Aryabhata I, the father of trigonometry and zero

Aryabhata I, the father of trigonometry and zero 
StuffThatMatters: Notable people 

One of the, if not the, greatest formulations in the human history is the Indo-Arabic system of numerals, that we currently use in most places. This elegant and simple system of numbering things, using just 10 symbols - 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 0 - it's an absolute beauty. And without it, had we been using the old western (Roman) system of numbering, with I, II, XI etc mathematics would certainly have taken a blow. 

This number system was first put in place by the great Indian mathematicians of the classical age of Indian history (during the Gupta dynasty). And arguably, much of the credit for formalizing this system can be given to Aryabhata I. 

Who was Aryabhata? 

Aryabhata, also called Aryabhata the first or Aryabhata the elder, to distinguish him from Aryabhata the second, is regarded as the greatest mathematician and astronomer of India'a classical age. Before him, there were great scholars like Baudhayana, Lagadha but it was Aryabhata who gave a formal touch to the subject of mathematics, founded trigonometry (especially the sine and versine functions), formally introduced 0 as a number and much more. 

Born in central India (although his birthplace is debated) in 476 AD, Aryabhata witnessed an age of tremendous scientific and mathematical progress in India, thanks to the patronage of the powerful Gupta emperors. He made use of this opportunity, joined the prestigious Nalanda University (an ancient seat of learning in eastern India) and authored some of the greatest mathematical works. 

Contributions to science and mathematics 

Without Aryabhata, many of the things we take for granted today, would not have been around. Whenever we dial a phone number, do some transactions or programming, use the sine function in calculus or just know that a new day has begun as soon as the clock struck 00:00 AM, we are unknowingly indebted to Aryabhata. 

As a mathematician, Aryabhata modified the already present Indian place-value system, and formalized it to make room for 0. This is open to debate, though, but it is usually accepted that Aryabhata was the first to develop the concept of Shunya in mathematics, which still serves as the Indian term for zero. 

Contributions in mathematics 

1. Zero and the place-value system: As already mentioned, Aryabhata modified and formalized the Indian number system. In the tabular arrangement of digits, he used the term Kha to denote zero. It is unclear though, whether Aryabhata crafted the symbol for 0, or it was done by some other Indian mathematician, but it is accepted that the symbol for zero is round because it is made to mimic the round depression in the sand left, after we take out a pebble from it (one of the ways the Indians loved to calculate). 

2. Discovery of the value of pi: Although pi, our beloved number, was discovered long before Aryabhata, and was used in cultures such as ancient Egypt, Aryabhata was the first to correctly estimate it to upto four decimal places, using his elegant and purely arithmetic approach. In his works, we find the value of π to be: 

 π = 62,832/20,000 = 3.1416 

3. Sine function and trigonometry: Although basics such as the Pythagorus Theorem existed before Aryabhata, it was he who virtually founded the discipline, giving rise to the very common functions sine, cosine, (Jya and Koti-jya). Using nothing other than Pythagorus theorem, Aryabhata wrote the first ever trigonometric tables, carefully calculating the values.

Using this great knowledge, later Indiana astronomers were able to estimate the relative distances of the sun and other planets, right from the earth itself.

4. Powered arithmetic series: Aryabhata formulated the equations for calculating the sum of any finite series of squares and cubes of natural numbers. The following equations appear in his work:

1^2 + 2^2 + 3^2 + ... + N^2 = N x ( N + 1 ) x ( 2N + 1 ) / 6,

1^3 + 2^3 + 3^3 + ... + N^3 = ( 1 + 2 + 3 + ... + N ) ^ 2 = { N x ( N + 1 ) / 2 } ^ 2

5. Aryabhata algotirhm and Diophantine equations: Aryabhata's work on the Diophantine equations (of the form ax + by = c) were referenced by later mathematicians like Bhaskara I, Mahavira and Brahmagupta. His method involved breaking down the first-order standard Diophantine equations into smaller and newer equations with smaller coefficients, thereby utilizing the method of continued fractions finally. 

Aryabhata referred to this method as Kuttaka (fragmenting into smaller pieces). This method is used in modern disciplines such as cryptology with great vigour. 

6. Ganita and the systematization of mathematics: Aryabhata used the term Ganita to refer to his mathematics, although Ganita was much more than that. It was essentially a philosophy of mathematics, which was subsequently carried by the Arabs of the Islamic empire to Europe. Many great early mathematicians were greatly influenced by this concept of Ganita, and Aryabhata's number system. Fibonacci, for example, had argued in the favour of the Indian number system as opposed to the Roman system, after virtually bringing this system to Europe. 

Contributions in astronomy 

Other than being a first-class mathematician, Aryabhata was also hugely interested in astronomy. Long before Newton would eomply the full might of mathematics to bind the heavens to our intuition, Aryabhata utilized simple observations and geometry to frame what would finally be the first truly Heliocentric theory of the solar system. 

1. Rotation of earth: Before Aryabhata, it was believed that the cosmic bodies like stars and planets themselves moved, while the earth stayed in the same place. But Aryabhata correctly explained the concept of terrestrial rotation, that the earth itself rotates on its axis daily, thereby causing the illusion of a rotating sky. In Aryabhatiya, he describes the development of this radical new theory, using the counterexample of asterisms. 

2. Explanations of eclipses: Aryabhata also gave a very detailed description of the eclipses, and established them as optical phenomena rather than events caused by pseudo-planetary nodes of the lunar orbit, called Ketu and Rahu. 

3. Geometric model of the solar system: Although Aryabhata's own model was geocentric, it contained the roots of Heliocentrism in it. In addition, Aryabhata developed an observationally accurate model of the solar system, which was also the first truly geometric model in the Indian subcontinent. This influenced Arabian thinkers of the middle ages, and helped them further improve the model. 

4. Times of rotational and revolution periods: Aryabhata estimated the time of earth's rotation to 23 hours 56 minutes 4.1 s, while the modern value is 23 hours 56 minutes 4.091 s! This would tell us how accurat and sophisticated his calculations were. 

