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Stories from SlateWednesday Is E Day. Can We Make It As Fun As Pi Day?
https://slate.com/technology/2018/02/how-to-celebrate-e-day-feb-7-2018-with-the-mathematical-constant.html
<p>Wednesday is E Day. That’s because the date, Feb. 7, 2018, is reminiscent of the mathematical constant <em>e, </em>which starts out 2.718281 (and goes on forever). As with <a href="http://www.slate.com/articles/health_and_science/science/2015/03/pi_day_2015_ten_digits_represented_on_march_14_at_9_26_53.html">Pi Day 2015</a>, the day, month, and year correspond to the first few digits of this constant, which means it’s time to party. (People who use the day/month/year convention can save this article and celebrate on July 2.)</p>
<p>How do you celebrate E Day? An eggplant-edamame enchilada would be one way—perhaps Pi Day has a bit of an edge on the culinary front. Instead, I think the first thing you should do is invest a bit of money in a savings account, because the first place students usually encounter the number <em>e</em> is in the formula for continuously compounded interest. That formula is A=Pe<sup>rt</sup>, where <em>A</em> is the amount of money in the account, <em>P</em> is the principal amount invested, <em>r</em> is the interest rate, and <em>t</em> is the length of time the money has been there.</p>
<p>If your high school interest rates lessons are a bit foggy, here’s a refresher. If you open an account with, say, $1,000 in it and a 10 percent annual interest rate (it’s nice to dream), the amount of money you have at the end of the year actually depends on how often the interest is compounded. If you compound it just once over the course of the year, you’ll end up with $1,100. If you compound it once after six months, the interest you earn in the first half of the year then goes on to earn more interest in the second half of the year. After six months, you earn $50 in interest. In the next six months, that total amount of $1,050 earns $52.50 in interest for a grand total of $1,102.50 at the end of the year. An extra $2.50 might not sound like much, but I’m sure <a href="https://slate.com/news-and-politics/2018/02/paul-ryan-deletes-tweet-after-hes-mocked-for-celebrating-usd1-50-benefit-from-tax-cut.html">Paul Ryan will be very excited for you</a>. If you compounded three times in the year instead of twice, you’d end up with $1,103.37. The more frequently you compound interest, the more money you end up with, though at some point you’re talking minute fractions of a penny.</p>
<p>Compound interest is the first unambiguous use of the number <em>e</em> that we learned about. (It had secretly made an appearance decades before in tables of natural logarithms, but no one had explicitly looked at it as a number in and of itself.) In 1683, Jacob Bernoulli was studying compound interest. He calculated that the formula for compounding interest a certain number of times over the course of the year, let’s say <em>n</em> times, will be A=P(1+r/n)<sup>n</sup>. Bernoulli found that if you wanted to compound interest continuously—that is you want every morsel of interest itself to earn interest immediately—you would take the limit of P(1+r/n)<sup>n </sup>as <em>n</em> went to infinity. The number <em>e</em> shows up when you set <em>r</em> to the value 1, so it is the limit of the expression (1+1/n)<sup>n</sup> as <em>n</em> goes to infinity. (As a side note, banks often tell you an annual percent <em>yield</em> that takes into account the way they compound interest rather than an annual percent <em>rate</em>.)</p>
<p>Bernoulli himself didn’t figure out <em>e</em>’s value beyond noting that it was between 2 and 3. But a few decades later, Leonhard Euler named it <em>e</em> and found that it was the limit of another expression, the infinite sum 1+1/1+1/2+1/6+1/24+1/120+… . That helped him compute <em>e</em> to 18 decimal places and show that, like <em>π</em>, it is irrational, that is, it can’t be written as a fraction. Though Euler didn’t actually choose the letter <em>e</em> to name it after himself, we now call it the Euler constant. (Sadly, Euler waited until April 15 to be born instead of making an appearance on E Day in 1707.)</p>
