frac{1} {2 + \sqrt{4 - x^2}} = \frac{2 - \sqrt{4 - x^2}} {x^2}$$

Tried with partial fractions but that does not work. I also tried to split the fraction but I don't know how to get the x^2 at the bottom. In short I'm completely lost and that looks like black magic to me. I'd appreciate it if anyone has a link to explain what's going on here.

Edit: Messed up the latex. If you click on it the correct code shows up. Not sure how I can refresh the originally posted one.

Last edited:
He just rationalized the denominator.

$$\frac{1} {2 + \sqrt{4 - x^2}} \ \frac{2-\sqrt{4 - x^2}}{2-\sqrt{4 - x^2}} = \frac{2 - \sqrt{4 - x^2}} {x^2}$$

Note the denominator is a difference of squares: $$a^{2}-b^{2}$$

Last edited:
Thanks a lot, I was stuck on that for a long time.

1. Why is it important to understand the first chapter of Apostol?

The first chapter of Apostol lays the foundation for understanding the rest of the book. It introduces important concepts and techniques that are used throughout the rest of the text.

2. What are some common stumbling blocks when reading the first chapter of Apostol?

Some readers may struggle with understanding the mathematical notation and terminology used in the first chapter. It may also be challenging to grasp the abstract concepts presented in this chapter.

3. How can I better understand the first chapter of Apostol?

It is helpful to have a strong foundation in basic calculus and linear algebra before reading this chapter. Additionally, taking the time to work through the examples and practice problems can aid in understanding the material.

4. What is the most important concept covered in the first chapter of Apostol?

The most important concept in this chapter is the notion of a vector space. Understanding this concept is crucial for understanding the rest of the book.

5. How can I apply the knowledge from the first chapter of Apostol to real-world problems?

The concepts and techniques introduced in the first chapter of Apostol have many applications in various fields such as physics, engineering, and computer science. They can be used to solve problems and model real-world situations.

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