HIPPARCHUS

     Hipparchus, better known as Hipparchos was a Greek mathematician born in 190 BC. Not much is known about Hipparchus’s life however it is deduced that his place of birth was Nicaea in Bithynia which is modern day Turkey. Though being one of the most influential mathematicians and astronomers, the details of his work are very scarce the most definite survived piece being his commentary on a poem by Aratus from the 3rd century the ‘Commentary on the Phainomena of Eudoxus and Aratus’. Also in the list of his contributions are his books on optics and arithmetic, writings concerning geography and astrology and a treatise called ‘On Objects Carried Down by their Weight’. 

Contribution to mathematics:

         His contributions to astronomy are believed to be of significant use in modern day applications of the field. Being the first to calculate a heliocentric system he left his work as according to his calculations the orbits were not truly circular as was the belief of science of that time. Hipparchus had observed the stars from a time span of 147 to 127 BC using an instrument called ‘dioptra’. Some historians suggest that he was the inventor of ‘Planispheric Astrolabe’, an astronomical device. It was none other than Hipparchus who raised important questions such as what the length of a year was and what the lunar distances were. Curious to find an answer, Hipparchus extensively studied the solar and lunar motions and their orbits using several calculations and techniques. He also determined the distances and sizes of both the sun and moon.

       INTERESTING  FACTS  OF  MATHEMATICS-1

1.   The first woman engaged in mathematics was Hypatia from Alexandria.

2.   Equal sign invented in 1557 by Robert Recorde.

3.   If you add numbers from 1 to 100, you get 5050.

4.   Residents of Taipei are officially allowed not to use the number four (4), because it is considered to be “unlucky number”, and its meaning incorporated with the word “death”. Therefore, there are many buildings with no 4th floor, and after the third one, there is a 5th floor.

5.   Muslim mathematician Al-Khwarizmi is considered to be “the Father of Algebra”

6.   Despite the fact that many educated people lived in the Roman Empire, there was no number 0 in their mathematics. It is hard to imagine how they managed their life without the number 0 (zero)

7.   You may have already heard this story somewhere before. George Dantzig when he was a doctoral candidate at the University of California, he was late for a lecture. Seeing on the board some equations, he mistakenly took them for homework. Arriving home, he solved them, although he found the task rather difficult. Bringing them to the next lesson, he learned that these were 2 math problems that had been considered unsolvable.

8.   Negative numbers first appeared in the “Nine Chapters on the Mathematical Art” in the period of Chinese Han Dynasty (202 BC- 220 AD)

            DIOCLES CONTRIBUTION TO MATHEMATICS

                 Although little is known about the life of Diocles, it is known that he was a contemporary of Apollonius and that he flourished sometime around the end of the 3rd century BC and the beginning of the 2nd century BC.

            Diocles is thought to be the first person to prove the focal property of the parabola. His name is associated with the geometric curve called the Cissoid of Diocles, which was used by Diocles to solve the problem of doubling the cube. The curve was alluded to by Proclus in his commentary on Euclid and attributed to Diocles by Geminus as early as the beginning of the 1st century.

           Fragments of a work by Diocles entitled On burning mirrors were preserved by Eutocius in his commentary of Archimedes' On the Sphere and the Cylinder. Historically, On burning mirrors had a large influence on Arabic mathematicians, particularly on al-Haytham, the 11th-century polymath of Cairo whom Europeans knew as "Alhazen". The treatise contains sixteen propositions that are proved by conic sections. One of the fragments contains propositions seven and eight, which is a solution to the problem of dividing a sphere by a plane so that the resulting two volumes are in a given ratio. Proposition ten gives a solution to the problem of doubling the cube. This is equivalent to solving a certain cubic equation. Another fragment contains propositions eleven and twelve, which use the cissoid to solve the problem of finding two mean proportionals in between two magnitudes. Since this treatise covers more topics than just burning mirrors, it may be the case that On burning mirrors is the aggregate of three shorter works by Diocles. In the same work, Diocles, just after demonstrating that the parabolic mirror could focus the rays in a single point, he mentioned that It is possible to obtain a lens with the same property.

               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 

        INTRODUCTION  TO  ALGEBRA

A Puzzle

What is the missing number?

−

2

=

4

OK, the answer is 6, right? Because 6 − 2 = 4. Easy stuff.

Well, in Algebra we don't use blank boxes, we use a letter (usually an x or y, but any letter is fine). So we write:

x

−

2

=

4

It is really that simple. The letter (in this case an x) just means "we don't know this yet", and is often called the unknown or the variable.

And when we solve it we write:

x

=

6