Tag: mathematics

Why Math Might Be Complete BS

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Mathematics is the backbone of all sciences, no theories or hypotheses are proven unless there is math to back it up. But there are many who believe that math isn’t real. In today’s video, we’ll break down the arguments.

From the Mathematical Physicalists to the Platonists to the Mathematical Fictionalists, we look at all the theories behind whether numbers actually exist, and what they mean.

The Largest Prime Number Was Discovered By A FedEx Employee

A FedEx employee in Tennessee has discovered the largest known prime number.

Germantown, Tenn., resident John Pace found the number through his volunteer work with the Great Internet Mersenne Prime Search (GIMPS), a project that crowd sources computing power to search for a subset of prime numbers called Mersenne primes.

Like a normal prime number, these can only be divided by themselves and one. What sets them apart is that they can all be expressed as the number 2 raised to a given power minus one.

The newly discovered Mersenne prime, called M77232917, can be expressed as 2 to the 77,232,917 power minus one. It’s the 50th Mersenne prime to be discovered and it’s more than 23 million digits long.

Pace might be the only person in history who went into math for the money.

He told NPR, “There was a $100,000 prize attached to finding the first prime that had a 10 million digit result, and I was like, ‘Well you know, I’ve got as much chance as anybody else.’

He has been participating in the program for 14 years and this is his first discovery.

The previous longest-known prime number was discovered in January of 2016 at the University of Central Missouri. It contains 22 million digits and is also a Mersenne prime.

Large prime numbers are important for the future of computing and cyber security, and the search is already on for larger numbers.

The Electronic Frontier Foundation is offering a prize of $150,000 for finding the first prime number with one hundred million digits and $250,000 for finding the first prime with one billion digits.

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Largest Known Prime Number Discovered With Over 23 Million Digits

A collaborative computational effort has uncovered the longest known prime number.

At over 23 million digits long, the new number has been given the name M77232917 for short.

Prime numbers are divisible only by themselves and one, and the search for ever-larger primes has long occupied maths enthusiasts.

However, the search requires complicated computer software and collaboration as the numbers get increasingly hard to find.

M77232917 was discovered on a computer belonging to Jonathan Pace, an electrical engineer from Tennessee who has been searching for big primes for 14 years.

Mr Pace discovered the new number as part of the Great Internet Mersenne Prime Search (GIMPS), a project started in 1996 to hunt for these massive numbers.

Mersenne primes – named after the 17th century French monk Marin Mersenne who studied them – are calculated by multiplying together many twos and then subtracting one.

Six days of non-stop computing in which 77,232,917 twos were multiplied together resulted in the latest discovery.

The number is the 50th Mersenne prime to be discovered, and the 16th to be discovered by the GIMPS project.

It is nearly one million digits longer than the previous record holder, which was identified as part of the same project at the beginning of 2016.

Mersenne primes are a particular focus for prime aficionados because there is a relatively straightforward way to check whether a number is one or not.

Nevertheless, the new prime has to be verified using four different computer programs on four different computers.

The process also relies on thousands of volunteers sifting through millions of non-prime candidates before the lucky individual chances upon their target.

Professor Caldwell runs an authoritative website on the largest prime numbers, with a focus on the history of Mersenne primes.

He emphasised the pure excitement that searching for prime numbers brings, describing the latest discovery as “a museum piece as opposed to something that industry would use”.

Besides the thrill of discovery, Mr Pace will receive a $3,000 (£2,211) GIMPS research discovery award.

GIMPS uses the power of thousands of ordinary computers to search for elusive primes, and the team behind it state that anybody with a reasonably powerful PC can download the necessary software and become a “big prime hunter”.

The next Mersenne prime discovery could be smaller or larger than the existing record holder, but the big target for the GIMPS team is to find a 100 million digit prime number.

The person who discovers such a number will be awarded $150,000 by the Electronic Frontier Foundation for their efforts.

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The 500-Page Proof That Only One Mathematician Can Understand

Nearly four years after Shinichi Mochizuki unveiled an imposing set of papers that could revolutionize the theory of numbers, other mathematicians have yet to understand his work or agree on its validity.

Although they have made modest progress.