He also estimated the time of earth's revolution to be 365.25858 days, as opposed to the modern value of 365.25636 days. These were the most accurate values for thousands of years. 

5. Midnight day: Although the prevalent Hindu model was to consider dawn to be the beginning of a new day, Aryabhata's genius was to first formally establish the notion of midnight days. 

Legacy 

Aryabhata's works Aryabhatiya and Arya-Siddhantam are widely famous in India, although the latter has not been recovered fully. It's only through commentaries on his work and references, that we can estimate what was in his second book. 

The Muslim invaders obviously ransacked the great Nalanda university, after defeating the Indian kings through the use of treachery (battle of Tarain, 1192). Thousands of scholars were beheaded and burnt alive by the invaders, while great books and texts were plundered and burned. This, according to many, destroyed much of the data on Aryabhata we could have recovered. 

However, Aryabhata, the immortal mathematical giant of India, continues to live today through his works, without which much of modern-day mathematics would have been obsolete. The Indian government named its first satellite Aryabhata in 1975, in the honour of this great personality. 

To translate a portion of his biography from a Bengali text: 

"It was the middle of 499 AD, and the great scholars of the Nalanda university had gathered together - to welcome a young 23-year-old, polite of speech, radiant as the midday sun, before whom, even the elderly thinkers seem to feel nervous. With the sound of conch shells, Aryabhata, pride of the _Aryaas, walked in front, lifted his hands to the midday sun, and began his great work Aryabhatiya, that would one day redefine how we view the universe itself...*

References 

http://ift.tt/1vjLp2s

http://ift.tt/1vjLq6t

http://ift.tt/1t7VATa

#Aryabhata #scientists #science #math #india 




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Infinitely many infinities: a proof using Hilbert's Hotel paradox

Infinitely many infinities: a proof using Hilbert's Hotel paradox 
StuffThatMatters: Mathematics 

Georg Cantor, the founding father of the wondrous discipline called Set Theory, was one of the greatest mathematicians of the 20th century, and as Prof. Marcus du Sautoy notes, maybe his mathematics was too ahead of his time, which contributed to its underestimated devaluation by his contemporaries. 

In here, let us discuss one of the greatest mathematical discoveries by Cantor, which is a rather mind-blowing conclusion for us all, and yet seemingly irrefutable (to date, at least). 

Taming the infinity 

Before Cantor, not many mathematicians worked with infinity, and it was widely thought that infinity was beyond the scope of mathematics. However, it was Cantor's genius that finally included the infinity, in fact, the infinities, into the paradigm of modern math. 

We are all pretty familiar with the symbol ∞, which is the symbolic representation of infinity. However, to think that there's only a single infinity, would be a premature conclusion. 

Hilbert's Hotel, a hotel with infinitely many rooms 

In his 1947 book, Cantor devised this beautiful and easily understandable veridical paradox (a statement with a very intuitive set of premises, but a bizzare and counter-intuitive conclusion), which he attributed to one of his mathematical heroes, David Hilbert. And this is what changed how we see infinity, forever. 

Premise: Let us assume that there's a hotel, run by Hilbert, which has infinitely many rooms! So, the room numbers can be written as the series below: 

1 , 2 , 3 ... ∞ 

To write it as a set, it'd be: 

{ x : x ∈ [ 1 , ∞ ) } 

Now, let us say that on a typical evening, Mr. David Hilbert sees that every room in the hotel is full, and there's no room left for anyone else. But then, 3 groups of guests arrive, and the fun starts. 

Group 1: Finite number of guests 

Ok, so at 7:00 PM, 5 new guests arrive at the hotel, and request Mr. Hilbert to reserve a room for each of them. Now, the hotel is already full, as we've mentioned. But Mr. Hilbert makes a clever solution. He shifts the occupants of each room n to the room n + 5. 

That is, the guy occupying room 1 shifts to room 6, the gal occupying room 2 shifts to room 7 and so on. 

Naturally, rooms 1 to 5 are now empty, and Mr. Hilbert easily books them for the new guests. 

Countably infinite guests 

Now, let us say that at 8:00 PM, another bunch of guests arrive, only they are infinite in number this time. So, the guests too, like the rooms, are numbers from 1 to ∞. They request rooms to Mr. Hilbert. 

Mr. Hilbert employs a yet cleverer solution, he makes each occupant at room n to the room 2n. So, the man at room 1 moves to room 2, the lady from room 2 shifts to room 4 and so on. Since n is a natural number, 2n excludes odd numbers from the list. 

So, all the odd-numbered rooms are now empty, and the guests now comfortably enter the rooms. 

Conclusion: Finite infinities can be equivalent. 

Uncountably infinite guests 

Ok, so at 10:00 PM, another group of infinitely many guests arrive, with each guest denoted by a real number (R) between 0 and 1. This time, there's trouble. 

Say, Mr. Hilbert empties the entire hotel for the newly arriving guests, and now all the rooms are empty. Then, he makes the following arrangement: 

Guest 0.1 moves to Room 1
Guest 0.2 moves to Room 2
... 
Guest 0.n moves to Room n 

Now, all the rooms are now full. 

But there are guests left! Between 0.1 and 0.2, there are infinitely many numbers, 0.11, 0.12, 0.1113646247 and 0.164924848246284638 and so on! So where are they going to stay now? 

Conclusion: This demonstrates a very important property of the infinities, real number infinities are larger than integer infinities. 

Infinitely many infinities: power sets to the rescue 

So, let us say that Mr. Hilbert now grows really wary, and decides to build an ever grander hotel, with uncountably infinite rooms. 

Now, let us say that the third group of guests are grouped together to form a set S. So, n(S) = ∞ (i.e the cardinality of the set, or the number of elements in the set, is infinite).  

But, what about P(S), i.e the power set of S? (I explain power sets at the end of this post, refer to it if you're not familiar with it). We know that the cardinality of a power-set is given as: 

n( P(S) ) = 2 ^ ( n(S) ). 

