Where Transcendental Numbers Hide in Everyday Math

Asia's Tech News Daily

The transcendental number π is as familiar as it is ubiquitous, but how does Euler’s number e transcend the ordinary?

Where Transcendental Numbers Hide in Everyday Math

James Round for Quanta Magazine

In everyday language, the word “transcendental” connotes something that’s beyond the ordinary, something that is hidden and mysterious, with almost magical or mystical powers. In mathematics, on the other hand, the meaning of the term “transcendental” is more mundane. It simply describes the class of the infinitely many numbers that cannot be solutions of polynomial equations like ax3 + bx2 + cx + d = 0, where the coefficients a, b, c, etc., are all rational and the highest power of x can be any positive integer. As the great mathematician Leonhard Euler put it, “They transcend the power of algebraic methods.”

However, the everyday connotations of the word “transcendental” ring true for the two most famous transcendental numbers — the universal constants π and e. These two numbers are indeed mysterious and powerful and exhibit an almost magical quality. They play a central role in many branches of mathematics, popping out of nowhere in the solutions of problems when you least expect them. Of these two powerful constants, π is much more familiar to most of us. Every student knows its approximate value and has used it in calculations. But the other one, Euler’s number e or 2.71828…, is not as well known to most of us. But in fact, e was the first non-constructed number to be proved transcendental, by Charles Hermite in 1873. We have to specify “non-constructed” because in 1850, Joseph Liouville provided the first example of a provably transcendental number — but it was a number he constructed for that sole purpose, and it did not arise naturally in any branch of mathematics. This, of course, makes it quite different from e, which arises naturally almost everywhere in mathematics. Many of us know that e is the base of natural logarithms and arises in the theory of compound interest and exponential growth and decay, but we can do computations in these fields without explicitly encountering e. Today we will examine some ordinary-looking problems in which e shows up unexpectedly, giving us a fleeting glimpse of its universality.

Like π and other transcendental numbers, e has an infinite decimal representation — its digits go on forever without any discernible pattern. Nevertheless, the first 15 digits of e have a pattern that is pleasingly regular and easy to remember using the following digit grouping: 2.7 1828 1828 45 90 45. This regularity is, of course, a coincidence — the rest of the digits, as expected, are completely random. But e has several amazing properties that make it unique among all numbers. You will be introduced to some of these properties in this suite of puzzles, some of which are classical but, of course, with some additions of my own. The number e arises naturally in all of them, but they can be enjoyed even if you understand the reason for e’s appearance only vaguely. The exact details of this transcendental visitation can be left to those with the requisite mathematical training. Let’s see who can express this connection in the simplest possible way.

Puzzle 1: Partition

Let’s take any number, such as 10. Divide it into some number of identical pieces, such as two 5s, and multiply them together: 5 × 5 = 25. Now, we could have divided 10 into three, four, five or six identical pieces and done the same. Here’s what happens to our product when we do so:

2 pieces: 5 × 5 = 25
3 pieces: 3.33 × 3.33 × 3.33 = 37.04
4 pieces: 2.5 × 2.5 × 2.5 × 2.5 = 39.06
5 pieces: 2 × 2 × 2 × 2 × 2 = 32
6 pieces: 1.67 × 1.67 × 1.67 × 1.67 × 1.67 × 1.67 = 21.43

You can see that the product increases, reaches what seems to be a maximum and then starts decreasing. Try doing the same with some other numbers such as 20 and 30. You’ll notice that the same thing happens in every case. This has nothing to do with the numbers themselves but is caused by a unique property of the number e.

a. See if you can figure out when the product reaches a maximum for a given number and what this has to do with e. If you’re stumped, click on the hint below.

Hint:

The product reaches a maximum when the value of the pieces is closest to e.

b. For the number 10, the largest product (39.06) is about 5.5% larger than the next largest (37.04). Without calculating the actual difference, can you guess which number less than 100 has the smallest percentage difference between the largest product and the next largest? Why should this be?

c. Can you explain why e arises in this apparently simple problem?

