thesciencenut

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The Beauty of Numbers

Hey there. This is - TheScienceNut. Well let me tell you that I am new to tumblr. This is my first post, and it is gonna be interesting.In this post, I am going to show you all the beauty that lies with numbers. One such number is - 101. 101 is a very very very special number. It is different from other numbers and has a uniqueness in it's properties.

So people, can I ask you all to perform a simple multiplication of the numbers -101 and 25. What is your answer? Is it 2525? Now multiply the numbers - 101 and 83. What do you get? Do you get 8383?. Now multiply 101 with any two digit number, say ab. Then you will observe that-

101 × ab(any 2 digit number) = abab. Isnt it fascinating? 101 is also called as a palindrome. What are palindromes? Palindromes are numbers which look the same from both directions i. e. from both right to left and left to right. Look the number 101 number from both sides. Aren't they the same. While many other numbers aren't palindromes. Some other examples of palindromes are -101010101 , 2020202, 1111111, 203020302 etc.

Manu such numbers are present in the vast number system. Do you wanna go deeper and deeper into the world of numbers, and explore the world of numbers? You have found the right guy. Follow me , and give as much of likes to this post as you can...........next special number will be posted soon.....

Till then, good bye.

[thesciencenut]

https://architadutta940.wixsite.com/themathsnut

Follow me and like my posts, on WiX as well for more interesting posts like the earlier ones 🤗.

[thesciencenut]

The fantastic number

This number is the most famous number. This number's speciality was discovered by the Indian mathematician ramanujan. 3435 is known as the fantastic number.

The speciality of the number 3435 is that it is equal to the sum of the power of its digits. Which means-

3435 = (cube of 3) + (4 to the power of 4) + (cube of 3) + (5 to the power of 5)

Isnt it interesting.....

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[thesciencenut]

The Armstrong Number

153 is known as the armstrong number. It's speciality is that it is the sum of the cubes of its digits. That is -

153 = (cube of 1) + (cube of 5) + (cube of 3)

153 = 1 + 125 + 27

153=153.

Interesting....isnt it?

[thesciencenut]

Continuation of the earlier post

In this post, I am going to talk about - ramanujan and shakuntala devi.

RAMANUJAN

was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Ramanujan initially developed his own mathematical research in isolation: according to Hans Eysenck: "He tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a postal partnership with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognizing Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that "defeated me completely; I had never seen anything in the least like them before", and some recently proven but highly advanced results.

SHAKUNTALA DEVI

Shakuntala devi was a great indian mathematician. Born in Bangalore, she.was given the title of "human computer " for her unbelievable calculation skills. She has been listed in the guiness book of world records for being able to multiply a 13 digit number by another 13 digit number, which is beyond human abilities. But her personal life was not very easy. She went through a very bad period in her life, which affected her calculation skills as well.

A biopic has been made on her as well.

Information credit-

Google

Photo credits -

Google

[thesciencenut]

Continuation of the earlier post.

In the earlier post, I was talking about the great mathematicians of the world. In this post, I am gonna talk about - Archimedes and Pythagoras.

ARCHIMEDES

Archimedes was the most-famous mathematician and inventor in ancient Greece. Archimedes is especially important for his discovery of the relation between the surface and volume of a sphere and its circumscribing cylinder.He is known for his formulation of a hydrostatic principle (known as Archimedes’ principle) and a device for raising water, still used in developing countries, known as the Archimedes screw. Archimedes probably spent some time in Egypt early in his career, but he resided for most of his life in Syracuse, the principal Greek city-state in Sicily, where he was on intimate terms with its king, Hieron II. Archimedes published his works in the form of correspondence with the principal mathematicians of his time, including the Alexandrian scholars Conon of Samos and Eratosthenes of Cyrene. He played an important role in the defense of Syracuse against the siege laid by the Romans in 213 BCE by constructing war machines so effective that they long delayed the capture of the city. When Syracuse eventually fell to the Roman general Marcus Claudius Marcellus in the autumn of 212 or spring of 211 BCE, Archimedes was killed in the sack of the city.