<p>The number <em>e</em> is good for a lot more than bank accounts. If you took calculus, you might remember your sighs of relief when you were asked to find the derivative of the function y=e<sup>x</sup>. The derivative is a measure of how quickly a function is changing at a particular point. Any exponential function has a derivative that is a multiple of itself. For example, the derivative of the function y=2<sup>x</sup> is approximately (0.69)2<sup>x</sup>. The derivative of the function y=4<sup>x</sup> is approximately (1.39)4<sup>x</sup>. Only when the base is <em>e</em> do you get to discard that clunky number at the beginning: The derivative of e<sup>x</sup> is just e<sup>x</sup>. To know how quickly this function is increasing at any given point, you just have to look at the value of the function at that point.</p>
<p>For example, here is a graph of the function y=e<sup>x</sup>. The slope of the graph at any point (x,e<sup>x</sup>) is e<sup>x</sup>:</p>
<p>That property of the function e<sup>x</sup> is why some people like to describe <em>e</em> as the natural base for growth, as James Grime explains in this Numberphile video:</p>
<p>Like <em>π</em>, the constant <em>e</em> seems to show up in all sorts of places in math. Both numbers are part of what is often described as the “most beautiful equation,” which says e<sup>πi</sup>+1=0. Here, e is indeed the base of the natural logarithm and π is the circle constant we know and love. The number <em>i</em> is the imaginary unit, a square root of -1.</p>
<p>I must admit I have my qualms about proclaiming e<sup>πi</sup>+1=0 to be the most beautiful equation. People are entranced by the fact that it contains five of the most important numbers—0, 1, <em>i</em>, <em>e</em>, and<em> </em><em>π</em>—but you can easily stuff 0 and 1 into any equation you want by adding 0 or multiplying by 1. A more natural way to write this expression, in my opinion, is e<sup>πi</sup>=-1. Somehow that doesn’t capture people’s imaginations the same way, even though it is a more compact way of writing the exact same thing. But beyond my cosmetic preference for this writing, the equation e<sup>πi</sup>=-1 also depends in a deep way on how we define an awful lot of things, and I think the <a href="https://www.youtube.com/watch?v=F_0yfvm0UoU">magic is in all of those definitions,</a> rather than one example.</p>
<p>But back to <em>e—</em>is there anything this great number can’t do? It finds its way into <a href="https://mathwithbaddrawings.com/2018/02/05/the-abc-book-of-e/">population equations, the famous bell curve, calculating probabilities of success in repeated experiments</a>, even making important life decisions about <a href="http://www.slate.com/articles/technology/technology/2014/12/the_secretary_problem_use_this_algorithm_to_determine_exactly_how_many_people.html">hiring, dating</a>, or what <a href="https://www.youtube.com/watch?v=ZWib5olGbQ0">public toilet to use at a music festival</a>. That leaves E Day celebrations wide open. Anything you want to do in life that involves change or growth probably has an <em>e</em> hiding in the background somewhere. So perhaps instead of activities like opening a savings account or going on a date, E<em> </em>Day can be a day for reassessing or renewing those resolutions that might have faded since Jan. 1. Change and growth, here we come!</p>Tue, 06 Feb 2018 21:22:31 GMThttps://slate.com/technology/2018/02/how-to-celebrate-e-day-feb-7-2018-with-the-mathematical-constant.htmlEvelyn Lamb2018-02-06T21:22:31ZFeb. 7, 2018, is the calendar date of the mathematical constant 2.718, and we have several ways to celebrate.TechnologyWednesday Is E Day, for the Mathematical Constant 2.718. Here’s How to Celebrate.171748mathon instagramEvelyn LambSciencehttps://slate.com/technology/2018/02/how-to-celebrate-e-day-feb-7-2018-with-the-mathematical-constant.htmlfalsefalsefalseWednesday is E Day, for the mathematical constant 2.718. Here’s how to celebrate:Anything you want to do in life that involves change or growth probably involves an e hiding in the background somewhere.The Largest Known Prime Number
http://www.slate.com/articles/health_and_science/science/2016/01/the_world_s_largest_prime_number_has_22_338_618_digits_here_s_why_you_should.html
<p><strong><em>Update, Jan. 4, 2018: </em></strong>On Wednesday, the <a href="https://www.mersenne.org/primes/press/M77232917.html">Great Internet Mersenne Prime Search announced</a> that a computer owned by Jonathan Pace in Germantown, Tennessee, discovered a new prime number. At 23,249,425 digits, the number, known as M77232917, is now the largest known prime.</p>
<p>In 2016, I wrote the following article about the previous largest known prime, which is now the second largest known prime. Its name is M74207281, and it’s about a million digits shorter than the shiny new prime. But other than a few details about whose computer found it and exactly how long it is, I could have written this article today about the new prime. So we’re sharing it with you again.</p>