Puzzle 2: Union

The unmarried heir to a billionaire’s fortune is in a bind. According to the terms of his inheritance, he needs to be married within 21 weeks or else he will forfeit his share. Despite the short deadline, he is determined to choose the best possible partner, so he signs up for a matrimonial app called e-marriage, which has the following rules. The app’s proprietary algorithms will match you with a highly compatible potential life-partner right away, and then every two weeks thereafter. In those two weeks, you must meet the candidate, get to know them, and either accept or reject them. Once candidates are rejected, they cannot be recalled.

The heir reasons that he can meet 10 already well-matched prospective candidates in 20 weeks and get married in the last week before the deadline. But he still wants to maximize his chances of choosing the best partner. So as he gets to know the candidates, he assigns a rank to each one. The problem is, he cannot predict the rankings of the candidates he hasn’t met or the final rankings for the candidates he has. If he accepts a candidate too early, he could miss out on a potentially better one. If he waits too long, he may have already rejected the best one.

a. How can the heir maximize his chances of choosing the best candidate, assuming there are no ties?

b. How do the heir’s chances change if there is a 10% chance of a tie for first place?

c. This problem is a classical one whose solution has something to do with e. Can you explain how e enters the picture?

While the classical problem does indeed focus on maximizing the heir’s chances of choosing the best life partner, not even e can guarantee transcendental bliss. That’s because, if the best candidate appears early and has been rejected, the heir may be stuck with a significantly lower-ranked candidate at the end of the 20th week. A more practical method is required if the goal is to pick one of the best candidates, but not necessarily the very best one. If we assume that the 10 candidates are ranked from 1 to 10, with 1 as the highest rank, we would like a way to pick, say, one of the top three or four most of the time.

d. How can the heir achieve the highest expected rank of his chosen candidate in this more practical choosing scenario?

Finally, for those who just cannot get enough of these e-puzzles, here’s a last, tougher one.

Puzzle 3: Togetherness

 Let us suppose that the heir’s mission was successful and that he did find marital bliss and inherited his large fortune as well. The happy couple decide to celebrate with a getaway to a couples-only resort, where an exclusive concert has been scheduled in a large auditorium. The entry is on a first-come, first-served basis, and the audience consists of couples only, of course. When a couple enters the auditorium, they pick at random a pair of seats next to each other. Each new couple does the same, and in many cases this results in empty single seats between couples. The seating continues until only single seats are left. Then the auditorium is declared full, and the show starts.

a. What proportion of seats are expected to be left unfilled when the seating is stopped?

b. How does e enter this theater of togetherness?

That’s it for our transcendental suite. I hope you enjoy solving these puzzles and perhaps learn something about the amazing number e that you didn’t know before.

So, happy puzzling, and here’s wishing you e-harmony in your transcendental meditations.

Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a Quanta Magazine T-shirt or one of the two Quanta books, Alice and Bob Meet the Wall of Fire or The Prime Number Conspiracy (winner’s choice). And if you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION.” (It will not appear online, so solutions to the puzzle above should be submitted separately.)

Internet Explorer Channel Network
Asia's Tech News Daily
News Related

OTHER NEWS

Paolo Banchero, No. 5 Duke knock off No. 1 Gonzaga in thrilling battle

Paulo Banchero, left, and No. 5 Duke knocked off Chet Holmren, right, and No. 1 Gonzaga on Friday night in Las Vegas. (Brian Rothmuller/Icon Sportswire/Getty Images) The battle between the ... Read more »

WHO names new variant of concern 'omicron,' as scientists advise caution

The World Health Organization on Friday classified a new variant first identified in southern Africa as a “variant of concern.” As the world continues to learn more about the new ... Read more »

Man killed by falling tree as Storm Arwen arrives to batter UK with high winds

A man in Northern Ireland was killed by a falling tree as Storm Arwen brought high winds, rain and snow to the UK. The man died after his car was ... Read more »

I'm a Celebrity: Naughty Boy and Frankie Bridge clash over correct way to cook rice

The tensions occurred after the music producer felt the rice in camp wasn’t being cooked correctly. Read more »

How has the tube strike affected some London passengers?

