Topics

31

Pure Maths / The Pythagorean Theorem

« on: April 22, 2015, 01:27:58 PM »

 Theorem: The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.

There are many different proofs of the theorem (even one supplied by President Garfield in 1876!), and we know that the Babylonians knew about the Pythagorean theorem about 1000 years before the time of Pythagoras (born in 572 B.C.). Nonetheless, a rigorous, general proof of the theorem requires the development of deductive geometry, and thus it is thought that Pythagoras probably supplied the first proof. Most math historians credit him with a proof by dissection, which relies on the use of two squares, one inscribed inside the other. The Indian astronomer Bhaskara (1114-1185) developed this proof(attachment):

32

Pure Maths / The Pythagorean Theorem

« on: April 22, 2015, 01:25:38 PM »

 Theorem: The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.

There are many different proofs of the theorem (even one supplied by President Garfield in 1876!), and we know that the Babylonians knew about the Pythagorean theorem about 1000 years before the time of Pythagoras (born in 572 B.C.). Nonetheless, a rigorous, general proof of the theorem requires the development of deductive geometry, and thus it is thought that Pythagoras probably supplied the first proof. Most math historians credit him with a proof by dissection, which relies on the use of two squares, one inscribed inside the other. The Indian astronomer Bhaskara (1114-1185) developed this proof(attachment):

33

Interesting Maths / Finding the Value of Pi

« on: April 22, 2015, 01:20:18 PM »

Historians estimate that by 2000 B.C. humans had noticed that the ratio of circumference to diameter was the same for all circles. This discovery hinged on the idea of proportion - in this case humans noticed that if you double the distance "across" a circle, then you double the distance "around" it. In today's algebraic notation this implied the formula
where Pi was constant. (It wasn't until 1706 that this notation, using the Greek letter seen in the above equation - often written Pi and pronounced like the English 'pie' - was introduced by William Jones).

pi=Circumference/diameter

The significance of this discovery is clear: Circles are everywhere - in the sun, the moon, the pupils of our eyes, the most basic religious rituals and the earliest man-made structures. Achieving a greater mathematical understanding of Pi would lead to scientific and technological advances that would further the development of civilization, as well as creating some very interesting problems in pure mathematics.

34

History of Mathematics / History of Leonardo Pisano Blgollo

« on: April 22, 2015, 12:03:03 AM »

Leonardo Pisano Blgollo:

Blgollo, also known as Leonardo Fibonacci, is perhaps one of the middle ages greatest mathematicians. Living from 1170 to 1250, he is best known for introducing the infamous Fibonacci Series to the western world. Although known to Indian mathematicians since approximately 200 BC, it was, nonetheless, a truly insightful sequence, appearing in biological systems frequently. In addition, from this Fibonacci also contributed greatly to the introduction of the Arabic numbering system. Something he is often forgotten for.

Haven spent a large portion of his childhood within North Africa he learned the Arabic numbering system, and upon realizing it was far simpler and more efficient then the bulky Roman numerals, decided to travel the Arab world learning from the leading mathematicians of the day. Upon returning to Italy in 1202, he published his Liber Abaci, whereupon the Arabic numbers were introduced and applied to many world situations to further advocate their use. As a result of his work the system was gradually adopted and today he is considered a major player in the development of modern mathematics.

35

History of Mathematics / History of Isaac Newton and Wilhelm Leibniz

« on: April 22, 2015, 12:02:11 AM »

Isaac Newton and Wilhelm Leibniz:

I have placed these two together as they are both often given the honor of being the ‘inventor’ of modern infinitesimal calculus, and as such have both made monolithic contributions to the field. To start, Leibniz is often given the credit for introducing modern standard notation, notably the integral sign. He made large contributions to the field of Topology. Whereas all round genius Isaac Newton has, because of the grand scientific epic Principia, generally become the primary man hailed by most to be the actual inventor of calculus. Nonetheless, what can be said is that both men made considerable vast contributions in their own manner.

36

History of Mathematics / History of Andrew Wiles

« on: April 22, 2015, 12:01:19 AM »

Andrew Wiles:
The only currently living mathematician on this list, Andrew Wiles is most well known for his proof of Fermat’s Last Theorem: That no positive integers, a, b and c can satisfy the equation a^n+b^n=c^n For n greater then 2. (If n=2 it is the Pythagoras Formula). Although the contributions to math are not, perhaps, as grand as other on this list, he did ‘invent’ large portions of new mathematics for his proof of the theorem. Besides, his dedication is often admired by most, as he quite literally shut himself away for 7 years to formulate a solution. When it was found that the solution contained an error, he returned to solitude for a further year before the solution was accepted. To put in perspective how ground breaking and new the math was, it had been said that you could count the number of mathematicians in the world on one hand who, at the time, could understand and validate his proof. Nonetheless, the effects of such are likely to only increase as time passes (and more and more people can understand it).