<p>It’s exciting to find a new largest known prime number, but this is another verse of the same song. Both numbers, like nine of the 10 largest known prime numbers, have a special form and are called Mersenne primes. We find them because that’s where we keep looking. The light is better there. Between these two largest known primes lie an unfathomable number of monstrously large primes; we may never find even one.</p>
<p><strong><em>Original, Jan. 22, 2016:</em></strong> Earlier this week, BBC News reported an important mathematical finding, sharing the news under the headline “<a href="http://www.bbc.com/news/technology-35361090">Largest Known Prime Number Discovered in Missouri</a>.” That phrasing makes it sound a bit like this new prime number—it’s 2<sup>74,207,281</sup>-1, by the way—was found in the middle of some road, underneath a street lamp. That’s actually not a bad way to think about it.</p>
<p>We know about this enormous prime number thanks to <a href="http://www.mersenne.org/m49/74207281.htm">the Great Internet Mersenne Prime Search</a>. The Mersenne hunt, known as GIMPS, is a large distributed computing project in which volunteers run software to search for prime numbers. Perhaps the best-known analogue is <a href="http://setiathome.ssl.berkeley.edu/">SETI@home</a>, which searches for signs of extraterrestrial life. GIMPS has had a bit more tangible success than SETI, with 15 primes discovered so far. The shiny new prime, charmingly nicknamed M74207281, is the fourth found by University of Central Missouri mathematician Curtis Cooper using GIMPS software. This one is 22,338,618 digits long.</p>
<p>A prime number is a whole number whose only factors are 1 and itself. The numbers 2, 3, 5, and 7 are prime, but 4 is not because it can be factored as 2 x 2. (For reasons of convenience, we don’t consider 1 to be a prime.) The M in GIMPS and in M74207281 stands for <a href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Mersenne.html">Marin Mersenne</a>, a 17<sup>th</sup>-century French friar who studied the numbers that bear his name. Mersenne numbers are 1 less than a power of 2. Mersenne primes, logically enough, are Mersenne numbers that are also prime. The number 3 is a Mersenne prime because it’s one less than 2<sup>2</sup>, which is 4. The next few Mersenne primes are 7, 31, and 127.</p>
<p>M74207281 is the 49<sup>th</sup> known Mersenne prime. The next largest known prime, 2<sup>57,885,161</sup>-1, is also a Mersenne prime. So is the one after that. And the next one. And the next one. All in all, the 11 largest known primes are Mersenne.</p>
<p>Why do we know about so many large Mersenne primes and so few large non-Mersenne ones? It’s not because large Mersenne primes are particularly common, and it’s not a spectacular coincidence. That brings us back to the road and the street lamp. There are several different versions of <a href="http://quoteinvestigator.com/2013/04/11/better-light/">the story</a>. A man, perhaps he’s drunk, is on his hands and knees underneath a streetlight. A kind passerby, perhaps a police officer, stops to ask what he’s doing. “I’m looking for my keys,” the man replies. “Did you lose them over here?” the officer asks. “No, I lost them down the street,” the man says, “but the light is better here.”<a></a></p>
<p>We keep finding large Mersenne primes because the light is better there.</p>
<p>First, we know that only a few Mersenne numbers are even candidates for being Mersenne primes. The exponent <em>n</em> in 2<em><sup>n</sup></em>-1 needs to be prime, so we don’t need to bother to check 2<sup>6</sup>-1, for example.<a>*</a> There are a few other <a href="http://primes.utm.edu/glossary/xpage/NewMersenneConjecture.html">technical conditions</a> that make certain prime exponents more enticing to try. Finally, there’s a particular test of primeness—<a href="https://www.youtube.com/watch?v=lEvXcTYqtKU">the Lucas–Lehmer test</a>—that can only be used on Mersenne numbers.</p>
<p>To understand why the test even exists, let’s take a detour to explore why we bother finding primes in the first place. There are infinitely many of them, so it’s not like we’re going to eventually find the biggest one. But aside from being interesting in a “math for math’s sake” kind of way, finding primes is good business. <a href="http://www.slate.com/articles/health_and_science/science/2013/06/online_credit_card_security_the_rsa_algorithm_prime_numbers_and_pierre_fermat.html">RSA encryption</a>, one of the standard ways to scramble data online, requires the user (perhaps your bank or Amazon) to come up with two big primes and multiply them together. Assuming the encryption is implemented correctly, the difficulty of factoring the resulting product is the only thing between hackers and your credit card number.</p>