Now, the same group of guests arrive again, and demand rooms. Mr. Hilbert can now easily accommodate them all, with the following process: 

Guest 0.1 shifted to Room { 0.1 } 
Guest 0.172646 shifted to Room { 0.172646 }
... 
Guest x ∈ ( 0 , 1 ),  x ∈ R shifted to Room { y : y ∈ ( 0 , 1), y ∈ R } 

So, since 2 ^ n(S) > n(S), there must be rooms still left, even after accommodating these guests. That is, when each guest is shifter to the room whose name is the singleton set containing the number associated with the guest, all those elements of the power-set containing non-singleton sets, such as { 0.1 , 0.2 }, { 0.1 , 0.4 , 0.6 }  etc are now empty! 

Naturally, infinitely many rooms are still empty. 

Now, there's a power-set of the power-set itself, so if a group of people now arrive at the hotel, with each person associated with an element of the power-set of the power-set set of the real-number infinity between 0 and 1, the hotel will be blocked once again, with infinitely many people left without rooms. 

Conclusion: So, there are infinitely many infinities. At first, we see that the power-set of the real-number infinity set is larger than the latter, but then again, every power-set can have its own power-set. And since there can be indefinitely many, infinitely many power-sets, there are infinitely many infinities (proved). 

A bit of explanation 

Although set theory is the very basic thing behind this, non-mathematics students may not be familiar with it. So, here's a brief discussion. 

Set: A set is roughly any group of things, known as its elements. These can be numbers, letters, names and anything else. For example, S = { 1 , 2 , 3 } is a set of 3 elements. If an element e is in a set, it is written as e ∈ S (i.e e belongs to S). 

Cardinality: The cardinality of a set is the number of elements it contains. It may be finite or infinite. It's written as n(S). 

Subsets and supersets: Subsets are sets such that, every element of the subset P are present in the original set S. But not vice versa, usually. They are written as P ⊂ S. 
Likewise, supersets are sets such that, every element of the set S are present in the superset Q. They are written as S ⊂ Q. 

Power-sets: A power-set, P(S), is a collection of all possible subsets of a set S. It has a number of elements given as 2^n, where n is the number of elements on the original set. For example, let S = { 1 , 2 , 3 }. So, all possible subsets of S are given below: 

{ } (empty set, with no elements) 
{ 1 } 
{ 2 } 
{ 3 }
{ 1 , 2 }
{ 1 , 3 }
{ 2 , 3 } 
{ 1 , 2 , 3 }

Naturally, the power-set is a set containing all these. So, 

P(S) = { { } , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 } } 

And clearly, n = 3, so number of elements in P(S) = 8 = 2^3. 

Set theory is a very broad theory, and so there's a lot more to explain, but I guess this can do for our purpose here. 

References 

http://ift.tt/1DCKHRD 

http://ift.tt/1DCKG0b 

#math   #friday   #infinity   #interesting   #science   #paradox  




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5 great Computer Algebra Systems that you can get for free

5 great Computer Algebra Systems that you can get for free 
StuffThatMatters: Mathematics & Computing 

For mathematicians, chemists, engineers and physicists alike, calculations are a must. And doing lengthy calculations by hand is both tiresome and error-prone. This is why Computer Algebra Systems (CAS) are important. 

What are Computer Algebra Systems?

But for starters, here's an introduction to CAS. Basically, a Computer Algebra System is a piece of software that can work with equations, matrices, graphs and other mathematical constructs, because a CAS is specially designed for one single purpose - manipulation of mathematical constructs.

You can always calculate a large but straightforward arithmetic problem with nothing but a calculator, or the usual software on your computer, but what about the roots of an equation, say x^3 + 7x^2 + 9x + 10? Or, what if I ask you to invert a 4 x 4 matrix? This is where CAS software come into play, it can just take the equation, solve it instantly and return you the result, and even tell you the steps (in some commercial software). 

Some examples of what a CAS can do for you 

1. Solving linear, quadratic, differential equations etc. 

2. Differentiating/integrating functions. 

3. Expanding/simplifying expressions. 

4. Plotting/graphing functions. 

5. Manipulate special types of data, such as matrices, arrays, intervals etc. 

Free CAS software you can download now 

Now, for those who use CAS, Mathematica and Maple are like the Holy Grails of CAS. But not everyone can afford them, especially since those software are pretty aggressively priced. And being an advocate of open-source and software freedom, I always look out for better, open-source alternatives. 

Of course, there are many, many features which Mathematica and Maple offer, that will never be found in the free software I mention below, but unless you're really, really serious about your mathematical computation, I believe things like Maxima, Sage etc will work flawlessly for you. Be warned though, there's a bit of learning curve in each of them, but plenty of online documentation and volunteers are available to help you. 

So, here they are. 

1. Maxima 

Maxima is my personal favourite, when it comes to open source CAS software. It's fast, it's comprehensive and it's expandable, plus there's a great deal of information about it online. 

Maxima was originally based on Macsyma (hence the name), but the original CAS got discontinued, and Maxima survives today. It includes a powerful mathematical programming language, that you can use to code scripts (executable via maxima interpreter). 

Learning it is pretty straightforward, plus it's available on a variety of platforms. Be sure to use a GUI if you plan to install Maxima, such as wxMaxima, GMaxima (on GTK+ interface) etc. Also, for Linux users, KDE's fantastic front-end Cantor can be linked with Maxima, which is perhaps the best combination. 

Download: http://ift.tt/1gN7Yno (check the Downloads section) 

2. Sage (System for Algebra and Geometry Experimentation 

Sage is in itself a complete package, and much more complete than Maxima so far as the integrated GUI is concerned. But here's the catch - instead of packing a GUI of its own, Sage makes use of your existing browser to give you a notebook-like interface, that you can use for all your calculation needs. 

The notebooks of Sage are a delight to use, and there's an integrated tutorial that guides you well. Even better, you can keep the entire installation as a binary on your pen-drive, and then access it on each machine you own. Recently, an online version of Sage is also being developed, with a native Android app for easy access. 

One problem, however, is the lack of any Windows version of Sage. 