Passengers outside Stratford Station have shared their experiences of their journey this morning after London Underground services were disrupted on Friday. It comes after drivers launched a 24-hour strike in ... Read more »

Pharmaceutical firms work to tweak vaccines against new ‘Omicron' variant

Work is underway to look at tweaking vaccines against the new concerning strain of coronavirus that has sparked travel bans. The variant, named Omicron and designated a “variant of concern” ... Read more »

Climate activists target Amazon on Black Friday

Climate activists targeted Amazon depots in Europe on Friday (November 26). Members of the Extinction Rebellion group blocked access to one warehouse near London. The move was timed to coincide ... Read more »

‘This is no life': Migrants in bitterly cold Dunkirk camp ask why England isn't doing more to help

Migrants living in the bitter cold in a Dunkirk campsite have asked why the UK does not do more to help them, just days after 27 people died crossing the ... Read more »

Queen Letizia's handbag contents revealed as she suffers major slip-up during royal tour

Queen Letizia suffered a relatable slip-up during her royal tour of Sweden as she dropped her clutch bag during a photo call on Thursday. The contents of the queen's clutch ... Read more »

Omicron variant: WHO names 'extremely concerning' new variant identified in South Africa

The new coronavirus variant was detected in South Africa. Scientists have raised concerns about the large number of mutations, which could mean that vaccines are less effective. (Reuters) The World ... Read more »

Health Secretary warns new COVID variant 'could pose substantial risk to public health'

Sajid Javid told MPs in a Commons statement that the new variant identified in South Africa is ‘of huge international concern’, amid fears the strain, which ‘could defeat vaccines’ may ... Read more »

Bryan Adams Remains Hospitalized In Italy After Testing Positive For Covid Again – Update

UPDATE FRIDAY 1:30 PM: Bryan Adams posted an update from the hospital in Milan, Italy where he was taken after testing positive for Covid upon arrival at the Milan Malpensa ... Read more »

French block British boat amid fishing row

The fishermen held aloft red flares as they circled their boats outside Saint-Malo to block the Normandy Trader’s path – a prelude to a planned blockade later on Friday of ... Read more »

Meet the variant hunters on the frontline

(Getty Images) Professor Sharon Peacock is acutely aware that in her line of work there is “a fine balance between calling it right and unduly panicking people”. She is the ... Read more »

The Queen to leave Windsor Castle next month?

The Queen is expected to travel up to her Sandringham estate next month, where she will reportedly host her family for Christmas this year. While Buckingham Palace is yet to ... Read more »

'Tune out the world': These Sony noise-canceling wireless headphones are nearly 60 percent off for Black Friday

We may receive commission from purchases made via links on this page. Pricing and availability are subject to change. They don’t really float in the air like this; but they’ll ... Read more »

Roombas Are as Little as $175 Thanks to These Epic Black Friday Deals

vacuuums Amazon/Walmart If your most time-consuming chore is keeping your floors clean, then it's a good idea to invest in a high-quality robot vacuum cleaner. And there's no better time ... Read more »

New virus variant emerges in Africa, stokes worldwide fears

1/8Virus Outbreak New VariantPeople wait to get vaccinated at a shopping mall, in Johannesburg, South Africa, Friday Nov. 26, 2021. Advisers to the World Health Organization are holding a special ... Read more »

Coleen Nolan's jaw-dropping Christmas tree leaves Loose Women co-stars amazed

Loose Women star Coleen Nolan has the most incredible Christmas tree – and even her co-stars Christine Lampard, Jane Moore and Andrea McLean agree! RELATED: 15 best Black Friday homeware ... Read more »

Covid variant causing ‘huge international concern' found in Belgium

A worrying new variant of coronavirus has been found in Belgium as the UK’s Health Secretary Sajid Javid said there was “huge international concern” over the strain. Belgium became the ... Read more »

Boot up the best laptop deals Black Friday's got to offer, from $99: Apple, Samsung, Microsoft and more!