<p>This new Mersenne prime is not going to be used for encryption any time soon. Currently we only need primes that are a few hundred digits long to keep our secrets safe, so the millions of digits in M74207281 are overkill, for now.</p>
<p>You can’t just look up a 300-digit prime in a table. (There are about 10<sup>297</sup> of them. Even if we wanted to, we physically could not write them all down.) To find large primes to use in RSA encryption, we need to test randomly generated numbers for primality. One way to do this is trial division: Divide the number by smaller numbers and see if you ever get a whole number back. For large primes, this takes way too long. Hence there are primality tests that can determine whether a number is prime without actually factoring it. The Lucas-Lehmer test is one of the best.</p>
<p>The heat death of the universe would occur before we could get even a fraction of the way through trial division of a number with 22 million digits. It only took a month, however, for a computer to use the Lucas-Lehmer test to determine that M74207281 is prime. There are no other primality tests that run nearly as quickly for arbitrary 22 million–digit numbers.</p>
<p>How many primes have we missed by looking for them mostly under the Lucas-Lehmer street lamp? We don’t know the exact answer, but the <a href="http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html">prime number theorem</a> gets us close enough. It makes sense that primes get less common as we stroll out on the number line. Fully 40 percent of one-digit numbers are prime, 22 percent of two-digit numbers are prime, and only 16 percent of three-digit numbers are. The prime number theorem, first proved in the late 1800s, quantifies that decline. It says that in general, the number of primes less than <em>n</em> tends to <em>n/ln(n)</em> as <em>n</em> increases. (Here <em>ln</em> is the <a href="https://en.wikipedia.org/wiki/Natural_logarithm">natural logarithm</a>.)</p>
<p>We can use the prime number theorem to estimate how many missing primes there are between M74207281 and the next smallest prime. We just plug 2<sup>74,207,281</sup>-1 into <em>n/ln(n) </em>and get, well, a <a href="http://www.wolframalpha.com/input/?i=%282%5E74%2C207%2C281-1%29%2F%28ln%282%5E74%2C207%2C281-1%29%29">really big number</a>. We can write it most compactly by stacking exponents: 10<sup>10<sup>7.349</sup></sup>. This number has about 22,338,610 digits, give or take a couple, so we can also write it as 10<sup>22,338,610</sup>.</p>
<p>Another visit to the prime number theorem shows there are approximately 10<sup>17,425,163</sup> primes less than the next-largest known prime. That sounds impressive until you realize 10<sup>17,425,163</sup> is less than 0.000000000001 percent of 10<sup>22,338,610</sup>.</p>
<p><a></a>Stop and think about that for a moment. There are about 10<sup>22,338,610</sup> primes less than M74207281, and approximately all of them are between it and the next-largest known prime. If you want to be charitable, you could say we have some gaps in our knowledge of prime numbers. But really, it makes more sense to say we have gaps in our lack of knowledge. The <a href="http://primes.utm.edu/lists/small/millions/">millions upon millions</a> of prime numbers we’ve already found make up approximately 0 percent of the primes that are less than M74207281. Each one is a little grain of sand, a speck that does little to cover up our overwhelming ignorance of exactly where the prime numbers live.</p>
<p><em><strong>Correction, Jan. 22, 2016: </strong>This story originally referred to a possible prime number as 2<sup>n-1</sup>. That number should have been rendered as 2<sup>n</sup>-1. (<a>Return</a>.)</em></p>Thu, 04 Jan 2018 20:54:00 GMThttp://www.slate.com/articles/health_and_science/science/2016/01/the_world_s_largest_prime_number_has_22_338_618_digits_here_s_why_you_should.htmlEvelyn Lamb2018-01-04T20:54:00ZIt’s 23,249,425 digits long. That’s incredibly exciting.Health and ScienceWhy You Should Care About a Prime Number That’s 23,249,425 Digits Long100160122018sciencemathEvelyn LambSciencehttp://www.slate.com/articles/health_and_science/science/2016/01/the_world_s_largest_prime_number_has_22_338_618_digits_here_s_why_you_should.htmlfalsefalsefalseWhy you should care about a prime number that’s 23,249,425 digits long:Introducing the newest longest known prime number.1519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t460449660800147041282720011519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t460449660800147041282720011519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t460449660800147041282720011519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t46044966080014704128272001Eyematrix/ThinkstockRSA encryption, one of the standard ways to scramble data online, requires the user to come up with two big primes and multiply them together.An Election Season Reminder That Voting Is Mathematically Flawed