37

History of Mathematics / History if Mathmatician (Pythagoras of Samos)

« on: April 21, 2015, 11:58:31 PM »

Greek Mathematician Pythagoras is considered by some to be one of the first great mathematicians. Living around 570 to 495 BC, in modern day Greece, he is known to have founded the Pythagorean cult, who were noted by Aristotle to be one of the first groups to actively study and advance mathematics. He is also commonly credited with the Pythagorean Theorem within trigonometry. However, some sources doubt that is was him who constructed the proof (Some attribute it to his students, or Baudhayana, who lived some 300 years earlier in India). Nonetheless, the effect of such, as with large portions of fundamental mathematics, is commonly felt today, with the theorem playing a large part in modern measurements and technological equipment, as well as being the base of a large portion of other areas and theorems in mathematics. But, unlike most ancient theories, it played a bearing on the development of geometry, as well as opening the door to the study of mathematics as a worthwhile endeavor. Thus, he could be called the founding father of modern mathematics.

38

Mathematics / Math Olympiad-6

« on: April 21, 2015, 11:54:04 PM »

6. A positive integer is called “Fantabulous” if there is another fantabulous positive integer smaller than it. Find the number of fantabulous integers.

Answer: Let x be the smallest Fantabulous number. Due to condition we know that it is a positive integer. The smallest positive integer is 1. So x=1. All other Fantabulous numbers will be valid if 1 is a valid Fantabulous number. But there is no smaller positive integers than 1. So 1=x can not be a Fantabulous number. So there are no valid Fantabulous numbers at all. Ans=0

39

Mathematics / Math Olympiad-5

« on: April 21, 2015, 11:51:19 PM »

5. There are some boys and girls in a class. Every boy is friends with exactly three girls, and every girl is friends with exactly three boys. If there are 13 boys in the class, how many girls are there? (Assume that friendship is mutual, i.e. if A is friend of B then B is also friend of A.)

Answer: Given there are 13 boys . Every boy have 3 girls as their friends . And every girl have 3 boys as their friends . Therefore , making one girl as the friend of 3 boys , the total number of girls , as friend of boys are 13*3 or . Taking one girl for three times .So the number of girls is 13 also .

40

Mathematics / Math Olympiad-4

« on: April 21, 2015, 11:49:45 PM »

4. Two isosceles triangles are possible with area of 120 square unit and length of edges integers. One of these two triangles has sides of lengths 1717  and 16. Determine the length of edges of second one.

[Hint: In ABC if AB=AC and AD is perpendicular to BC then BD=CD.]

Answer: The height of the isosceles triangle which has three sides of length 17,17,16, is =sqrt((17))−sqrt((8.0))=15  The area of this triangle is 120. Let x,y be the height and base of the new triangle respectively. if x=16/2=8 [Half of the base the first triangle] and y=15*2=30 [double of the height of the first triangle] then the area of the new triangle is 120 and the length of the two equal sides will be sqrt(15)+sqrt((8.0))=17  so the length of the sides of the new triangle are 17,17,30.

41

Mathematics / Math Olympiad-3

« on: April 21, 2015, 11:43:41 PM »

3. The English alphabets are arranged in 3 rows in a Keyboard. Now somebody presses one key in the first row in such a way that there are same number of keys on both sides of that key in that row. Now a second person presses a key in the second row in the same way and a third person also does the same in the third row. Show that it is impossible.

Answer: The patern of keyboard is 10:9:7.the term is when a word selected then there must be same number word beside 2side of row. So though the 2nd and 3rd row agree with this term but 1st don't

42

Mathematics / Math Olympiad-2

« on: April 21, 2015, 11:39:54 PM »

2. There are n cities in the country. Between any two cities there is at most one road. Suppose that the total number of roads is n. Prove that there is a city such that starting from there it is possible to come back to it without ever traveling the same road twice.

Answer: Let therebe C ways to take n points such that it is not possible to return to its original position. So, we can take 1 point in (n−1). 2 points be taken in (n−1)(n−2) ........... So n points can be taken (n−1)(n−2)(n−3)(n−n+1)(n−n)
C=(n−1)(n−2)(n−n+1)(n−n)=0.
So, there is 0 ways to take n points such that it is not possible to return to its original position. So,it is always possible to return to its original position.

43

Mathematics / Math olympiad-1

« on: April 21, 2015, 11:32:25 PM »

1. A group of 7 women takes 7 days to make 7 Nokshikatha. How many days will a group of 5 women take for making 5 Nokshikatha?