Download: http://sagemath.org/ (for Linux & Mac systems) 

3. Mathomatic 

Mathomatic is a general purpose CAS, and it excels with regards to its size. It's really lightweight and small, but this comes at a price. It can only deal with elementary algebra, and it doesn't have a GUI. 

Instead, mathomatic makes use of your terminal (DOS-like prompt in Windows), and requires you to use its specialized syntax. 

But yes, mathomatic is available for almost all OSes out there. Linux users can install a PPA to stay updated, and Windows users can just follow the old .exe download + installation process. 

Download: www.mathomatic.org 

4. Reduce 

Reduce was previously a commercial CAS, but since 2008 (IIRC) it's open-source. It's primarily meant for physics, especially astronomy-related calculations. It's a brilliantly programmed software, and has been developed for over a long time, since the 1960s. This gives it an edge over other, newer solutions. 

Reduce is also available for a variety of platforms.

Download: http://ift.tt/1vhjRsC 

5. OpenAxiom 

OpenAxiom, or simply Axiom (following the publication of its code). Like Reduce, it too has been in development for a long time. However, it's perhaps the most frequently developed one in the list. 

I confess, that unlike the other four in the list, I haven't used Axiom well enough. In fact, I just downloaded it day before yesterday, and decided to give it a shot. But from what I've seen, other than the lack of a GUI, it's quite powerful. Also, as the website states, work is under way, to develop the software into a fully usable GUI-backed solution for computational needs. 

Download: http://ift.tt/1mTdkoF 

Conclusion 

Well, these software are nowhere near Maple or Mathematica, but then again, the total price you'll have to pay for all these 5 software is $0.00. And that amounts to something, I guess. 

These glorious software demonstrate the beauty of open-source and the idea of software freedom. Thousands of volunteers are working hard so that we may use them, and the makers get nothing in return. In fact, I believe they are somewhat like me, since I maintain this page only for sharing what little I know with everyone, without any commercial interests. Only, they are much, much, much better than I am, because what they've designed is million times better than my act of spreading the word. 

If these software benefit you, please be sure to spread the word. 

References 

http://ift.tt/1mTdkoH 

#freesoftware   #linux   #math   #science   #opensourcesoftware  




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Mars & Beyond: A brief account of India's journey to space

Mars & Beyond: A brief account of India's journey to space 

Today, the 24th of September, 2014, will forever be remembered as one of the days that saw history being made. The Mars Orbiter Mission, also known as Mangalyaan is India, has successfully reached the Martian orbit, in its very first, maiden attempt to do so. This is truly spectacular, since it's the first time any space research body in the world, has been successful in reaching the Martian orbit on its own, at the first attempt. 

The Mangalyaan (Mars-craft) was launched on 5th November, 2013. It carried a suite of 5 kg of scientific equipments, and attempts to study the upper Martian atmosphere and minerals on the surface of Mars. It is being monitored and guided by the IDSN (Indian Deep Space Network), which is equivalent to the Deep Space Network operated by NASA. 

MOM, why the hype? 

Out of 51 missions sent to the Martian orbit, only 21 have been successful. Besides, no country had previously been successful in doing this feat in the first attempt, this is what makes the MOM so important. 

But in addition, MOM is also the most cost-effective and economical deep space mission in the recent times. The US's MAVEN craft, that was launched alongside MOM, costs around 10 times more than MOM. As Business Insider notes, "The $74 million bill for India’s Mars Orbiter Mission is the lowest compared to missions of other space agencies. For instance, NASA's Mars Atmosphere and Volatile EvolutioN's (MAVEN), which reached Mars on September 21, cost was $617 million". 

BBC additionally notes  that MOM is "the cheapest inter-planetary mission ever to be undertaken by the world".

This amount to only a meagre Rs. 4.00 per person. And it's less than the cost for organizing an IPL (Indian Premiere League, a nationwide cricket match series held annually). It's significantly less that budgets of hi-fi Hollywood movies like Pirates of the Caribbean III. 

A brief history of ISRO 

I969: ISRO is established on the 15th of August, and brought under the Department of Atomic Energy (DAE), India. 

1972: The Indian government establishes the Indian Space Commission and Department of Space (DoS). ISRO is brought under the umbrella of the DoS. DoS reports direct to the Prime MInister, and is an otherwise independent body, securing flexibility in the operations of ISRO. 

1975: ISRO is brought under direct control of the Prime Minister, and also launches its first ever indigenously built satellite, the Aryabhata, named after the great Indian mathematician who had developed the Indo-Arabic number system (that we use today). 

1979: India's second satellite, Bhaskara I, is launched. 

1980: ISRO successfully launches Rohini I on the indigenously built SLV (Satellite Launch Vehicle). 

1983: The Rohini 3 telecommunications satellite is launched successfully. 

1984: With Soviet collaboration, the Indian astronaut, Rakesh Sharma, becomes the first Indian to reach and spend 7 days in space.  

1992: Following two failed attempts to use the Augmented Satellite Launch Vehicle ASLV), ISRO tastes success with INSAT-2, the indigenously built geostationary satellite. 

1999: - More powerful Polar Satellite Launch Vehicle (PSLV), for the first time, launches 3 satellites in one ago. The PSLV could initially lift 1000 kg payload to a Sun-synchronous polar orbit. 

2001: The Geosynchronous Satellite Launch Vehicle (GSLV) is tested successfully, with the initial capacity to lift 2500 kg load to a geosynchronous orbit. GSLV enabled India to do lift a series of weather and remote sensing satellites, and it is what powered India's dreams of the Mars exploration. 

2008: Chandrayaan I, India's first unmanned Lunar orbiter, is successful, and it becomes the first orbiter to detect the existence of water on moon. 

2013: Mangalyaan, India's Mars Orbiter Mission, is launched. 

2014: Manganyaan, as of today, successfully reaches the Martian orbit, and creates history for the world. ISRO becomes the first organization to successfully reach the Martian orbit in the first attempt, as well as the first Asian organization ever to do so. 