http://www.slate.com/articles/health_and_science/science/2016/10/it_s_mathematically_impossible_to_vote_between_more_than_two_candidates.html
<p>If you had told me when I moved to Utah in 2013 that it would be a swing state in the next presidential election, I would have laughed and laughed. But in the midst of this not-so-funny election season, Utah is indeed making news as independent candidate Evan McMullin’s presidential campaign has gained traction here. As of Thursday, election reporting juggernaut <em>FiveThirtyEight</em> is currently giving him a <a href="http://projects.fivethirtyeight.com/2016-election-forecast/utah/">19.6 percent chance</a> of winning the state’s popular vote, and a sliver of a chance of <a href="http://fivethirtyeight.com/features/how-evan-mcmullin-could-win-utah-and-the-presidency/">throwing this election into chaos by stealing enough electoral votes that neither Hillary Clinton nor Donald Trump gets the requisite 270 to win the race outright</a>.</p>
<p>This is not an article about the intricacies of the Electoral College or why you should or should not vote for a third-party candidate this year. I assume you’ve seen enough <a href="http://qz.com/779147/whats-the-purpose-of-a-protest-vote/">articles</a> <a href="http://www.njherald.com/20161024/why-not-vote-for-third-party-candidate">like</a> <a href="http://www.nbcnews.com/politics/2016-election/dem-group-launches-effort-warning-about-third-party-votes-n670451">that</a> <a href="http://www.usatoday.com/story/news/politics/onpolitics/2016/10/25/johnson-stein-mcmullin-wasted-votes/92694606/">already</a>—or <a href="https://www.youtube.com/watch?v=k3o01efm5fu">heard John Oliver passionately argue against the idea</a>. No, in this <a href="http://www.huffingtonpost.com/entry/octavia-butler-predicted-make-america-great-again_us_5776d9dce4b0416464100242">Octavia Butler dystopia</a> of an election season, I want to encourage you to move your frustration beyond the two-party system and the Electoral College. Because those aren’t the only factors that are against us—the very fabric of mathematics itself falls down when it comes to trying to accurately assess voter preferences.</p>
<p>It’s not this election that’s rigged. It’s the entire idea of trying to vote.</p>
<p>If there are exactly two choices on a ballot and every voter votes for one of them, it’s easy to find the winner: Count the votes, and whoever has more triumphs. The winner must have more than half of the votes, so a majority of voters will have their preferred candidate win. That is about as fair as it gets.</p>
<p>But with three or more options, all bets are off. It’s not just a problem with the frequently used plurality-rule system—in which a candidate only needs more votes than others rather than a majority of votes to win. There is no system that can fairly decide the winner of an election with more than two candidates. It simply doesn't exist.</p>
<p>American voters are quite familiar with our system’s flaws vis-a-vis third parties. The pet examples are Ross Perot and Ralph Nader as “spoiler” candidates in the 1992/1996 and 2000 presidential elections—Nader in particular was blamed for Al Gore’s loss in 2000. The thinking goes that almost all Nader supporters would prefer Gore to George W. Bush, and in some states, notoriously Florida, the margin between Bush and Gore was smaller than Nader’s vote percentage. If Nader’s votes had gone to Gore, the outcome of the election would have been different. But because we do not have a preferential voting system, there was no way for Nader supporters to indicate that if their candidate lost, they would prefer Gore to Bush.</p>
