Characteristics Of A Theorem
The initial premises on which all theorems are based are called axioms. An axiom, or postulate, is a basic fact that is not subject to formal proof. For example, the statement that there is an infinite number of even integers is a simple axiom. Another is that two points can be joined to form a line. When developing a theorem, mathematicians choose axioms, which seem most reliable based on their experience. In this way, they can be certain that the theorems are proved as near to the truth as possible. However, absolute truth is not possible because axioms are not absolutely true.
To develop theorems, mathematicians also use definitions. Definitions state the meaning of lengthy concepts in a single word or phrase. In this way, when people talk about a figure made by the set of all points which are a certain distance from a central point, one can just use the word circle.
See alsoSymbolic logic.
Applications Of Pythagoras Theorem
Though it is necessary to learn the basic concepts such as theorem statements and their mathematical representation, we would be more curious in understanding the applications of the Pythagoras theorem which we face in day-to-day life situations.
Here are some of the applications of the Pythagoras Theorem
- Engineering and Construction fields
Most architects use the technique of the Pythagorean theorem to find the value as well as when length or breadth are known it is very easy to calculate the diameter of a particular sector. It is mainly used in two dimensions in engineering fields.
- Face recognition in security cameras
We are more familiar with face recognition nowadays it reduces the turmoil in investigating the crimes in the security areas. It undergoes the concept of the Pythagorean theorem that is, the distance between the security camera and the place where the person is noted is well projected through the lens using the concept.
- Woodworking and interior designing
As the main concept indicates if the cardboards being square can be made into a triangle easily by cutting diagonally then very easily the Pythagoras concept can be applied. Most woodworks are made on the strategy which makes it easier for designers to proceed.
It’s a very amazing fact but people traveling in the sea use this technique to find the shortest distance and route to proceed to their concerned places.
Brief Biography Of Pythagoras
The Greek mathematician Pythagoras has high name recognition, not only in the history of mathematics. Everyone who has studied geometry can recall, well after the high school years, some aspect of the Pythagorean Theorem. However, the story of Pythagoras and his famous theorem is not well known. Some story plot points are: the famous theorem goes by several names grounded in the behavior of the day , including the Pythagorean Theorem, Pythagoras Theorem and notably Euclid I 47. The Pythagorean Theorem is arguably the most famous statement in mathematics, and the fourth most beautiful equation.1, 2 There are well over 371 Pythagorean Theorem proofs originally collected by an eccentric mathematics teacher, who put them in a 1927 book, which includes those by a 12-year-old Einstein, Leonardo da Vinci and President of the United States James A. Garfield.3, 4, 5
Is seems that Pythagoras was the first person to define the consonant acoustic relationships between strings of proportional lengths. Specifically, strings of equal tension of proportional lengths create tones of proportional frequencies when plucked. For example, a string that is 2 feet long will vibrate x times per second , while a string that is 1 foot long will vibrate twice as fast: 2x. Furthermore, those two frequencies create a perfect octave.9
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History Of Pythagoras Theorem
Pythagoras theorem was introduced by the Greek Mathematician Pythagoras of Samos. He was an ancient Ionian Greek philosopher. He started a group of mathematicians who works religiously on numbers and lived like monks. Finally, the Greek Mathematician stated the theorem hence it is called by his name as the “Pythagoras theorem.” Though it was introduced many centuries ago its application in the current era is obligatory to deal with pragmatic situations.
Although Pythagoras introduced and popularised the theorem, there is sufficient evidence proving its existence in other civilizations, 1000 years before Pythagoras was born. The oldest known evidence dates back to between 20th to 16th Century B.C in the Old Babylonian Period.
Gdelian Arguments Against Mechanism
There have been repeated attempts to apply Gödels theoremsto demonstrate that the powers of the human mind outrun any mechanismor formal system. Such a Gödelian argument against mechanism wasconsidered, if only in order to refute it, already by Turing in thelate 1940s . An unqualified anti-mechanistconclusion was drawn from the incompleteness theorems in a widely readpopular exposition, Gödels Theorem, by Nagel andNewman . Shortly afterwards, J.R. Lucas famouslyproclaimed that Gödels incompleteness theorem
proves that Mechanism is false, that is, that minds cannot beexplained as machines.