We may receive commission from purchases made via links on this page. Pricing and availability are subject to change. Ready for a next-level laptop? Take advantage of these crazy early ... Read more »

Black Friday: retailers are forcing our heads into the sand to avoid facing climate realities

Black Friday traditionally marks the beginning of the Christmas retail season. Powhusku/Flickr, CC BY-NC Black Friday – widely considered the first day of the Christmas shopping season – is celebrated ... Read more »

Black Friday means unbeatable TV deals — starting at (what?!) $100

We may receive commission from purchases made via links on this page. Pricing and availability are subject to change. The screen sizes run the gamut; the prices are all fantastic. ... Read more »

Best Black Friday laptop deals UK - including Samsung, Huawei and more

We’ve rounded up all the best laptop deals for you. (Getty Images) Yahoo Entertainment's editors are committed to independently selecting wonderful products at great prices for you. We may receive ... Read more »

Best Samsung Black Friday 2021 deals on TVs, Chromebooks, phones and more

We may receive commission from purchases made via links on this page. Pricing and availability are subject to change. Samsung‘s top Black Friday deals include The Frame TV, the Z ... Read more »

‘Humiliated' French fishermen begin blockading British boats

French fishermen block the ‘Normandy Trader’ boat at the entrance of the port of St Malo (REUTERS) French fishermen who claim they have been ‘humiliated’ by Britain over post-Brexit operating ... Read more »

Black Friday 2021 UK: Don't miss these hugely reduced Christmas gift sets

Best gift sets to buy now. (Getty Images) Yahoo Life's editors are committed to independently selecting wonderful products at great prices for you. We may receive a share from purchases ... Read more »

Halo Infinite recaptures that original multiplayer magic... even for oldies like me

Halo Infinite I am old and grizzled enough to remember when Halo 2 was the ultimate vanguard of FPS multiplayer. When Call of Duty was a more niche PC pursuit ... Read more »

Best Black Friday laptop deals 2021 UK: Gaming and work laptops from Microsoft Surface to Lenovo

(Black Friday) When it comes to bagging a serious bargain, there’s really no better time than Black Friday. The annual sales event is one of the biggest days in the ... Read more »

Best Black Friday gaming deals UK - including Nintendo Switch, Playstation and more

If gaming’s your thing, make sure you don’t miss these great deals found on headphones, laptops and more this Black Friday. (Getty Images) Yahoo Entertainment's editors are committed to independently ... Read more »

Apple warns Thai activists calling for reform of monarchy that they are targets of ‘state-sponsored' attackers

Protesters hold up the three finger salute outside the Constitutional Court of Thailand in Bangkok (AFP via Getty Images) At least seven activists who had called for the reform of ... Read more »

New Covid variant: How worried should we be?

(AFP via Getty Images) The sudden emergence of a new variant – expected to be called Nu by the WHO – has brought back harrowing flashbacks of winter 2019-2020, when ... Read more »

Scientists ‘stunned' by unexpected discovery of mammoth tusk during deep-sea exploration

Scientists assess the mammoth tusk before beginning the retrieval operation (Darrin Schultz) Scientists have unexpectedly discovered a mammoth tusk while exploring a seamount more than 3km deep and about 300km ... Read more »

Richard Madeley's daughter Chloe says father is ‘livid' after being forced to quit I'm a Celebrity

Richard Madeley isn’t happy that he’s been forced to leave I’m a Celebrity just days into the new series. The broadcaster was made to quit the ITV show after being ... Read more »

Prince William enjoys rare night out at star-studded Christmas party

Prince William excused himself from bedtime duties on Wednesday and enjoyed a night out at a star-studded Christmas party that took place at private members' club Oswald's in London. MORE: ... Read more »

New coronavirus variant is 'not doomsday', leading UK scientist says

A scientist has said that the new highly infectious and potentially vaccine resistant variant of Covid-19 found in South Africa is not ‘doomsday’. (PA Images) The new coronavirus variant is ... Read more »

15 Bose Black Friday deals not to miss — save up to $100

We may receive commission from purchases made via links on this page. Pricing and availability are subject to change. It’s a Bose bonanza! Score speakers, headphones, earbuds and more at ... Read more »

iPad pro Black Friday deal 2021: Get £50 off the ‘fastest and most powerful' Apple tablet

This is the brand’s latest model (The Independent) Black Friday 2021 is finally here. And we can confirm that retailers, including the likes of Amazon, Argos and John Lewis & ... Read more »

‘Most worrying' Covid variant could already be in the UK, expert says

It is possible that a new worrying variant of coronavirus is already in the UK though current signs are reassuring, an expert at the Health Security Agency (UKHSA) has said. ... Read more »
On free-english-test.com you will find lots of free English exam practice materials to help you improve your English skills: grammar, listening, reading, writing, ielts, toeic