ISRO's Future projects 

Several projects have been undertaken by the ISRO, and are under various stages of implementation. Many of these are uniquely important, such as the celebrated Aditya I mission that is set to be implemented by 2017. 

Some of these projects are: 

1. Chandrayaan II: Following the success of Chandrayaan I, ISRO aims to send a second probe to the Lunar surface, using the GSLV system. It will also include a Lunar rover, alongside a more sophisticated orbiter. The primary aim of this mission would be to study the gradual evolution of the earth's satellite, as well as its origins. 

2. Aditya I: Aditya I will be India's first SEP (Solar Exploration Probe). It will contain the coronagraph modules to study the CME (Coronal Mass Ejections) and the formation and development of CME (Coronal Magnetic Field). 

3. Venus Explorer: ISRO is on its way of launching the first Indian Venus explorer. It will be launched by 2016, to study the weather patterns and overall climatic system of Venus. 

4. Equipments: A number of new and more powerful launch vehicles and other scientific instruments are being designed by ISRO. Some of these are the GSLV Mk-III, RLV-TD (Reusable Launch Vehicle Technology Demonstrator) and SCRE II (Space Capsule Recovery Experiment II). 

References 

http://ift.tt/1B5InOQ 

http://ift.tt/XYIv5L

http://ift.tt/ZJXbrc

http://ift.tt/1og9WiI

http://ift.tt/XV6oLK

Image credits: TOI, , Business Standard, ISRO & Wikimedia

#ISRO   #mangalyaan   #marsorbitermission   #india  




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Thanks to +Bonnie Clydes, the Power Circle is now online. I hope it benefits you all who would like ...

Thanks to +Bonnie Clydes, the Power Circle is now online. I hope it benefits you all who would like more interactive and engaging audience. 

Instructions 

1. Add Bonnie Clydes to your circles:http://ift.tt/1uUC2G0
2. Publicly share this circle to public, your circles and extended circles.
3. If possible, leave a comment on this post.


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A simple formula to calculate what day (Mon-Sun) any given date is/was/will be

A simple formula to calculate what day (Mon-Sun) any given date is/was/will be 

Karl Friedrich Gauss, one of the greatest mathematical physicists ever, and also known as the prince of mathematics (quite aptly so), had originally developed a formula using nothing other than simple arithmetic, to calculate the weekday of the 1st of January of any possible year. Even better, his formula can be used for both the Gregorian and Julian calendars, but since we're using Gregorian one anyway, let's stick to that. 

The formula, published after the death of Gauss, can be modified easily to calculate the weekday of any given date, which is pretty straightforward and neat. 

The formula 

Without further ado, here it is: 

W = { D + ( 2.6M - 0.2 ) + 5 Rem ( Y , 4 ) + 4 Rem ( Y , 100 ) + 6 Rem ( Y , 400 ) } mod 7

So, it requires a bit explanation at first. 

1. W = the weekday constant, which is to be matched against a rather simple table, here it is: 

W = 0 -> Sunday
W = 1 -> Monday
... 
W = 6 -> Saturday 

2. D = the date of the month, aka 22 for today, 23 for tomorrow etc. 

3. M = f(m), a function of the month's number, given as M = f(m) = ( m - 2 ) mod 12, i.e subtract 2 from the month's number (1-12), divide it by 12 and grab the remainder. Also, it's 12 for February, and not 0. 

Still, for simplicity, here's the list: 

M (January) = -1
M (February) = 12
M (March) = 1
...
M (December) = 10 

4. Y = the year. Y = 2014 for this year. 

5. Rem is a remainder function, i.e Rem ( a , b) = remainder when b divides a. 

Example for today 
 
Unlike other sciences that require experimental verification, the beauty of math is that, you only need pen, paper and some brainpower. That's why, let's calculate the weekday for today, and see if the formula works. 

So, for today, the year is 2014, date is 22nd and the month is September, i.e the 9th month. Naturally, 

W = { 22 + ( 2.6 * 7 - 0.2 ) + 5 Rem ( 2014 , 4 ) + 4 Rem (2014 , 100 ) + 6 Rem (2014 , 400 ) } mod 7

= { 22 + 18 + 5 x 2 + 4 x 14 + 6 x 14 } mod 7

= 190 mod 7 

= 1 (since 190 = 7 x 27 + 1 , evidently). 

Now, W = 1 -> Day = Monday! 

Further simplification and a second example 

One problem with this formula is that, calculations get lengthy at times. For example, if you intend to calculate the weekday of India's independence (August 15, 1947), then calculations using this formula will get long, and error-prone. Naturally, it can be simplified further with a simple modification:

Y = y + 100C,

where y = last two digits of the year (i.e YEAR mod 100), and C = the current century, i.e 20 for this one.

Then, W = { D + ( 2.6 x M - 0.2 ) + 5 Rem ( y , 4 ) + 3 Rem ( y , 7 ) + 5 Rem ( C , 4 ) } mod 7  

Let's verify the weekday of India's independence, 15th August, 1947. We know that it was a Friday, and let's check it. 

So, W = { 15 + ( 2.6 x 6 - 0.2 ) + 5 Rem ( 47 , 4 ) + 3 Rem ( 47 , 7 ) + 5 Rem ( 19 , 4 ) } 

= { 15 + 15.4 + 5 x 3 + 3 x 5 + 5 x 3 } mod 7 

= 75.4 mod 7

= 75 mod 7 (75.4 rounded off) 

= 5, 

And naturally, W = 5 -> Day = Friday. 

Conclusion 

There are other methods for the same purpose. I posted one last year, which can be found here: http://goo.gl/dUPXZV (same guy here, just posting as my page this time) 

Also, there's Kraitchik's algorithm, but Gauss's method is better than both because it doesn't need us to remember any table of constants. This is why this simple algorithm continues to be my favourite. 

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Trichotomy of control: a Stoic method to make our lives better

Trichotomy of control: a Stoic method to make our lives better 
StuffThatMatters: Philosophy & Self-help  

One of the things I personally believe in life, is the Socratic assumption that good thoughts, belief-systems, values etc can heal our lives. This view has been challenged, for being overly simplistic, by many philosophers, thinkers, scientists and especially psychiatrists (for business reasons, obviously), but I have discovered time and again, that this view works to a great extent. Yes, your views can't change your whole life, or make obstacles magically vanish, but they will definitely have a positive impact on the very way you think, behave and react, if applied wisely and refined correctly. 