Answer:  If 7 people need 7 days to make 7 katha, then it will take 77=49 days for 1 person to make 7 katha (since the workload will 7 times more)
So it will take that person 749=7 days to sew each katha. (7 times less workload)
Therefore it takes 57=35 days for the person to sew 5 katha. (5 times more workload)
Now if we increase the number of sewers to 5 , the workload will be reduced 5 times.
So, the time needed for 5 people to sew 5 katha will be 535=7 days.

44

Faculty Forum / ইন্টারেস্টিং মাস ফেব্রুয়ারি !

« on: February 25, 2015, 03:50:37 PM »

 ২০১৫ সালের ফেব্রুয়ারি মাসটি ব্যতিক্রম একটি মাস। এমাসে একটি ইন্টারেস্টিং বিষয় রয়েছে। সেটা হয়তো অনেকেই খেয়াল করে দেখেননি। এমন বিষয় ৮২৩ বছর পর আরেকবার আসবে। চলুন বিষয়টি ধরিয়ে দেই…
চলতি বছর
ফ্রেব্রুয়ারি মাসে রয়েছে–
৪টি শনিবার
৪টি রবিবার
৪টি সোমবার
৪টি মঙ্গলবার
৪টি বুধবার
৪টি বৃহস্পতিবার
৪টি শুক্রবার
এটি মূলত ৮২৩বছরে একবার ঘটে”!!
[সংগৃহীত: মামুন খান]

Source:campuslive24.com

45

Faculty Forum / মাত্র ১ বছরে প্রতিবন্ধী শিশুর হিফয

« on: February 23, 2015, 02:35:05 PM »

মাত্র ১ বছরে প্রতিবন্ধী শিশুর হিফয

 
যুদ্ধবিদ্ধস্ত দেশ ফিলিস্তিনের গাজায় এখন শুধুই গোলাবারুদের পোড়া গন্ধ। তবুও গাজাবাসী বেঁচে আছে প্রতিদিন আকাশের বুকে নতুন এক একটি সূর্যের আগমন দেখার জন্য। রাতে ঘুমানোর আগেও কেও জানেনা আগামীকাল সে বেঁচে থাকবে কিনা। তবুও মেধা আর যোগ্যতার দিক দিয়ে যেন পিছিয়ে নেই ফিলিস্তিন।
প্রতিমুহূর্তে শত প্রতিবন্ধকতাকে জয় করে সামনের দিকে এগিয়ে চলেছে সাহসী এই জাতি। সেই গাজার এক বিস্ময় বালক খালেদ আবু মুসা। জন্ম থেকেই অটিজমে ভুগছে খালেদ। মানসিকভাবে প্রতিবন্ধী শিশুরা নাকি বিভিন্ন দিক দিয়ে থাকে অসাধারন! তাই তাদের বলা হয় বিশেষ শিশু। আবারো যেন তা প্রমাণ করল খালেদ আবু মুসা।
নিজের অসাধারণ শ্রবণশক্তি আর মুখস্তশক্তিকে কাজে লাগিয়ে মাত্র এক বছরের মধ্যে ৩০ পাড়া কোরআন আত্মস্থ করে বিশ্বের বুকে তাক লাগিয়ে দিয়েছে বিস্ময় বালক খালেদ আবু মুসা।
তুরস্কের জনপ্রিয় পত্রিকা ‘ডেইলি সাবাহ’ ওয়েব সাইটে এমনই একটি খবর দিয়েছে। খবরে প্রকাশ, গাজার অটিস্টিক স্কুলের শিক্ষা বিভাগের প্রধান নাদভাহ আবদুল আল জানিয়েছেন, ওই স্কুলের ১০ বছরের বালক খালেদ আবু মুসা পবিত্র কোরআনের আয়াত শ্রবণের মাধ্যমে পুরো কোরআন শরীফ মুখস্থ করতে সক্ষম হয়েছে। শিশু খালেদ মুসাকে তার শিক্ষকরা কোরআনে আয়াত কয়েকবার করে তেলাওয়াত করে শোনাত। এভাবে বারংবার তার উদ্দেশ্যে কৃত কোরআন তেলাওয়াত শোনে সে সম্পূর্ণ কোরআন মুখস্থ করতে সক্ষম হয়েছে।
খালেদ মুসা বর্তমানে লিখতে ও পড়তে পারে। অথচ অটিস্ট শিশুদের জন্য এ ধরনের কাজ একটু কষ্টসাধ্য ও দূরহ বিষয়। হয়ত ভবিষ্যতে বিশ্বের বুকে আবারো নতুন কোন দৃষ্টান্ত স্থাপন করবে এই শিশু।
Source: campuslive24.com