Deriving the Formula of the Vertex of Quadratic Functions

In getting the vertex of the quadratic function in general form f(x) = ax^2 + bx + c, we usually need to convert it to the vertex form f(x) = a(x - h)^2 + k. In the latter form, the vertex of the parabola is at (h,k). For example, the function in the general form

f(x) = 2x^2 - 12x + 22

can be rewritten in the vertex form as

f(x) = 2(x - 3)^2 + 4.

In the latter form, it is easy to see that the vertex is at (h,k) = (3,4).

Aside from this method, we can also use the ordered pair

\left ( \dfrac{-b}{2a}, \dfrac{4ac - b^2}{4a} \right )

in place of (h,k)Continue reading…

 

GIMPS Discovers The Largest Known Prime Number Yet

The Great Internet Mersenne Prime Search (GIMPS) discovered the largest prime number yet on January 7, 2016. The number is 2^{74, 207, 281} - 1. It contains 22,338,618 digits. If you are wondering how long it is, suppose that you can write an average of 2 digits per second, you will be able to finish writing this number in about 129 days without eating, sleeping, or toilet break. Using the same rate, if you are going to write this number for 6 hours a day, then you will finish it in about 517 days (roughly one and a half years).

GIMPS has been calculating large prime numbers since 1996 and has discovered 15 of the largest known prime numbers. As of this writing, the 11 largest prime numbers are Mersenne primes. Mersenne primes are prime numbers of the form 2^n - 1 for some integer n. Three of the smallest Mersenne primes are 2^2 - 1 = 3, 2^3 - 1 = 7, and 2^5 - 1 = 31 .

Question: Are all positive integers of the form 2^n - 1 prime numbers?

For the non-math persons, prime numbers are positive integers that can only be divided by 1 and itself. For example, 5 is a prime number since you cannot divide 5 by any number except one and itself. On the other hand, 8 is not a prime number because 8 can be divided by 1, 2, 4, and 8.

It was already proven by Euclid (some 2300 years ago) that there is no largest prime number, so the search for large prime numbers will never end.

Some of the most useful application of prime numbers is cryptography, particularly internet security. It’s what makes your password safe. It is what makes shopping safe. Well, you still have to be careful though.

 

Take Notes and Scribble with Inkflow Visual Notebook

I usually solve problems and organize my thoughts by taking notes  while on the go, so I have been looking for a free note taking, scribbling, or doodling phone application for some time. I have downloaded and tested several applications, but I was not really satisfied by any of them.

A week ago, I purchased Inkflow Visual Notebook for iOS and I’m very happy with it, although I personally think that $7.99 is quite expensive. What attracted me the most is its ink which really feels like a smooth fountain pen.

Inkflow is simple but user-friendly. For example, its selection tool, is very handy to use: select the part that you want to edit and two icons appear which allow you to easily enlarge, reduce, or rotate the selection. Also, Inkflow uses vector graphics, so the ink stays sharp even if it is zoomed in. Further, I have not experienced any lag although I have not really created a lot of notes yet.  Continue reading…

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