Category Mathematics

In mathematics, how do you know when you have proven a theorem?

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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What is the story of taxicab numbers?

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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What is the product of all the numbers that appear in the dial pad of our mobile phones?

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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What is a zero in math?

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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Pythagoras was a Greek living in the sixth century BC. He was a mathematician and scientist who are now best remembered for Pythagoras’ Theorem, a formula for calculating the length of one side of a right-angled triangle if the other sides are known. However, this theorem was, in fact, already known hundreds of years earlier by Egyptian and Babylonian mathematicians.

Pythagoras was a Greek philosopher who was born in Samos in the sixth century B.C. he was a great mathematician who explained everything with the help of numbers. He gave the Pythagorean Theorem. The Pythagorean Theorem states that the sum of the squares of the lengths of legs of any right angled triangle is equal to the square of the length of its hypotenuse. The hypotenuse is known to be the longest side and is always equal opposite to the right angle.

The theorem can be written as an equation where lengths of the sides can be a, b and c. The Pythagorean equation is a2 + b2 = c2 where c is the length of the hypotenuse and a, b are lengths of the two sides of the triangle. The Pythagorean equation simplifies the relation of the sides of the right triangle to each other in such a way that if the length of any of the two sides of the right triangle is known, then the third side can be easily found.

To generalise this theorem, there is the law of cosines which helps in calculation of the length of any of the sides of the triangle when the other two lengths for the two sides are given along with the angle between them. When the angle between the other sides turns out to be a right angle, then the law of the cosines becomes the Pythagorean Theorem. The converse of this theorem is also true. It is that for any triangle with sides a, b and c, if a2 + b2 = c2, then the angle between the two sides a and b would turn out to be 900.

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An abacus is a frame of beads used in China and neighbouring countries for making calculations. A skilled abacus user can produce answers to some calculations almost as quickly as someone using an electronic calculator.

The word abacus is derived from the Latin word abax, which means a flat surface, board or tablet. As such, an abacus is a calculating table or tablet. The abacus is the oldest device in history to be used for arithmetic purposes, such as counting. It is typically an open wooden rectangular shape with wooden beads on vertical rods. Each bead can represent a different number. For simple arithmetic purposes, each bead can represent one number. So, as a person moves beads from one side to the other, they would count, ‘one, two, three’, etc.

An abacus can be used to calculate large numbers, as well. The columns of beads could represent different place values. For example, one column may represent numbers in the hundreds, while another column may represent numbers in the thousands.

One of the most popular kinds of abacuses is the Chinese abacus, also known as the suanpan. Rules on how to use the suanpan have dated all the way back to the 13th century.

On a Chinese abacus, the rod or column to the far right is in the ones place. The one to the left of that is in the tens place, then the hundreds, etc. So, the columns are different place values and the beads are used to represent different numbers within those place values. For addition, beads on the suanpan are moved up towards the beam in the middle. For subtraction, they are moved down towards the bottom or outer edge of the suanpan. The rules of use are a bit more intricate and complicated, but this is the general idea of how one is used.

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Geometry is the branch of mathematics that is concerned with points, lines, surfaces and solids, and their relation to each other. Shapes, both flat and three-dimensional, are an important part of geometry. When we describe something as geometric, we mean that it has a regular, often angular pattern of lines or shapes.

Geometry is a term used to refer to a branch in mathematics that deals with geometrical objects such as straight lines, points and circles and other shapes. However, circles are the most elementary of geometric objects. The term geometry was derived from a Greek word, ‘geo’ which means earth and metron, meaning measure. These words reflect its actual roots. However, Plato knew how to differentiate the process of mensuration as used in construction from the philosophical implication of Geometry. In essence, Geometry in Greek implies earth measurements. Geometry was first organized by Euclid a mathematician who was able to arrange more than 400 geometric suggestions. Being one of the early sciences, it is the substance of most developments and it was believed that it has been in use way before in Egypt. Evidence shows that geometry dates back to the days of Mesopotamia in 3000 BC and is attributed to numerous developments since its discovery.

Geometry is not just a math topic created to make your life harder. It is a topic that was developed to answer questions about shapes and space related to construction and surveying. It answers questions about all the different shapes we see, such as how much space an object or shape can hold. Geometry even has application in the field of astronomy, as it is used to calculate the position of stars and planets. Over time, different people contributed new and different things to grow geometry from its basic beginnings to the geometry we know, use and study today.

The first written record that we have of geometry comes from Egypt back in 2000 BC. Some of the earliest texts that have been discovered include the Egyptian Rhind papyrus, Moscow papyrus and some Babylonian clay tablets, such as the Plimpton 322. These early geometry works included formulas for calculating lengths, areas and volumes of various shapes, including those of a pyramid.

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Mathematical formulae are useful rules expressed using symbols or letters. In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula. The informal use of the term formula in science refers to the general construct of relationship between given quantities. The plural of formula can be spelled either as formulas (from the most common English plural noun form) or, under the influence of scientific Latin, formulae (from the original Latin).