He stated that
given any machine which is consistent and capable of doing simplearithmetic, there is a formula it is incapable of producing as beingtrue but which we can see to be true.
More recently, very similar claims have been put forward by RogerPenrose . John Searle has joined the discussion andpartly defended Penrose against his critics. Crispin Wright has endorsed related ideas from an intuitionistic point of view. They all insist thatGödels theorems imply that the human mind infinitelysurpasses the power of any finite machine or formal system.
Fundamental Theorem Of Arithmetic
The function prime_factorization is defined for any factorial semiring. It returns the factorization as a multiset that fulfils the following properties:
lemmain_prime_factors_iff:"p set_mset x 0 p dvd x prime p"lemmaprod_mset_prime_factorization:assumes"x 0"shows"prod_mset = normalize x"lemmaprime_factorization_unique:assumes"x 0""y 0"shows"prime_factorization x = prime_factorization y normalize x = normalize y"
The normalize function is required because associated elements have the same factorization for natural numbers, it is the identity.
Another Amazingly Simple Proof
Here is one of the oldest proofs that the square on the long side has the same area as the other squares.
Watch the animation, and pay attention when the triangles start sliding around.
You may want to watch the animation a few times to understand what is happening.
The purple triangle is the important one.
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Pythagorean Theorem In The General Theory Of Relativity
From the latest results of the theory of relativity, it is probable that our three-dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry. According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. I wished to show that space time is not necessarily something to which one can ascribe to a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept empty space loses its meaning. The fact that such a metric is called Euclidean is connected with the following. The postulation of such a metric in a three-dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean Geometry. The defining equation of the metric is then nothing but the Pythagorean Theorem applied to the differentials of the co-ordinates.25
Gdel And Benacerraf On Mechanism And Platonism
Interestingly, Gödel himself also presented an anti-mechanistargument although it was more cautious and only published posthumously. That is, in his1951 Gibbs lecture, Gödel drew the following disjunctiveconclusion from the incompleteness theorems:
either the human mind infinitely surpasses the power of any finite machine, orelse there exist absolutely unsolvable diophantine problems.
Gödel speaks about this statement as a mathematicallyestablished fact .According to Gödel, the second alternative
seems to disprove the view that mathematics is only our own creation that mathematical objects and facts existobjectively and independently of our mental acts and decisions.
Gödel was nonetheless inclined to deny the possibility ofabsolutely unsolvable problems, and although he did believe inmathematical Platonism, his reasons for this conviction weredifferent, and he did not maintain that the incompleteness theoremsalone establish Platonism. Thus Gödel believed in the firstdisjunct, that the human mind infinitely surpasses the power of anyfinite machine. Still, this conclusion of Gödel follows, asGödel himself clearly explains, only if one denies, as doesGödel, the possibility of humanly unsolvable problems. It is nota necessary consequence of incompleteness theorems.
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The History And Early Reception Of The Incompleteness Theorems
Gödels results were certainly surprising, but some sort ofincompleteness phenomenon was not totally unexpected. The possibilityof incompleteness in the context of set theory was discussed byBernays and Tarski already in 1928, and von Neumann, in contrast tothe dominant spirit in Hilberts program, had considered itpossible that logic and mathematics were not decidable. Gödelhimself had mentioned the possibility of an undecidable problemconcerning real numbers in his thesis in 1929 .Hilbert , on the other hand, had assumed that Peano Arithmeticand other standard theories were complete. Apparently Gödel wasalso impressed by Brouwer, who in his lecture in Vienna in 1928 hadsuggested that mathematics is inexhaustible and cannot be completelyformalized .
However, the intellectual environment of Gödel was that of theVienna Circle with its radically anti-metaphysical attitude. Inparticular, even the notion of truth was considered as suspicious oreven nonsensical at the time, at least by some logical positivists. Therefore, Gödel worked hard toeliminate any appeal to the notion of truth and attempted to dowithout it. He therefore introduced the notion of \-consistency,which can be defined rigorously and purely syntactically. This led tothe incompleteness theorems in the form that they are now known.