Some four dozen mathematicians converged last week for a rare opportunity to hear Mochizuki present his own work at a conference on his home turf, Kyoto University’s Research Institute for Mathematical Sciences (RIMS).

Mochizuki is “less isolated than he was before the process got started”, says Kiran Kedlaya, a number theorist at the University of California, San Diego.

Although at first Mochizuki’s papers, which stretch over more than 500 pages, seemed like an impenetrable jungle of formulae.

Experts have slowly discerned a strategy in the proof that the papers describe, and have been able to zero in on particular passages that seem crucial, he says.

Mochizuki’s theorem aims to prove the important abc conjecture, which dates back to 1985 and relates to prime numbers — whole numbers that cannot be evenly divided by any smaller number except by 1.

The conjecture comes in a number of different forms, but explains how the primes that divide two numbers, a and b, are related to those that divide their sum, c.

If Mochizuki’s proof is correct, it would have repercussions across the entire field, says Dimitrov.

When you work in number theory, you cannot ignore the abc conjecture,” he says.

This is why all number theorists eagerly wanted to know about Mochizuki’s approach.”

For example, Dimitrov showed in January how, assuming the correctness of Mochizuki’s proof, one might be able to derive many other important results, including a completely independent proof of the celebrated Fermat’s last theorem.

But the purported proof, which Mochizuki first posted on his webpage in August 2012, builds on more than a decade of previous work in which Mochizuki worked in virtual isolation and developed a novel and extremely abstract branch of mathematics.

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He’s Gambling Obsession Spurred Him To Invent Two Of The Most Important Theories In Math

Girolamo or Hieronimo Cardano‘s name was Hieronymus Cardanus in Latin and he is sometimes known by the English version of his name Jerome Cardan.

Girolamo Cardano was the illegitimate child of Fazio Cardano and Chiara Micheria. His father was a lawyer in Milan but his expertise in mathematics was such that he was consulted by Leonardo da Vinci on questions of geometry.

In addition to his law practice, Fazio lectured on geometry, both at the University of Pavia and, for a longer spell, at the Piatti foundation in Milan.

When he was in his fifties, Fazio met Chiara Micheria, who was a young widow in her thirties, struggling to raise three children.

Chiara became pregnant but, before she was due to give birth, the plague hit Milan and she was persuaded to leave the city for the relative safety of nearby Pavia to stay with wealthy friends of Fazio.

Thus Cardan was born in Pavia but his mother’s joy was short lived when she received news that her first three children had died of the plague in Milan.

Chiara lived apart from Fazio for many years but, later in life, they did marry.

Cardan at first became his father’s assistant but he was a sickly child and Fazio had to get help from two nephews when the work became too much for Cardan.

However, Cardan began to wish for greater things than an assistant to his father. Fazio had taught his son mathematics and Cardan began to think of an academic career.

After an argument, Fazio allowed Cardan to go university and he entered Pavia University, where his father had studied, to read medicine despite his father’s wish that he should study law.

When war broke out, the university was forced to close and Cardan moved to the University of Padua to complete his studies.

Shortly after this move, his father died but by this time Cardan was in the middle of a campaign to become rector of the university. He was a brilliant student but, outspoken and highly critical, Cardan was not well liked.

However, his campaign for rector was successful since he beat his rival by a single vote.

Cardan squandered the small bequest from his father and turned to gambling to boost his finances. Card games, dice and chess were the methods he used to make a living.

Cardan’s understanding of probability meant he had an advantage over his opponents and, in general, he won more than he lost. He had to keep dubious company for his gambling.

Once, when he thought he was being cheated at cards, Cardan, who always carried a knife, slashed the face of his opponent.

Gambling became an addiction that was to last many years and rob Cardan of valuable time, money and reputation.

Cardan was awarded his doctorate in medicine in 1525 and applied to join the College of Physicians in Milan, where his mother still lived.

The College did not wish to admit him for, despite the respect he had gained as an exceptional student, he had a reputation as a difficult man, whose unconventional, uncompromising opinions were aggressively put forward with little tact or thought for the consequences.

The discovery of Cardan’s illegitimate birth gave the College a reason to reject his application.

Cardan, on the advice of a friend, went to Sacco, a small village 15km from Padua. He set up a small, and not very successful, medical practice.