<p>Preferential voting systems seek to solve the Nader problem by allowing voters to provide a ranked list of their preferences. There are many possible ways to allocate votes based on preferential ordering. One, called instant runoff voting, eliminates the candidate with the fewest first-place votes. Then the race is repeated for the remaining candidates. As a simplistic example, say 25 percent of voters had the preference rank ABC, 35 percent voted BCA, and 40 percent voted CAB. Candidate A got the fewest first-place votes, so we take A away entirely. Now 60 percent of people have the preference order BC over 40 percent for CB, so B wins. This is frustrating because more people ranked candidate C as their first choice than any other first choice. And going back to the original rankings shows that if C wasn’t going to win, more people would have preferred A to B. But the runoff system sticks us with candidate B.</p>
<p>Another way to score that election is called a Borda count, where candidates get more votes when they are placed higher by more people. In this example, we will give each candidate 3 points for every first-place vote, 2 points for every second-place vote, and 1 point for every third-place vote. Here, we’ll pretend we have 100 voters to make the numbers nice and round. A gets 75 points from the voters who ranked them ABC, 35 from the BCA voters, and 80 from the CAB voters for a grand total of 190 points. Candidate B gets 195 points and C gets 215, so C is elected. That seems fair, right?</p>
<p>But what if things were a bit more complicated—instead of 40 percent of voters having the preference CAB, 30 percent prefer CBA and 10 percent prefer CAB? Voters’ relative preference between B and C has not changed but now C gets 210 points and B gets 220, so B is elected instead of C.</p>
<p>Both of these methods of deciding the winner in this election seem flawed. Is this just because we haven’t figured out the right preferential voting system? That would be nice, but the answer instead is that we’re running into a mathematical obstacle called Arrow’s impossibility theorem. It states that any preferential voting system must fail some fairly basic test of fairness. The exact criteria are a bit technical, but the big sticking point is called the independence of irrelevant alternatives: If no one’s relative preference between two candidates changes, then the relative rank of the two candidates should not change. But in the Borda count example above, changing voters’ preferences between A and B changed the relative ranks of B and C even though C wasn’t involved in the preference changes. In 1951, Kenneth Arrow proved that problems like that are a feature of all preferential voting systems.</p>
<p>In my original example, 65 percent of people preferred candidate A to candidate B, 60 percent preferred candidate B to candidate C, and 75 percent preferred candidate C to candidate A. If you’ve played rock, paper, scissors, you understand the problem: No single move is always going to win, it all hinges on what you’re up against. (Mathematicians call preferences like this <em>nontransitive</em>.) No matter what system we use to choose between A, B, and C, a majority of voters will prefer a different candidate to the one who is chosen. If C is our choice, most people will wish B had been elected instead. If it’s B, most people will prefer A. If it’s A, most people will prefer C. In some elections, voters’ preferences will be better-behaved, with one candidate who beats every other candidate head-to-head. But whenever the voters’ preferences are non-transitive, there’s no way to be sure our choice of candidate doesn’t violate one of the fairness criteria. (For a more in-depth explanation and explicit examples of different voting systems and their foibles, I highly recommend “There is no such thing as public opinion,” a chapter in Jordan Ellenberg’s book <a href="http://www.amazon.com/dp/0143127535/?tag=slatmaga-20">How Not to Be Wrong</a>.)</p>
<p>When I suggested this story to my editor, she asked if Arrow’s theorem is how we can end up with unpopular candidates winning major party nominations. After all, the primaries have a greater number of viable candidates than the general elections do, so they’re especially susceptible to problems arising from Arrow’s theorem. Frustratingly, even though we know this is a mathematical possibility, we don’t know how often real voters’ preferences are nontransitive and therefore problematic. Political polls don’t ask respondents to rank candidates, so we just don’t have the data. In 2012, <a href="https://www.princeton.edu/~cuff/voting/index.html">researchers at Princeton University ran several polls</a> to compare different voting methods on real data from voters and found that there was usually a winner who did satisfy the basic fairness criteria. As far as I can tell, there are no similar polls from this year’s primary season or other political contests that would allow us to assess this.</p>
