ERATOSTHENES

          The mathematician is also known as Eratosthenes of Cyrene, he was also classified as being a poet, astronomer, geographer, athlete and music theorist.

        Not only is he known to figure out the circumference of the earth by making use of stadiums during that time, but he also calculated the tilt of the Earth’s axis. Using the knowledge available regarding geography during that time period, he presented the first map of the world, additionally he was the first person to make use of the term ‘geography’ and brought forth the system of latitude and longitude.

Life Details

       Eratosthenes was born in 276 BC in what is now known as ‘Libya’. He studied in Alexandria and in 236 BC he was chosen as librarian in the Alexandrian Library. He was also the third chief librarian of the Great Library of Alexandria. It is also known that he was good friends with Archimedes.

Contributions to Mathematics

       One of the most remarkable things done by Eratosthenes included his effort to calculate the Earth’s circumference without leaving the comfortable perimeters of Egypt. Eratosthenes also discovered an algorithm to determine prime numbers which is known as the Sieve of Eratosthenes. He also invented the armillary sphere.

Writing an algebraic expression

Example #1

6 more than n

Key word : more than

More than indicates addition. Add the first number 6 to the second number n

The algebraic expression is  n + 6

Example #2

The difference of x and 9.

Key word : difference

Difference indicates subtraction. Start with the first number x. Then, subtract the second number 9.

The algebraic expression is x - 9

Example #3

two less than m.

Key word : less than

Less than indicates subtraction. Subtract the first number 2 from the second number m

The algebraic expression is  m - 2

                        PASCAL’S  TRIANGLE

The Pascal's triangle, named after Blaise Pascal, a famous french mathematician and philosopher, is shown below with 5 rows. 

 Some Important things to notice      

·         The first row starts with 1

·         Starting with row #2, the row starts and ends with 1.

·         The number 2  in the third row is found by 1 and 1 in the second row

·         The number 3 in the fourth row is found by adding 1 and 2 in the third row.

·         The number 4 in the fifth row is found by adding 1 and 3 in the fourth row. The number 6 in the fifth row in found by adding 3 and 3 in the fourth row. 

You can indeed keep building more rows by doing just that. It is not that complicated.

                                                                 1

                                                         1                1

                                                 1                2                 1

                                      1                 3                   3                 1 

                            1                 4                  6                   4                 1  

The Pascal's triangle can also be obtained by pulling out the coefficients of (a + b)ⁿ

We show the expansion of 4 binomials when n = 0, 1, 2, and 3.

You can really see the coefficients when n = 1. When n = 0, it is just the number 1 that we put on top on row #1.

(a + b)⁰ = 1

(a + b)¹ = a + b = 1a + 1b

(a + b)² = a² + 2ab + b² = 1a² + 2ab + 1b²

a³ + 3a²b + 3ab² + b³ = 1a³ + 3a²b + 3ab² + 1b³