Genetic Testing Done In Parallel With Other Risk Factor Identification
Bayesian analysis can be done using phenotypic information associated with a genetic condition, and when combined with genetic testing this analysis becomes much more complicated. Cystic Fibrosis, for example, can be identified in a fetus through an ultrasound looking for an echogenic bowel, meaning one that appears brighter than normal on a scan2. This is not a foolproof test, as an echogenic bowel can be present in a perfectly healthy fetus. Parental genetic testing is very influential in this case, where a phenotypic facet can be overly influential in probability calculation. In the case of a fetus with an echogenic bowel, with a mother who has been tested and is known to be a CF carrier, the posterior probability that the fetus actually has the disease is very high . However, once the father has tested negative for CF, the posterior probability drops significantly .
Risk factor calculation is a powerful tool in genetic counseling and reproductive planning, but it cannot be treated as the only important factor to consider. As above, incomplete testing can yield falsely high probability of carrier status, and testing can be financially inaccessible or unfeasible when a parent is not present.
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The Factor And Remainder Theorems
lemmalong_div_theorem:assumes"g carrier P"and"f carrier P"and"g \< ^bsub> P\< ^esub> "shows"q r . \< ^bsub> R\< ^esub> k \< ^bsub> P\< ^esub> f = g \< ^bsub> P\< ^esub> q \< ^bsub> P\< ^esub> r "
Researchers Create Ai That Can Invent Brand New Math Theorems
Why do we need to study math? Once upon a time, the answer was generally something like well you wont have a calculator in your pocket all the time! but smartphones solved that one. Nowadays, if you ask a professional mathematician to justify their existence, theyll probably say something about needing human intuition to solve the greatest mysteries of the universe. A calculator may be able to compute the cube root of 14.7 in under a second, theyll say, but it takes something special to prove the four color theorem.
Well, sucks to be them, because DeepMind researchers have now created an artificial intelligence capable of proving and even suggesting abstract mathematical theorems. DeepMind is the team of computer scientists that made history in 2016 with AlphaGo, the first computer program to successfully defeat a world champion in the game of Go.
While mathematicians have used machine learning to assist in the analysis of complex data sets, this is the first time we have used computers to help us formulate conjectures or suggest possible lines of attack for unproven ideas in mathematics, said mathematician Geordie Williamson, co-author of a paper on the AI mathmo that was published today in the journal Nature.
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What Is The Theorem That Has The Most Proofs
Classical theorems like the irrationality of $\sqrt$ or the infinitude of the primes have lots of proofs. But one theorem in particular, which I studied years ago in an introductory course of Number Theory, called the Quadratic Reciprocity Law, has tons of proofs. Gauss himself provided some of them.
And the question is: Is here a theorem that has more proofs than the Quadratic Reciprocity Law? If you know other theorems that have lots of different proofs, please list below.
Maybe we can divide the theorems by area.
In Euclidean Geometry, the Pythagorean Theorem won the game.
In Classical Number Theory, the Quadratic Reciprocity Law won the game .
Now, what about Calculus? Algebraic number theory? Real Analysis? Commutative Algebra? Topology? Differential Geometry? Differential Equations? Probability? And so on.
Well, there was a book with 367 proofs of the Pythagorean Theorem published. I’m sure there’s more.
- 5$\begingroup$Maybe there will be a time when the number of proofs for the Riemann Hypothesis exceeeds this number. :-)$\endgroup$
Proof 2: Induction on FacesProof 3: Induction on VerticesProof 4: Induction on EdgesProof 5: Divide and ConquerProof 6: Electrical ChargeProof 7: Dual Electrical ChargeProof 8: Sum of AnglesProof 9: Spherical AnglesProof 16: Binary Space PartitionProof 17: Valuations
There are also a lot of ways to prove that there are infinitely many primes.