TOP QUESTIONS

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Study how turning a helix enclosed in a circular pipe raises water in an Archimedes screw

Study how turning a helix enclosed in a circular pipe raises water in an Archimedes screw

An animation of Archimedes screw.

Encyclopædia Britannica, Inc.

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Far more details survive about the life of Archimedes than about any other ancient scientist, but they are largely anecdotal, reflecting the impression that his mechanical genius made on the popular imagination. Thus, he is credited with inventing the Archimedes screw, and he is supposed to have made two “spheres” that Marcellus took back to Rome—one a star globe and the other a device (the details of which are uncertain) for mechanically representing the motions of the Sun, the Moon, and the planets. The story that he determined the proportion of gold and silver in a wreath made for Hieron by weighing it in water is probably true, but the version that has him leaping from the bath in which he supposedly got the idea and running naked through the streets shouting “Heurēka!” (“I have found it!”) is popular embellishment. Equally apocryphal are the stories that he used a huge array of mirrors to burn the Roman ships besieging Syracuse; that he said, “Give me a place to stand and I will move the Earth”; and that a Roman soldier killed him because he refused to leave his mathematical diagrams—although all are popular reflections of his real interest in catoptrics (the branch of optics dealing with the reflection of light from mirrors, plane or curved), mechanics, and pure mathematics.

According to Plutarch (c. 46–119 CE), Archimedes had so low an opinion of the kind of practical invention at which he excelled and to which he owed his contemporary fame that he left no written work on such subjects. While it is true that—apart from a dubious reference to a treatise, “On Sphere-Making”—all of his known works were of a theoretical character, his interest in mechanics nevertheless deeply influenced his mathematical thinking. Not only did he write works on theoretical mechanics and hydrostatics, but his treatise Method Concerning Mechanical Theorems shows that he used mechanical reasoning as a heuristic device for the discovery of new mathematical theorems.

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His Works

There are nine extant treatises by Archimedes in Greek. The principal results in On the Sphere and Cylinder (in two books) are that the surface area of any sphere of radius r is four times that of its greatest circle (in modern notation, S = 4πr2) and that the volume of a sphere is two-thirds that of the cylinder in which it is inscribed (leading immediately to the formula for the volume, V = 4/3πr3). Archimedes was proud enough of the latter discovery to leave instructions for his tomb to be marked with a sphere inscribed in a cylinder. Marcus Tullius Cicero (106–43 BCE) found the tomb, overgrown with vegetation, a century and a half after Archimedes’ death.

sphere with circumscribing cylinder

sphere with circumscribing cylinder

The volume of a sphere is 4πr3/3, and the volume of the circumscribing cylinder is 2πr3. The surface area of a sphere is 4πr2, and the surface area of the circumscribing cylinder is 6πr2. Hence, any sphere has both two-thirds the volume and two-thirds the surface area of its circumscribing cylinder.

Encyclopædia Britannica, Inc.

Measurement of the Circle is a fragment of a longer work in which π (pi), the ratio of the circumference to the diameter of a circle, is shown to lie between the limits of 3 10/71 and 3 1/7. Archimedes’ approach to determining π, which consists of inscribing and circumscribing regular polygons with a large number of sides, was followed by everyone until the development of infinite series expansions in India during the 15th century and in Europe during the 17th century. That work also contains accurate approximations (expressed as ratios of integers) to the square roots of 3 and several large numbers.

On Conoids and Spheroids deals with determining the volumes of the segments of solids formed by the revolution of a conic section (circle, ellipse, parabola, or hyperbola) about its axis. In modern terms, those are problems of integration. (See calculus.) On Spirals develops many properties of tangents to, and areas associated with, the spiral of Archimedes—i.e., the locus of a point moving with uniform speed along a straight line that itself is rotating with uniform speed about a fixed point. It was one of only a few curves beyond the straight line and the conic sections known in antiquity.

