Category Mathematics

Narendra Krishna Karmarkar a famous Indian mathematician is the one behind Karmarkar’s algorithm. An algorithm is a step-by- step solution to a problem. You can call it a recipe book for mathematics.

Karmarkar’s algorithm helped to solve problems in linear programming in a novel way. He found this method and published the results while working for Bell Laboratories in New Jersey.

Karmarkar did his B.Tech in Electrical Engineering from IIT Bombay and M.S. from the California Institute of Technology. He then took Ph.D. in computer science the University of California, Berkeley.

After that, Karmarkar joined the Tata Institute of Fundamental Research, Mumbai. He continues to work on new architecture for supercomputing. The digital library, IEEE Xplore, has published some of his works.

He received the prestigious Paris Kanellakis Award from the Association for Computing Machinery in 2000. The Prime Minister of India also presented him the Srinivasa Ramanujan Birth Centenary Award for 1999.

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Two things: You learn that you don’t know, and you learn that deep inside, you do.

When you find, or compose, or are moonstruck by a good proof, there’s a sense of inevitability, of innate truth. You understand that the thing is true, and you understand why, and you see that it can’t be any other way. It’s like falling in love. How do you know that you’ve fallen in love? You just do.

Such proofs may be incomplete, or even downright wrong. It doesn’t matter. They have a true core, and you know it, you see it, and from there it’s only a matter of filling the gaps, cleaning things up, eliminating redundancy, finding shortcuts, rearranging arguments, organizing lemmas, generalizing, generalizing more, realizing that you’ve overgeneralized and backtracking, writing it all neatly in a paper, showing it around, and having someone show you that your brilliant proof is simply wrong.

And this is where you either realize that you’ve completely fooled yourself because you so wanted to be in love, which happens more often when you’re young and inexperienced, or you realize that it’s merely technically wrong and the core is still there, pulsing with beauty. You fix it, and everything is good with the world again.

Experience, discipline, intuition, trust and the passage of time are the things that make the latter more likely than the former. When do you know for sure? You never know for sure. I have papers I wrote in 1995 that I’m still afraid to look at because I don’t know what I’ll find there, and there’s a girl I thought I loved in 7th grade and I don’t know if that was really love or just teenage folly. You never know.

Fortunately, with mathematical proofs, you can have people peer into your soul and tell you if it’s real or not, something that’s harder to arrange with crushes. That’s the only way, of course. The best mathematicians need that process in order to know for sure. Someone mentioned Andrew Wiles; his was one of the most famous instances of public failure, but it’s far from unique. I don’t think any mathematician never had a colleague demolish their wonderful creation.

Breaking proofs into steps (called lemmas) can help immensely, because the truth of the lemmas can be verified independently. If you’re disciplined, you work hard to disprove your lemmas, to find counterexamples, to encourage others to find counterexamples, to critique your own lemmas as though they belonged to someone else. This is the very old and very useful idea of modularization: split up your Scala code, or your engineering project, or your proof, or what have you, into meaningful pieces and wrestle with each one independently. This way, even if your proof is broken, it’s perhaps just one lemma that’s broken, and if the lemma is actually true and it’s just your proof that’s wrong, you can still salvage everything by re-proving the lemma.

Or not. Maybe the lemma is harder than your theorem. Maybe it’s unprovable. Maybe it’s wrong and you’re not seeing it. Harsh mistress she is, math, and this is a long battle. It may takes weeks, or months, or years, and in the end it may not feel at all like having created a masterpiece; it may feel more like a house of sand and fog, with rooms and walls that you only vaguely believe are standing firm. So you send it for publication and await the responses.

Peer reviewers sometimes write: this step is wrong, but I don’t think it’s a big deal, you can fix it. They themselves may not even know how to fix it, but they have the experience and the intuition to know that it’s fine, and fixing it is just work. They ask you politely to do the work, and they may even accept the paper for publication pending the clean up of such details.

There are, sometimes, errors in published papers. It happens. We’re all human. Proofs that are central have been redone so many times that they are more infallible than anything of value, and we can be as certain of them as we are certain of anything. Proofs that are marginal and minor are more likely to be occasionally wrong.

So when do you know for sure? When reviewers reviewed, and time passes, and people redo your work and build on it and expand it, and over time it becomes absolutely clear that the underlying truth is unassailable. Then you know. It doesn’t happen overnight, but eventually you know.

And if you’re good, it just reaffirms what you knew, deep inside, from the very beginning.