Methods of proof :
Using Pedigree To Calculate Probabilities
|/ = 1/17||/ = 16/17|
Example of a Bayesian analysis table for a female individual’s risk for a disease based on the knowledge that the disease is present in her siblings but not in her parents or any of her four children. Based solely on the status of the subjects siblings and parents, she is equally likely to be a carrier as to be a non-carrier . However, the probability that the subjects four sons would all be unaffected is 1/16 if she is a carrier, about 1 if she is a non-carrier . The Joint Probability reconciles these two predictions by multiplying them together. The last line is calculated by dividing the Joint Probability for each hypothesis by the sum of both joint probabilities.
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Changing The Qed Symbol
The symbol printed at the end of a proof is called the âQED symbolâ. To quote the meaning of QED from Wikipedia:
QED is an initialism of the Latin phrase quod erat demonstrandum, meaning âthus it has been demonstratedâ
It is straightforward to use a symbol, or wording, of your choice to represent the QED symbol. The command
can be used to replace the default white square for a black square printed by $\blacksquare$, the parameter inside the braces. Or, you can write the word QED explicitly:
Here is an example to demonstrate both options:
\documentclass\usepackage\usepackage\usepackage\newtheorem\newtheorem\begin\section\beginGiven two line segments whose lengths are \ and \ respectively there is a real number \ such that \.\end\renewcommand\qedsymbol\beginTo prove it by contradiction try and assume that the statement is false,proceed from there and at some point you will arrive to a contradiction.\end\renewcommand\qedsymbol\beginTo prove it by contradiction try and assume that the statement is false,proceed from there and at some point you will arrive to a contradiction.\end\end
This example produces the following output:
The Solution Of The General Quartic Equation
lemmaquartic:"^3 - b * y^2 + * y - a^2 * d + 4 * b * d - c^2 = 0 R^2 = a^2 / 4 - b + y s^2 = y^2 - 4 * d / )) / )) x^4 + a * x^3 + b * x^2 + c * x + d = 0 x = -a / 4 + R / 2 + D / 2 x = -a / 4 + R / 2 - D / 2 x = -a / 4 - R / 2 + E / 2 x = -a / 4 - R / 2 - E / 2"
The Central Limit Theorem
theoremcentral_limit_theorem:fixesX::"nat 'a real"and::"real measure"and::realandS::"nat 'a real"assumesX_indep:"indep_vars X UNIV"andX_integrable:"n. integrable M "andX_mean_0:"n. expectation = 0"and_pos:"> 0"andX_square_integrable:"n. integrable M \< ^sup> 2)"andX_variance:"n. variance = \< ^sup> 2"andX_distrib:"n. distr M borel = "defines"S n x. i< n. X i x"shows"weak_conv_m )) "
How To Prove The Mean Value Theorem
Let’s use $$f$$ to represent the function and $$\ell$$ to represent the secant line connecting the endpoints.
$$ \begin \frac} & = f’\\ \ell’} & = f’ \end $$
Now, if we subtract one side from the other we get
$$ \begin f’ – \ell’} = 0. \end $$
If we define a new function as $$h = f – \ell$$ then
$$ h’ = f’ – \ell’ $$
So, equation can be written
$$ \beginh’ = 0.\end $$
This function we’ve defined, $$h = f – \ell$$, meets all three criteria for Rolle’s Theorem:
- Continuity: We know $$f$$ is continuous and $$\ell$$ is continuous since it is just a linear function. Therefore, $$h$$ is continuous since it is the difference between two continuous functions.
- Differentiability: Likewise, both $$f$$ and $$\ell$$ are differentiable, so $$h$$) is too.
- Finally, $$h = f – \ell = 0$$ and $$h = f-\ell = 0$$ so $$h = h$$.
Now we use this line of reasoning in reverse to prove the MVT:
- Start with a function,$$f$$, that is continuous and differentiable over the interval.
- Define two new functions: $$\ell$$ is the equation of the secant line and $$h = f – \ell$$.
- Note that $$h$$ meets the conditions for Rolle’s Theorem.
- That means $$h’ = 0$$ for some $$c\in$$.
- We know $$h’ = f’ – \ell’$$.
- This means
$$ \begin f’ – \ell’ & = 0\\ f’ & = \ell’\\ f’ & = \frac \end $$
And we’re done.
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