This is where stoicism comes into effect. For a long time, I had been searching for a philosophy I could relate to, and I devised a personal philosophical view of the world, which, although accurate (subjectively speaking, anyway), is a pessimistic approach to life. That's why I turned to ancient ethical philosophies of life, and discovered stoicism. 

What is stoicism? 

Since I'm writing this post for both philosophy-nerds like me, and common users as well, I guess it'd be better if I explain stoicism briefly at first. 

Stoicism is one of the great Hellenistic philosophies that developed after the BIG 3 - Socrates, Plato and Aristotle. It was first devised formally by Zeno of Citium, and then modified by a number of other thinkers. It grew immensely popular in the Roman empire, and under the watchful guidance of great Roman Stoics like Seneca, Rufus, Epictetus etc it developed into a reasonable, logical and powerful philosophy of life. Even the Roman emperor Marcus Aurelius was a practising Stoic, as evident from his writing Meditations, which survives today as one of the greatest Stoic texts ever. 

The trichotomy of control 

Describing stoicism as a whole would take hundreds, if not thousands, of pages, which is not the objective of this post. But the best part of this great philosophy is that, you can take bits of it and apply them to your life in a practical way, without having to know stoicism by heart. 

In this way, stoicism's teachings are in many ways standalone systems of assumptions. 

At the heart of stoicism, especially the version of it popularized by Musonius Rufus and his greatest student Epictetus, is the dichotomy of control. Epictetus, born a slave, showed great promise in learning, and was allowed by his master Epaphroditus to join Rufus's school. 

The dichotomy of control is simple - it's a logical premise that there are two categories of things (for want of a better term) around us, namely: 

1. Things we can control

and 

2. Things out of our control

However, later Stoics changed it into a trichotomy, adding a third category, an uncertain, grey area between the two extremes. Then the system became: 

1. Things we can control (A)

2. Things we can partly control (B)

and 

3. Things out of our control (C)

Here, the second category includes all the happenings, in which our actions may or may not have a limited impact on the outcomes. The best way to describe this system is through modern exemplification. And so, here they are: 

Example (A): Say, the AC if off, and it's very hot. You can obviously lie down and continue complaining, feeling miserable, accusing God/nature for the extreme heat etc, or you can just get up and turn the AC on (this is a very rough example, I admit). 

Example (B): Say, you want to develop good muscles. You can exercise, follow a healthy lifestyle, plan and maintain a diet, eat less junk food etc, but ultimately it's not certain whether you'll get those six pack abs. However, if you follow a logical sequence of activities, your chances of getting them get better, with time. 

Example (C): You get out for a stroll, and immediately it starts raining heavily. Other than grabbing an umbrella, there's virtually no respite from the rain. 

Almost everything we face can be categorized into this system of trichotomy, and once we do that, stoicism advises us to choose how to react to them. 

The trichotomy of actions 

For everything in A, we should learn to take responsibility. Basically, it is assumed that we are solely responsible for how we think, feel and decide. 

For things in B, we should employ internalization as opposed to externalization of goals. What this means is simple, back to the abs examples. So you decide to workout for a year, and still you might not get as fit as you wanted to be. Here, instead of setting your goal to be getting those abs, it would be much wiser to change it to just getting a bit more fit than I am currently. Then, even if you don't get those abs, you'll feel better, with a sense of achievement. 

Internalization of goals is, in a way, to set goals in such a manner that they are easier to achieve, within our control, and reasonable to think about. Giving it our best is an example of this great stoic advice. 

Now, for things in category C, the stoic suggestion is simple - try your best to avoid the situations (i.e don't allow them to needlessly damage yourself), but accept your limited capacity and don't let them affect your inner tranquillity. In the rain example, we don't have any control over the process, and so it's best to let it be as it is, while trying out best, in a logical manner, to stop it from damaging us physically. Mentally speaking, we shouldn't allow the external disturbance to affect our mind. 

Conclusion 

The standalone teachings of stoicism, if applied correctly, can add great value to our lives. And for opponents of this great line of thinking, I'd say that I exercise a pragmatic view regarding it - we can debate it forever, but it's better to just try and test it, see if it works, and then accept/leave it. 

References 

A guide to good life: the ancient art of Stoic joy (W. Irvine)

Meditations (Marcus Aurelius, also thanks to its free LibreVox recording)

Philosophy for life and other dangerous situations (Jules Evans) 

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Ancient water technologies that changed the history of the planet

Ancient water technologies that changed the history of the planet
StuffThatMatters: Science & Technology

Water is one of the most crucial resources for any civilization, and the world as a whole. Cultures have perished in history, because of drying up of rivers, while whole nations crumbled because of excessive rainfall, Tsunamis and oceanic disasters. That is why, the ancient cultures of the world had to rely on sophisticated technologies and engineering, to make efficient use of water, or risk drying up themselves.

Here, I list 5 really simple, but amazingly innovative technologies, that our ancients had developed for water management. I don't include the very common water wheel stuff, for obvious reasons. And obviously, the size limit of the post doesn't give me room to describe each and every technology, but these 5 technologies were definitely among the topmost innovations of the ancient period.

1. Aqueducts (India, Greece, Roman empire)

The earliest aquedacts have been discovered in the Indian subcontinent, centred around the Tungabhadra river, as well as around the IVC (Indus Valley Civilization) region in the north-western part of the subcontinent. The IVC was a very sophisticated civilization in terms of technology and innovation, and their city layouts consisted of a central citadel and a surrounding lower town or down town. Rectangular layout of the IVC cities helped build and sustain large aqueducts, which made use of descending gradients to carry waste water away from the city.

The largest early aqueducts in India were as long as 25 km in length, centred around modern day Hampi. Similar designs had also been built independently (this is debatable, though) in west Asia, Egypt and Greece back then.