On the Equilibrium of Planes (or Centres of Gravity of Planes; in two books) is mainly concerned with establishing the centres of gravity of various rectilinear plane figures and segments of the parabola and the paraboloid. The first book purports to establish the “law of the lever” (magnitudes balance at distances from the fulcrum in inverse ratio to their weights), and it is mainly on the basis of that treatise that Archimedes has been called the founder of theoretical mechanics. Much of that book, however, is undoubtedly not authentic, consisting as it does of inept later additions or reworkings, and it seems likely that the basic principle of the law of the lever and—possibly—the concept of the centre of gravity were established on a mathematical basis by scholars earlier than Archimedes. His contribution was rather to extend those concepts to conic sections.

Quadrature of the Parabola demonstrates, first by “mechanical” means (as in Method, discussed below) and then by conventional geometric methods, that the area of any segment of a parabola is 4/3 of the area of the triangle having the same base and height as that segment. That is, again, a problem in integration.

The Sand-Reckoner is a small treatise that is a jeu d’esprit written for the layman—it is addressed to Gelon, son of Hieron—that nevertheless contains some profoundly original mathematics. Its object is to remedy the inadequacies of the Greek numerical notation system by showing how to express a huge number—the number of grains of sand that it would take to fill the whole of the universe. What Archimedes does, in effect, is to create a place-value system of notation, with a base of 100,000,000. (That was apparently a completely original idea, since he had no knowledge of the contemporary Babylonian place-value system with base 60.) The work is also of interest because it gives the most detailed surviving description of the heliocentric system of Aristarchus of Samos (c. 310–230 BCE) and because it contains an account of an ingenious procedure that Archimedes used to determine the Sun’s apparent diameter by observation with an instrument.

Method Concerning Mechanical Theorems describes a process of discovery in mathematics. It is the sole surviving work from antiquity, and one of the few from any period, that deals with this topic. In it Archimedes recounts how he used a “mechanical” method to arrive at some of his key discoveries, including the area of a parabolic segment and the surface area and volume of a sphere. The technique consists of dividing each of two figures into an infinite but equal number of infinitesimally thin strips, then “weighing” each corresponding pair of these strips against each other on a notional balance to obtain the ratio of the two original figures. Archimedes emphasizes that, though useful as a heuristic method, this procedure does not constitute a rigorous proof.

On Floating Bodies (in two books) survives only partly in Greek, the rest in medieval Latin translation from the Greek. It is the first known work on hydrostatics, of which Archimedes is recognized as the founder. Its purpose is to determine the positions that various solids will assume when floating in a fluid, according to their form and the variation in their specific gravities. In the first book various general principles are established, notably what has come to be known as Archimedes’ principle: a solid denser than a fluid will, when immersed in that fluid, be lighter by the weight of the fluid it displaces. The second book is a mathematical tour de force unmatched in antiquity and rarely equaled since. In it Archimedes determines the different positions of stability that a right paraboloid of revolution assumes when floating in a fluid of greater specific gravity, according to geometric and hydrostatic variations.

Archimedes is known, from references of later authors, to have written a number of other works that have not survived. Of particular interest are treatises on catoptrics, in which he discussed, among other things, the phenomenon of refraction; on the 13 semiregular (Archimedean) polyhedra (those bodies bounded by regular polygons, not necessarily all of the same type, that can be inscribed in a sphere); and the “Cattle Problem” (preserved in a Greek epigram), which poses a problem in indeterminate analysis, with eight unknowns. In addition to those, there survive several works in Arabic translation ascribed to Archimedes that cannot have been composed by him in their present form, although they may contain “Archimedean” elements. Those include a work on inscribing the regular heptagon in a circle; a collection of lemmas (propositions assumed to be true that are used to prove a theorem) and a book, On Touching Circles, both having to do with elementary plane geometry; and the Stomachion (parts of which also survive in Greek), dealing with a square divided into 14 pieces for a game or puzzle.