In late 1531 Cardan married Lucia, the daughter of a neighbour Aldobello Bandarini, a captain of the local militia.

ardan’s practice in Sacco did not provide enough income for him to support a wife so, in April 1532, he moved to Gallarate, near Milan.

He applied again to the College of Physicians in Milan but again was not allowed membership.

Unable to practise medicine, Cardan reverted, in 1533, to gambling to pay his way, but things went so badly that he was forced to pawn his wife’s jewellery and even some of his furniture.

Desperately seeking a change of fortune, the Cardans moved to Milan, but here they fared even worse and they had to ignominiously enter the poorhouse.

Cardan was fortunate to obtain Fazio’s former post of lecturer in mathematics at the Piatti Foundation in Milan which gave him plenty of free time and he used some of this to treat a few patients, despite not being a member of the College of Physicians.

Cardan achieved some near miraculous cures and his growing reputation as a doctor led to his being consulted by members of the College.

His grateful patients and their relatives became whole hearted supporters and in this way, Cardan was able to build up a base of influential backers.

Cardan was still furious at his continuing exclusion from the College and, in 1536, he rashly published a book attacking not only the College’s medical ability but their character.

This was not the way to gain entry to the College and not surprisingly Cardan’s application to join in 1537 was again rejected.

However, two years later, after much pressure from his admirers, the College modified the clause regarding legitimate birth and admitted Cardan.

In the same year, Cardan’s first two mathematical books were published, the second The Practice of Arithmetic and Simple Mensuration was a sign of greater things to come.

This was the beginning of Cardan’s prolific literary career writing on a diversity of topics medicine, philosophy, astronomy and theology in addition to mathematics.

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Pass it on: New Scientist

3,700-Year-Old Babylonian Tablet Rewrites The History Of Math

A 3,700-year-old clay tablet has proven that the Babylonians developed trigonometry 1,500 years before the Greeks and were using a sophisticated method of mathematics which could change how we calculate today.

The tablet, known as Plimpton 332, was discovered in the early 1900s in Southern Iraq by the American archaeologist and diplomat Edgar Banks, who was the inspiration for Indiana Jones.

The true meaning of the tablet has eluded experts until now but new research by the University of New South Wales, Australia, has shown it is the world’s oldest and most accurate trigonometric table, which was probably used by ancient architects to construct temples, palaces and canals.

However unlike today’s trigonometry, Babylonian mathematics used a base 60, or sexagesimal system, rather than the 10 which is used today.

Because 60 is far easier to divide by three, experts studying the tablet, found that the calculations are far more accurate.

Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles,” said Dr Daniel Mansfield of the School of Mathematics and Statistics in the UNSW Faculty of Science.

It is a fascinating mathematical work that demonstrates undoubted genius. The tablet not only contains the world’s oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.”

This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3000 years, but it has possible practical applications in surveying, computer graphics and education.

The Greek astronomer Hipparchus, who lived around 120BC, has long been regarded as the father of trigonometry, with his ‘table of chords’ on a circle considered the oldest trigonometric table.

A trigonometric table allows a user to determine two unknown ratios of a right-angled triangle using just one known ratio.

But the tablet is far older than Hipparchus, demonstrating that the Babylonians were already well advanced in complex mathematics far earlier.

The tablet, which is thought to have come from the ancient Sumerian city of Larsa, has been dated to between 1822 and 1762 BC. It is now in the Rare Book and Manuscript Library at Columbia University in New York.

Plimpton 322 predates Hipparchus by more than 1000 years,” says Dr Wildberger.

It opens up new possibilities not just for modern mathematics research, but also for mathematics education. With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own.”

A treasure-trove of Babylonian tablets exists, but only a fraction of them have been studied yet. The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us.”

The 15 rows on the tablet describe a sequence of 15 right-angle triangles, which are steadily decreasing in inclination.

The left-hand edge of the tablet is broken but the researchers believe t there were originally six columns and that the tablet was meant to be completed with 38 rows.

Plimpton 322 was a powerful tool that could have been used for surveying fields or making architectural calculations to build palaces, temples or step pyramids,” added Dr Mansfield.

The new study is published in Historia Mathematica, the official journal of the International Commission on the History of Mathematics.

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