However, the aqueduct technology reached its peak with the Roman empire. Romans built some of the largest aqueducts of the ancient world, breaking the Indian record (although this was at least 1500 years after the first Indian aqueducts). More than 10 aqueducts served the capital region of Rome during the peak of imperial Rome, and carried water from nearly 92 kilometers away, into the city. And this was wholly done with the force of gravity alone, with distribution tasks built to act as supply storages in between. The slope of these channels could be around 30:1, i.e every 30 metres would see a 1 m descent.

2. Shaduf (Egypt)

In ancient Egypt, the Nile was the source of most of the water required by the people. In the fertile valleys on the twin sides of the Nile, agriculture demanded a fast, easy and effective method of irrigation. For this, the Shaduf was developed.

The Shaduf is actually a modification of the lever, and employs a very large mechanical advantage. It consists of a pole balanced on a crossbeam, with a bucket attached to one end of the pole with rope. The farmer would pull the rope to bring the bucket down, immerse the bucket in water and loosely hold the rope as the pole balances back to its original position, thereby lifting the bucket-full of water. This allowed the ancient farmers to irrigate their fields without much effort, as opposed to manually lifting the bucket.

In drier seasons, when the water level in the Nile became lower, longer poles and ropes could be used for immersing the bucket. In later models, the poles could also be swung around, to empty the water in another part of the farm.

3. Stepwells (India)

Since the foundation of the Indo-Aryan civilization in north-western India, the Indians faced a tremendous problem of water-scarcity in the region, especially because of the low rainfall in the arid parts. This forced the ancient Indian engineers to develop what are called stepwells, cylindrical tanks that extend deep into the water table below, and protected from the scorching heat with the help of constructed roofs.

The earliest stepwells were developed in the 2nd century AD, although Moenjodaro of the IVC did have a few structures that were similar to stepwells in design.

The stepwells are named so, because they have stones cut in the form of steps that lead to the central well. During the monsoon months, rainwater is naturally harvested, and it replinishes the water level. Covers are built to protect the water from external agents. When water level rises sufficiently, the we have to descend less, to get the water.

Other than acting as fresh water reserves, stepwells also had great religious and cultural significance. People of the rural communities often gathered in the buildings to enjoy the cooler summer noons, thanks to the dampening and cooling effect of the site.

4. Artificial canals (China)

In the 4th century BCE, Chinese rulers saw the need to connect the natural waterways of China, especially the Yangtze river in the south, and the Huai river in the north. This made the Chinese builders find a way, through marshes and lakes in between, to construct the world's first full-fledged artifically built canal connecting the two rivers. This provided the much needed waterway for transport in China.

Today, the Grand Canal of China is the longest artificially built canal in the world, covering more 1776 kilometres of extent.

5. Inverse siphons (Roman empire)

The previously mentioned aqueducts were of great important to the Romans, especially those dwelling in Rome itself. Since Rome contained lots of public bathhouses, fountains etc it needed a large amount of water, not all of which was readily available in the neighbourhood. As such, the Romans built giant aqueducts with inverse siphon mechanisms and collecting tanks, to use a practical version of Pascal's principle.

The inverse siphons were put in place where the natural terrain did not support building continuous aqueducts. At first, water from the higher portion of the aqueduct would deposit at the first collecting tank, which would then build up more pressure, forcing water into the pipes connecting the first collecting tank to the second. First, the water would descend, and then it would rise to the level of the second collecting tank, and then continue its journey along the remaining extent of the aqueduct.

Conclusion

As in one history channel documentary mentioned, the ancients were no fools. What they built may seem too childish and simple to us anyway, but back in those periods, those inventions and innovations were what kept the forward march of the human civilization intact. In fact, it would be better for us to learn from their simpler approaches to life, and incorporate this element of simplicity into our designs.

Maybe it's remotely similar to a practical version of Occam's razor.

References

http://ift.tt/1tFLAo9

http://ift.tt/1tFLAoc

http://ift.tt/1tFLCfO

#history   #technology   #interesting   #science   #water

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The Earth Goddesses - Prithvi, Gaia & Sif

In many ancient religions, the #earth is often regarded as a mother-deity, since the earth is the source of life-giving boons of nature such as food (grains, vegetation), water the sustainer of life, a solid base to stand on, etc. 

All these are the attributes that are generally linked with a motherly figure. Possibly this is why we mostly have earth-goddesses, and not gods. There are exceptions, such as Geb in Egyptian #mythology, but usually in the Indo-European religions, the opposite is the case. 

Prithvi 

The goddess #Prithvi, as the personified form of the planet itself, is hailed as the provider of food, bestower of vegetation and protector of the Aryan settlements. In ancient Vedic texts, Prithvi is one of the main goddesses, although her concept remains more of an abstract one. 

In later Hinduism, especially during the era of the epics Ramayana andMahabharata, Prithvi gained a more personified image - as the mother of the terrifying demon Narakasura. 

Gaia/Terra 

In many other religions of the Indo-European pantheon, we have earth-goddesses. For example, the religion of the ancient Greeks worshipped #Gaia, the mother of all gods, and the first Titan. She is seen as the embodiment of the earth, and the creation itself in a more extensive sense. 

Gaia gave birth to her sons Ouranos (Uranus) and Oceaneous, although Ouranos is seen as her equal. Together, they gave birth to the first Titan deities, including Cronus (Saturn) and Rhea, the parents of the Olympian gods Zeus, Poseidon and Hades. 

Terra is the Roman equivalent of Gaia, although she is usually regarded in an even more abstract sense. It is from Terra that the word terrain is derived. 

Sif 

In the Norse pantheon, Sif is the personification of the earth, and the wife of the mighty #Thor, the god of lightning and thunder. Her golden hair is said to represent the golden wheat fields, which she is said to bounteously provide for the people of earth. 

NOTE: Since we post on a multitude of topics (because interesting facts can't be kept confined to any specific genre), we have decided to add a subtitle to each post, such as StuffThatMatters - Science or StuffThatMatters - Literature etc to help you quickly find out what the whole post is about. 