Mathematical proofs can be formalized, using various logical frameworks (syntactic languages, axiom systems, inference rules). In that they are different from various other human endeavors.

It’s important to realize, however, that actual working mathematicians almost never write down formal versions of their proofs. Open any paper in any math journal and you’ll invariably find prose, a story told in some human language (usually English, sometimes French or German). There are certainly lots of math symbols and nomenclature, but the arguments are still communicated in English.

In recent decades, tremendous progress has been made on practical formalizations of real proofs. With systems like Coq, HOL, Flyspeck and others, it has become possible to write down a completely formal list of steps for proving a theorem, and have a computer verify those steps and issue a formal certificate that the proof is, indeed, correct.

The motivation for setting up those systems is, at least in part, precisely the desire to remove the human, personal aspects I described and make it unambiguously clear if a proof is correct or not.

One of the key proponents of those systems is Thomas Hales, who developed an immensely complex proof of the Kepler Conjecture and was driven by a strong desire to know whether it’s correct or not. I’m fairly certain he wanted, first and foremost, to know the answer to that question himself. Hales couldn’t tell, by himself, if his own proof is correct.

It is possible that in the coming decades the process will become entirely mechanized, although it won’t happen overnight. As of 2016, the vast majority of proofs are still developed, communicated and verified in a very social, human way, as they were for hundreds of years, with all the hope, faith, imprecision, failure and joy that human endeavors entail.

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Are you aware of numbers that are called as taxicab numbers? The nth taxicab number is the smallest number representable in n different ways as a sum of two positive integer cubes. These numbers are also called as the Hardy-Ramanujan number. The name taxicab numbers, in fact is derived from a story told about Indian mathematician Srinivasa Ramanujan by English mathematician GH Hardy. Here is the story, as told by Hardy I remember once going to see him (Ramanujan) when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number, it is the smallest number expressible as the sum of two [positive] cubes in two different ways.”

1729, naturally, is the most popular taxicab number. 1729 can be expressed as the sum of both 12^3 and 1^3 (1728+1) and as the sum of 10 and 9 (1000+729).

While the story involving Ramanujan made these numbers famous and also gave it its name. these numbers were actually known earlier. The first mention of this concept can be traced back to the 17th Century.

2 (1^3 + 1^3) is the first taxicab number and 1729 is the second. The numbers after 1729 have been found out using computers and six taxicab numbers are known so far.

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Since one of the numbers on the dial of a telephone is zero, so the product of all the numbers on it is 0.

The layout of the digit keys is different from that commonly appearing on calculators and numeric keypads. This layout was chosen after extensive human factors testing at Bell Labs. At the time (late 1950s), mechanical calculators were not widespread, and few people had experience with them. Indeed, calculators were only just starting to settle on a common layout; a 1955 paper states “Of the several calculating devices we have been able to look at… Two other calculators have keysets resembling [the layout that would become the most common layout]…. Most other calculators have their keys reading upward in vertical rows of ten,” while a 1960 paper, just five years later, refers to today’s common calculator layout as “the arrangement frequently found in ten-key adding machines”. In any case, Bell Labs testing found that the telephone layout with 1, 2, and 3 in the top row, was slightly faster than the calculator layout with them in the bottom row.

The key labeled ? was officially named the “star” key. The original design used a symbol with six points, but an asterisk (*) with five points commonly appears in printing.[citation needed] The key labeled # is officially called the “number sign” key, but other names such as “pound”, “hash”, “hex”, “octothorpe”, “gate”, “lattice”, and “square”, are common, depending on national or personal preference. The Greek symbols alpha and omega had been planned originally.

These can be used for special functions. For example, in the UK, users can order a 7:30 am alarm call from a BT telephone exchange by dialing: *55*0730#.

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Do you have a mathematics teacher who writes a big fat zero that occupies the entire blackboard whenever an answer boils down to it? None of us wish to see it on our answer sheets (unless, of course, it is for 100). but zero fascinates and frustrates maths lovers and haters in equal measures. Even though civilisations have always understood the concept of nothing or having nothing. India is generally credited with developing the numerical zero. It is hard for us to imagine a world without zero, and it is no wonder therefore that giving zero a symbol is seen as one of the greatest innovations in human history. Without this zero, modem mathematics, physics and technology would all probably zero down to nothing! The philosophy of emptiness or shunya (shunya is zero in Sanskrit) is believed to have been an important cultural factor for the development of zero in India. The concept is said to have been fully developed by the 5th Century. and maybe even earlier.