Archimedes’ mathematical proofs and presentation exhibit great boldness and originality of thought on the one hand and extreme rigour on the other, meeting the highest standards of contemporary geometry. While the Method shows that he arrived at the formulas for the surface area and volume of a sphere by “mechanical” reasoning involving infinitesimals, in his actual proofs of the results in Sphere and Cylinder he uses only the rigorous methods of successive finite approximation that had been invented by Eudoxus of Cnidus in the 4th century BCE. These methods, of which Archimedes was a master, are the standard procedure in all his works on higher geometry that deal with proving results about areas and volumes. Their mathematical rigour stands in strong contrast to the “proofs” of the first practitioners of integral calculus in the 17th century, when infinitesimals were reintroduced into mathematics. Yet Archimedes’ results are no less impressive than theirs. The same freedom from conventional ways of thinking is apparent in the arithmetical field in Sand-Reckoner, which shows a deep understanding of the nature of the numerical system.

In antiquity Archimedes was also known as an outstanding astronomer: his observations of solstices were used by Hipparchus (flourished c. 140 BCE), the foremost ancient astronomer. Very little is known of this side of Archimedes’ activity, although Sand-Reckoner reveals his keen astronomical interest and practical observational ability. There has, however, been handed down a set of numbers attributed to him giving the distances of the various heavenly bodies from Earth, which has been shown to be based not on observed astronomical data but on a “Pythagorean” theory associating the spatial intervals between the planets with musical intervals. Surprising though it is to find those metaphysical speculations in the work of a practicing astronomer, there is good reason to believe that their attribution to Archimedes is correct.

PYTHAGORAS

Pythagoras of Samos[a] (c. 570 – c. 495 BC)[b] was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Knowledge of his life is clouded by legend, but he appears to have been the son of Mnesarchus, a gem-engraver on the island of Samos. Modern scholars disagree regarding Pythagoras's education and influences, but they do agree that, around 530 BC, he travelled to Croton in southern Italy, where he founded a school in which initiates were sworn to secrecy and lived a communal, ascetic lifestyle. This lifestyle entailed a number of dietary prohibitions, traditionally said to have included vegetarianism, although modern scholars doubt that he ever advocated for complete vegetarianism.

In the next post, I will be talking about ramanujan and shakuntala devi.

Follow me, stay tuned, next post will be uploaded soon.

[thesciencenut]

About the great mathematicians of the world

Hi! Welcome back to #thesciencenut. In this post, I am gonna tell y'all about some of the greatest mathematicians of all times.

Following are the great mathematicians I am gonna talk about-

1) Euclid

2)Leonhard euler

3)Archimedes

4)Pythagoras

5)Ramanujan

6)Shakuntala Devi

EUCLID

Euclid was a greek mathematician. He has made great contributions to the field of geometry and hence is known as the founder of geometry. He lived in Alexandria, Egypt around 300 BCE. At that time, ptolmey ruled Alexandria, Egypt.He probably studied for a time at Plato’s Academy in Athens but, by Euclid’s time, Alexandria, under the patronage of the Ptolemies and with its prestigious and comprehensive Library, had already become a worthy rival to the great Academy. He wrote perhaps the most important and successful mathematical textbook of all time, the “Stoicheion” or “Elements”, which represents the culmination of the mathematical revolution which had taken place in Greece up to that time.

LEONHARD EULER

Leonhard Euler was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer. He made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory.

Learn about archimedes, pythagoras, ramanujan and shakuntala devi in the next posts.

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[thesciencenut]

Miss Revolving number 🤓

Heyo there ! Today, I am gonna introduce you to miss revolving number. Ye you heard it right, miss revolving number.

You must have heard of this number in the movie which recently flashed up in the Indian Cinema- "Shakuntala devi ". She indeed was a great mathematician and had got the title -"Human computer" for she is known to beat the fastest computer of the world.

This revolving number is 142857.

Now, if you multiply 142857 by 1 you get 142857. If you multiply 142857 by 2 you get 285714. If you multiply 142857 by 3 you get 428571. If you multiply 142857 by 4 you get 571428. If you multiply 142857 by 5 you get 714285. So you see, why is this known as "miss revolving number "?

Isnt it interesting ?

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[thesciencenut]

1729, The Hardy Ramanujan Number

In the earlier post, I had told that I will be talking about the speciality of number 1729. 1729 is the smallest number which can be expressed as sum of two different in two different ways.

1729= sum of cubes of 10 and 9. Also-

1729= sum of cubes of 12 and 1.