Thanks for reading!

How stars (yeah, cosmic ones) work

This is a rather crude, but easy to understand #infographic  on the Stellar Evolution(evolution of the stars). While it is nowhere close to the famed Hertzsprung-Russel diagram, the infographic gives us a quick look at what might be in store for our #sun in the future. 

The formation of a star

As we know (and to state in very simple language), a star is formed from interstellar clouds of dust, matter and gas - that contracts gradually because of its own gravitational attraction, and finally reaches a point-of-no-return, when the intense pressure inside leads to a rise in temperature and kickstarts the process of nuclear fusion. 

Hydrogen is the most common element found in nature, because it's also the simplest element. In nuclear fusion, the fist step usually involves the fusion of hydrogen nuclei (i.e protons) to form deuterium, where one of the protons changes to neutron. From this deuterium or heavy hydrogen, the light helium of He3 (2p+1n) is formed when a third hydrogen nucleus absorbs some extra mass/energy to become a neutron. In this step, powerful gamma rays are usually released to compensate for the extra energy. 

And finally, the third step sees a collision between two light He3 nuclei, which gives rise to 2 free protons to continue the chain, and 1 He4 or normal Helium nucleus. 

This is the secret behind the sun's tremendous energy. Usually this is called proton-proton chain. 

The stellar evolution

No matter how massive a star is, or how much hydrogen it has in store, one day it's hydrogen content will mostly have changed to helium. Then, instead of hydrogen, helium becomes the main nuclear fuel for the star. This begins the second step instellar evolution, which is called Helium Flash. 

In helium flash, basic steps usually involve tri-alpha process, where three He4 nuclei fuse to form 1 C12 nucleus. This carbon, that is generated in stellar evolution, is what constitutes your body and ours. Naturally, we certainly are children of the stars in a way. 

This also begins the C-N-O cycle in some stars, which involves the fusion between a C12 nucleus and a He4 nucleus, giving rise to #oxygen  O16 nucleus. And this gives birth to the precious oxygen that we are currently breathing. 

Death of a star 

These processes continue, giving rise to ever heavier elements, and releasing energy. This energy is crucial to keep the star itself in balance - since it counterbalances the tremendous gravitational pressure that tries to pull the star inwards. In heavier stars, when too much energy is released, the gravitational pull is temporarily defeated and matter ejects out of the star's surface. 

But this process cannot go on forever. So, when massive stars form iron, a deadlock is met. 

The funny thing about iron is that, nuclear fusion forming elements lighter than iron give rise to more energy that is consumed in the process. But once iron is formed, to form more massive elements would mean depositing and losing extra energy. It's like a movie that doesn't manage to raise its full budget, and is termed a flop. 

This gives rise to a sort of energy crisis inside the star. Temperatures rise further, as gravitational pull brings everything closer. In the case of most light stars, this is more or less the end, since they lack further energy or mass. They turn into white drawfs, which are virtually dead stars. 

Larger stars, however, meet a more deadly end. Because of their immense mass, the #supernova  explosion ejects out most of their matter, and following that, they collapse inwards. 

This collapse produces either a neutron star (if the star's core has < 3.2 x solar mass) or a #blackhole  if that isn't the case. 

References: Multiple sources, primarilyhttp://cosmos.phy.tufts.edu/~zirbel/ast21/handouts/StellarEvolution.PDF 

Ritchie vs Jobs - the saga of a forgotten pioneer

In October, 2011 the whole world was shocked, pained, thunderstruck and what not, to see Steve Jobs die. A visionary genius who had pioneered the persona computerrevolution, Jobs was a man of many talents. Despite having been an adopted child and a misguided teenager who had even turned into a hippie and fled to India, the to-be-founder of #Apple  had returned and restarted his life, bringing about one of the greatest visions the world had ever seen. 

Without Jobs, we would not have the Macintosh, iPhone, iPod, iPad, iTunes and Apple itself! But pardon us for saying that Apple Inc isn't the apple of everyone's eye, and for good reasons. His visions and ideas were revolutionary, but as far as genuine technological contributions to the field of computing in general, Jobs ranks well below the second man featured in the image. 

Dennis Ritchie?! Who's he? 

This is what we get to hear, whenever we mention him to most technology users of this generation, sadly. 

Dennis Ritchie was the man who had designed the C programming language, without which, much of what sophistication we see, wouldn't have been around. C has been hailed as God's own programming language, thanks to its virtual omnipotence when it comes to designing software from scratch.

And we must remember the time and the purpose for the designing of C - with his colleague Ken Thompson, Ritchie had wanted to design an industrial-gradeoperating system, that would be called UNIX. But the original UNIX was scrapped, and they needed a better way (and language) to build the UNIX kernel (the original kernel was coded with a machine-dependent assembly language) a better version of UNIX. The result? God's own programming language - C. 

It is analogically similar to when Newton had discovered that the mathematics of his day wasn't sophisticated enough for his physics, and therefore he developedcalculus to compensate for that. 

The UNIX operating system 

The UNIX operating system can be called the granddaddy of virtually everything you have today, be it an Android or iOS phone, a Windows-powered PC, a Macbook pro or a Linux/BSD box. 

Following UNIX's success, Prof. Andrew Tanenbaum had devised a free clone of UNIX, called Minix, which gradually inspired Linus Torvalds to build the Linux kernel, which forms the core and foundation of hundreds, if not thousands, of GNU/Linux operating systems (distributions) like Ubuntu, Gentoo, Debian and Slackware to name a few. 

Similarly, FreeBSD (Berkerley Software Distribution) contained original UNIX code, and it is (together with other BSDs) a robust OS for server-side capabilities and eve day-to-day needs. It is mostly from FreeBSD and NetBSD that the first Mac OS was developed, with the Mach kernel. 

And even the Windows itself has been influenced by Xenix, Microsoft's own UNIX-like solution that was later discontinued. 

So, if you're using an iPhone or an Android to read this post, please take a moment to remember Dennis Ritchie, the man without whom none of this would be possible.