The Bakhshali manuscript, discovered in a field in 1881, is currently seen as the earliest recorded use of a symbol for zero. Dating techniques place this manuscript to be written anywhere between the 3rd and 9th Century.

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               Infinity is a number, or value, which is so huge that it cannot be measured. For instance, the distance to the end of the Universe is called infinity, because if there is an end, it is so far away that it could never be measured.

               Sometimes infinity can be the result of a mathematical calculation. For example, the formula for calculating the distance around the outside of a circle is pi times the diameter. Pi is a Greek letter, and it represents a value of approximately 3.14159. Pi can never be fully calculated, because you would finish up with a string of numbers extending to infinity.

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                    Geometry is a part of mathematics that deals with the shape, position and size of things of geometric forms such as squares, triangles, cubes and cones. Its name comes from the Greek words meaning ‘earth measuring’, because it was probably originally invented as a means of surveying and measuring land.

                   The ancient Egyptians also used geometry when constructing buildings and tombs. Nowadays, geometry is important to engineers and architects. It is also essential in navigation because geometry is used to follow charts and maps.

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               Our numbers are called Arabic numerals, although they probably appeared first in India around AD600. By the 800s the Arab numbering system was use throughout Europe because it was much easier to use than the old Roman system. At first the numbers varied in the way they were written, but with the invention of printing the numbers became standardized. The basic Arab numbers are 0 to 9, and can be used to write any combination of numbers.

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                 In some ways, Roman numbering worked like the modern Arabic numeral system where, starting from the left, there are thousands, hundreds, tens and individual units. However, Roman numerals are quite different. One thousand is written as M, five hundred as D, one hundred as C, fifty as L, ten as X and five as V.

               The decimal system describes a numbering system for calculation based on multiples of ten. Multiplying or dividing a number by ten is very easy because only the decimal point needs to be moved. In the decimal system, each number has a value ten times that of the next number to the right. For instance, 5,283 means five thousands, two hundreds, eight tens and three ones. The decimal point simply separates the main number from numbers less than one. The number ten has always been important in mathematics — you can easily count it on your fingers.

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               The metric system is a group of various units of measurement. Its name comes originally from the metre. It is a decimal system, which means that each unit is ten times bigger, or smaller, than the next unit. Previously, measurements were difficult to calculate; in measurements of length, for example, there were 12 inches in one foot and three feet in one yard, while weight was calculated using ounces and pounds (16 ounces in one pound).

               The metric system was devised in the 1790s in France, and is now used in most countries in the world for all scientific and technical measurements.

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               Numbers are used to describe the amount of things. We can express numbers in words, by hand gestures or in writing, using symbols or numerals. When we talk about a number we use words (‘five’) rather than the numeral (5), but when we write we use both words and numerals.

               Numbers can describe how many objects there are or their position among lots of objects, such as 1st or 5th for example. Other types of numbers describe how many units of something there are, for example how many kilograms (weight) or metres (length).Numbers are just a convenient way to describe ideas.

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               All numbers are made from ten figures. These are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. All decimal numbers are also made from the above ten figures.

               Now the question arises, why are there only ten numbers with single figure? Although it cannot be precisely explained, yet some had concluded that this is because ancient people began counting by using their ten fingers. This was easier for them to learn how to handle numbers, as they could use their ten fingers to count.

               It is an established fact that right from ancient times, people tried various systems to write numbers. They showed numbers by marking separate indications. For a number like 10 for instance, they showed 10 lines or 10 drawings of birds or animals. This system was used by Egyptians which are still to be found in ancient monuments or structures. In fact, all civilizations have had their own way of writing and using numbers. Romans invented special signs or alphabets for counting. Such signs are called numerals. For centuries, people used Roman numerals. But they are rather clumsy. They did not have zero.

               But counting became easier after invention of zero. Zero was invented in 6th century in India and was brought to Europe by Arab travellers at a later stage. Having a figure for zero, any number can be made in a simpler way. It may continue in decimal system. Fractions like or  may be written as decimal fractions like 0.50, 0.25 or 0.75. In fact there are many ways of expressing numbers. In everyday terms numbers are used with units like kg; litre or metre and so on.

               It is most interesting to note that computers use a system that use only two digits — 1 and 0. This is called binary system. They play a key role in a modern digital computer. A computer changes numbers and words into codes made up of 0s and 1s. It makes calculation with these codes.

               The earliest known written numbers were those used by Babylonians about 5000 years ago. The symbols used today in English for the ten numbers (0, 1, 2,…) are called Arabic or Hindu-Arabic numerals because they came from India through the Arab countries and reached Europe in the 12th century.