In mathematics, a formula generally refers to an identity which equates one mathematical expression to another with the most important ones being mathematical theorems. Syntactically, a formula is an entity which is constructed using the symbols and formation rules of a given logical language. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion. However, having done this once in terms of some parameter (the radius for example), mathematicians have produced a formula to describe the volume of a sphere in terms of its radius: V = 4/3nr3

Having obtained this result, the volume of any sphere can be computed as long as its radius is known. Here, notice that the volume V and the radius rare expressed as single letters instead of words or phrases. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate formulas which are larger and more complex. Mathematical formulas are often algebraic, analytical or in closed form.

In modern chemistry, a chemical formula is a way of expressing information about the proportions of atoms that constitute a particular chemical compound, using a single line of chemical element symbols, numbers, and sometimes other symbols, such as parentheses, brackets, and plus (+) and minus (?) signs. For example, H2O is the chemical formula for water, specifying that each molecule consists of two hydrogen (H) atoms and one oxygen (O) atom. Similarly, O?
denotes an ozone molecule consisting of three oxygen atoms and a net negative charge.

In a general context, formulas are a manifestation of mathematical model to real world phenomena, and as such can be used to provide solution (or approximated solution) to real world problems, with some being more general than others. For example, the formula F = ma is an expression of Newton’s second law, and is applicable to a wide range of physical situations. Other formulas, such as the use of the equation of a sine curve to model the movement of the tides in a bay, may be created to solve a particular problem. In all cases, however, formulas form the basis for calculations.

Expressions are distinct from formulas in that they cannot contain an equal’s sign (=). Expressions can be liken to phrases the same way formulas can be liken to grammatical sentences.

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Decimal numbers use 10 digits, which are combined to make numbers of any size. The position of the digit determines what it means in any number. For example, the 2 in the number 200 is ten times the size of the 2 in the number 20. Each position of a number gives a value ten times higher than the position to its right. So 9867 means 7 units, plus 6 x 10, plus 8 x 10 x 10, plus 9 x 10 x 10 x 10. As decimal numbers are based on the number 10, we say that this is a base -10 number system.

We have learnt that the decimals are an extension of our number system. We also know that decimals can be considered as fractions whose denominators are 10, 100, 1000, etc. The numbers expressed in the decimal form are called decimal numbers or decimals.

For example: 5.1, 4.09, 13.83, etc.

A decimal has two parts:

(a) Whole number part

(b) Decimal part

These parts are separated by a dot (.) called the decimal point.

  • The digits lying to the left of the decimal point form the whole number part. The places begin with ones, then tens, then hundreds, then thousands and so on.
  • The decimal point together with the digits lying on the right of decimal point form the decimal part. The places begin with tenths, then hundredths, then thousandths and so on…


(i) In the decimal number 211.35; the whole number part is 211 and the decimal part is .35

(ii) In the decimal number 57.031; the whole number part is 57 and the decimal part is .031

(iii) In the decimal number 197.73; the whole number part is 197 and the decimal part is .73

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The binary system is another way of counting. Instead of being a base-10 system, it is a base-2 system, using only two digits: 0 and 1. Again, the position of a digit gives it a particular value. 1010101 means 1 unit, plus 0 x 2, plus 1 x 2 x 2, plus 0 x 2 x 2 x2, plus1 x 2 x 2 x 2 x 2, plus 0 x 2x 2 x 2 x 2x 2,plus 1 x 2 x 2 x 2 x 2 x 2 x 2. 1010101 is the same as 85 in decimal numbers.

When you learn math at school, you use a base-10 number system. That means your number system consists of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When you add one to nine, you move the 1 one spot to the left into the tens place and put a 0 in the ones place: 10. The binary system, on the other hand, is a base-2 number system. That means it only uses two numbers: 0 and 1. When you add one to one, you move the 1 one spot to the left into the twos place and put a 0 in the ones place: 10. So, in a base-10 system, 10 equal ten. In a base-2 system, 10 equal two.

In the base-10 system you’re familiar with, the place values start with ones and move to tens, hundreds, and thousands as you move to the left. That’s because the system is based upon powers of 10. Likewise, in a base-2 system, the place values start with ones and move to twos, fours, and eights as you move to the left. That’s because the base-2 system is based upon powers of two. Each binary digit is known as a bit.

Don’t worry if the binary system seems confusing right now. It’s fairly easy to pick up once you work with it a while. It just seems confusing at first because all numbers are made up of only 0s and 1s. The familiar base-10 system is as easy as 1-2-3, while the base-2 binary system is as easy as 1-10-11.

You may WONDER why computers use the binary system. Computers and other electronic systems work faster and more efficiently using the binary system, because the system’s use of only two numbers is easy to duplicate with an on/off system. Electricity is either on or off, so devices can use an on/off switch within electric circuits to process binary information easily. For example, off can equal 0 and on can equal 1.

Every letter, number, and symbol on a keyboard is represented by an eight-bit binary number. For example, the letter A is actually 01000001 as far as your computer is concerned! To help you develop a better understanding of the binary system and how it relates to the decimal system you’re familiar with, here’s how the decimal numbers 1-10 look in binary:

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