Tag: Math

Math Explains Why Your Bus Route Seems So Unreliable

Ever heard of the Inspection Paradox? In some contexts it just explains why you have to wait a long time for the bus. But if we started working with it, we could make the world a more extraordinary place – at least, on paper.

When you first hear about the Inspection Paradox, you might confuse it for something confirmed by MISPWOSO, The Maximegalon Institute for Slowly and Painfully Working Out the Surprisingly Obvious – from Douglas Adams’ books.

The math behind it shows that, for any periodically occurring event, the average wait time for the next event will always be longer than the average wait time for the next event.

Let’s make it simpler. Buses, trains, or shuttles will come a certain number of times per hour, say once every fifteen minutes. That means that your wait time for the next bus should be, on average, seven and a half minutes.

Anyone who has been on public transport will not be surprised to note that their average wait time is longer than the average wait time. They’ll also probably guess at the answer as to why.

Whenever there’s a long wait for a bus, it arrives packed to the roof with disgruntled people.

And it is about to be packed further, as anyone who has had a long wait time at a stop will be alone at first, then joined by more and more people until there is a crowd.

The longer the wait between buses, the more likely people are going to show up and be annoyed by the longer wait.

In short, you are more likely to experience a longer-than-average wait instead of a shorter-than-average wait, because you are more likely to show up during a long stretch of wait time than during a short stretch of wait time.

The bus riders are considered “inspectors” because they show up at random intervals and check out the wait time for the next bus.

They don’t sit all day long and survey the time between buses. Inspectors are always more likely than average to experience longer-than-average waits, because delays are likely to take up more than the average amount of time.

But the paradox doesn’t always work against you. Your light bulb is probably going to last longer than the average light bulb. Your watch will probably last longer than the average watch.

So will your laptop. If you were to measure various things in your life, you would probably think that your life was startlingly atypical, with the various appliances, delays, waits, and stop-gap solutions lasting longer than they should.

Instead, your life is typically paradoxical. You are always more likely to stumble into a longer situation than you should, on average, stumble into.

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Pass it on: Popular Science

Honey Bees Can Understand Nothing

Zero, zilch, nothing, is a pretty hard concept to understand. Children generally can’t grasp it until kindergarten. And it’s a concept that may not be innate but rather learned through culture and education.

Throughout human history, civilizations have had varying representations for it. Yet our closest animal relative, the chimpanzee, can understand it.

And now researchers in Australia writing in the journal Science say the humble honey bee can be taught to understand that zero is less than one.

The result is kind of astounding, considering how tiny bee brains are. Humans have around 100 billion neurons. The bee brain? Fewer than 1 million.

The findings suggest that the ability to fathom zero may be more widespread than previously thought in the animal kingdom — something that evolved long ago and in more branches of life.

It’s also possible that in deconstructing how the bees compute numbers, we could make better, more efficient computers one day.

Our computers are electricity-guzzling machines. The bee, however, “is doing fairly high-level cognitive tasks with a tiny drop of nectar,” says Adrian Dyer, a Royal Melbourne Institute of Technology researcher and co-author on the study.

Their brains are probably processing information in a very clever [i.e., efficient] way.”

But before we can deconstruct the bee brain, we need to know that it can do the complex math in the first place.

How to teach a bee the concept of zero

Bees are fantastic learners. They spend hours foraging for nectar in among flowers, can remember where the juiciest flowers are, and even have a form of communication to inform their hive mates of where food is to be found.

Researchers train bees like they train many animals: with food. “You have a drop of sucrose associated with a color or a shape, and they will learn to reliably go back to” that color or shape, Dyer explains.

With this simple process, you can start teaching bees rules. In this case, the researchers wanted to teach 10 bees the basic rules of arithmetic.

So they put out a series of sheets of paper that had differing numbers of objects printed on them. Using sugar as a reward, the researchers taught the bees to always fly to the sheet that had the fewest objects printed on it.

Once the bees learned this rule, they could reliably figure out that two shapes are less than four shapes, that one shape is smaller than three. And they’d keep doing this even when a sugary reward was not waiting for them.

And then came the challenge: What happens when a sheet with no objects at all was presented to the bees? Would they understand that a blank sheet — which represented the concept of zero in this experiment — was less than three, less than one?

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The Formula For The Perfect Free-Throw

Improving your free-throw percentage is a simple matter of mathematics, according to researchers Drs. Chau Tran and Larry Silverberg of North Carolina State University.

Using three-dimensional computer simulations of hundreds of thousands of basketball trajectories, the two engineers determined the ideal characteristics of a free-throw shot.

They based their data on the assumption of a 6’6” player who would release the ball (assumed to be a men’s basketball) at a height of 7 feet.

The first variable Tran and Silverberg examined was spin. According to them, you should release the ball with about three hertz of backspin – or, so that the ball makes roughly three full backwards rotations before reaching the hoop.

This slows the ball upon contact with the backboard or rim, making it more likely that the shot will go in.

The ball should also be released at 52 degrees to the horizontal, making the peak of its arc only a few inches higher than the top of the backboard.

For aiming, they found the most successful methods put the ball towards the back of the rim, either two inches to the left or two inches to the right of the place where the rim meets the backboard.

How a mathematician sees a free throw.

Their simulation data showed that aiming straight for the center of the backboard decreases the success rate by almost three percent.

Tran and Silverberg also recommend free-throw shooters should release the ball as high above the ground as possible with a smooth, consistent release speed for best results.

Our recommendations might make even the worst free-throw shooters – you know who you are, Shaquille O’Neal and Ben Wallace – break 60 percent from the free-throw line,” Silverberg joked.

Their work is just another example of how mathematical questions can crop up in the most unexpected areas.

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Pass it on: Popular Science

Why Math Might Be Complete BS

Set up a free Brilliant account at http://www.brilliant.org/answerswithjoe/ And the first 295 to sign up for a premium account get 20% off every month!

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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Pass it on: New Scientist

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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Pass it on: New Scientist