It is also known as the hardy ramanujan number.......wanna know why???

Dated long long back, ramanujan, the famous mathematician of india, had got several injuries during a car accident. He was in the hospital. At that time, he was in london as he wanted to meet the famous mathematician- hardy. That is why he was admitted in a London hospital. Hardy came to meet him. At that time, ramanujan asked him the number ,present in the number plate of the car he used to come to the hospital.

Hardy told that it was 1729.

At that time , ramanujan told the speciality of the number 1729.

Hardy was shocked. He then understood that ramanujan indeed was the greatest mathematician of all times. Because the number's speciality was discovered in the presence of hardy and ramanujan, the number was named the hardy -ramanujan number.

Also, one more speciality of the number is that -

1+7+2+9 = 19 and if we multiply 19 with it's reverse i.e. 91, we get 1729 again.

Photo Credits- Google

[thesciencenut]

Arithmetic Progression

So, hi everybody! As you can see, today we are going to talk about - Arithmetic Progression. But first, let's understand what is a progression.

A progression is a list of numbers which follow a common pattern. An example is as follows -

19, 38, 57, 76, 95.............

In the above list of numbers, a common difference of 19 is present. This is the common pattern which every number is following. This type of a progression is known as Arithmetic progression.

A list of numbers having a common difference is known as an Arithmetic progression. There are many other progressions like - geometric progression, harmonic progression, arithmetico geometric progression. We will discuss about these progressions as well.

Now, Arithmetic progression has certain formulas. These formulas are used to solve questions related to Arithmetic progression .

{a means first term of an AP, d means common difference, n means number of term in the AP}

Sum of n terms in AP =( n/2 )×{2a + (n-1)d}

nth term of an AP = a+(n-1)d

Now, let's talk about transcendental numbers.

Transcendental Numbers

Transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial with rational coefficients. The best known transcendental numbers are π and e.

Now, you must be thinking why the hell am I taking two different topics in a single post. That is because, I wanted to tell you all something which is understandable only when you understand what is AP and transcendental numbers.

Now, I am gonna tell a very weird, unexpected and mysterious secret of mathematics. Students will be the most shocked on knowing this thing.

π is NOT equal to 22/7.

YES , YOU HEARD IT RIGHT.

And this is because π is a transcendental number, which means in its decimal part, the numbers present are not recurring. They are all different. While in the fraction 22/7, the digits are recurring after we perform the division for a long time.

And in the value of π , in its decimal part, its 1729th number is 1, then you will see that the next digit is 2 and the next digit is 3.

So-

1,2,3

Aren't they an Arithmetic progression, with a common difference 1?

And do you know 1729 is a very special number. It is known as the hardy Romanian number. I will discuss about its special property in my next post.

Till then, stay tuned by following me and joining "The Maths Nut" group chat.

And also give this post as much of likes as you can 🤗.

Bbye!

[thesciencenut]

The Beauty of Numbers

Hey there. This is - TheScienceNut. Well let me tell you that I am new to tumblr. This is my first post, and it is gonna be interesting.In this post, I am going to show you all the beauty that lies with numbers. One such number is - 101. 101 is a very very very special number. It is different from other numbers and has a uniqueness in it's properties.

So people, can I ask you all to perform a simple multiplication of the numbers -101 and 25. What is your answer? Is it 2525? Now multiply the numbers - 101 and 83. What do you get? Do you get 8383?. Now multiply 101 with any two digit number, say ab. Then you will observe that-

101 × ab(any 2 digit number) = abab. Isnt it fascinating? 101 is also called as a palindrome. What are palindromes? Palindromes are numbers which look the same from both directions i. e. from both right to left and left to right. Look the number 101 number from both sides. Aren't they the same. While many other numbers aren't palindromes. Some other examples of palindromes are -101010101 , 2020202, 1111111, 203020302 etc.

Manu such numbers are present in the vast number system. Do you wanna go deeper and deeper into the world of numbers, and explore the world of numbers? You have found the right guy. Follow me , and give as much of likes to this post as you can...........next special number will be posted soon.....

Till then, good bye.