<p>Of course, preferential voting is not our only option. In addition to the plurality system we use, there is approval voting where people can vote for, or approve, as many people as they want and the person who gets the most approvals wins. There are also systems where you rate candidates—5 stars for A, 4 for B, and 2 for C, perhaps. Arrow’s impossibility theorem does not apply to those systems, though they have their vulnerabilities as well, usually on the side of the voter. <a>Someone could decide not to </a>“approve” a candidate they thought was just fine because it could prevent their favorite candidate from winning; someone could rate all opposing candidates one out of five stars to boost their own candidate. All voting systems are subject to some sort of manipulation.</p>
<p>And maybe that’s the point. We have the voting system we have, and each of us works within that system to make our voice heard and elect the candidates we think are most deserving. Are there problems? Of course. But until we turn elections over to all-powerful, benevolent mind-reading robots, there will never be a voting system that perfectly reflects the will of the people, whatever that means.</p>
<p>Now take a deep breath and go vote!</p>Fri, 28 Oct 2016 09:52:00 GMThttp://www.slate.com/articles/health_and_science/science/2016/10/it_s_mathematically_impossible_to_vote_between_more_than_two_candidates.htmlEvelyn Lamb2016-10-28T09:52:00ZThere is no fair way of assessing a populations’ preferences when there are more than two candidates.Health and ScienceFair Voting Between More Than Two Candidates Is Mathematically Impossible1001610280042016 campaignmathematicsEvelyn LambSciencehttp://www.slate.com/articles/health_and_science/science/2016/10/it_s_mathematically_impossible_to_vote_between_more_than_two_candidates.htmlfalsefalsefalseReminder: Voting between more than two candidates is mathematically impossible.Lol, nothing matters.Eduardo Munoz Alvarez/Getty ImagesA man casts his ballot at polling station during New Jersey's primary elections on June 7 in Hoboken, New Jersey.No, There Aren’t Enough Academic Jobs
http://www.slate.com/articles/health_and_science/science/2016/08/discouraging_people_from_academia_could_have_a_devastating_effect_on_women.html
<p>A <em>New York Times</em> Upshot article has been making its way around my Facebook and Twitter feeds recently: “<a href="http://www.nytimes.com/2016/07/14/upshot/so-many-research-scientists-so-few-openings-as-professors.html?smid=fb-share&_r=1">So Many Research Scientists, So Few Openings as Professors</a>.” The U.S. is overproducing Ph.D.s in STEM fields, the story goes, and we need to talk about the fact that not all science, technology, engineering, and mathematics graduates end up as tenured professors at research universities.</p>
<p>That message certainly seems to ring true for me. The academic job market for mathematicians has been pretty grim the past few years: I went through two unsuccessful rounds of applications before getting a postdoc, and I know a lot of good mathematicians who had to leave academia because the jobs <em>just aren’t there</em>. I wish—and I think I’m not alone in this wish—I had known more about career options outside of academia when I was in graduate school and considering my next move. (I eventually left academia to pursue writing full time.)</p>
<p>The article offers a solution:</p>
<blockquote>
For those thinking of science as a career, said P. Kay Lund, director of the division of biomedical research workers at the National Institutes of Health, perhaps the best thing would be for a mentor to sit down with them and have a heart-to-heart talk, preferably when they’re still undergraduates.
</blockquote>
<p>That sounds OK on paper. One solution to the problem of too many applicants for too few jobs is to lower the number of applicants, and lowering that number from the get-go by decreasing the number of STEM graduate students would be one way to do it. But this suggestion could have some serious unintended consequences. Advisers who initiate these conversations with their undergraduate students are most likely doing it with only their students’ best interests at heart. But the people most likely to listen to those conversations and heed them? They are likely to be the people we really need to encourage to go into this field: women, people with disabilities, blacks, Latinos, and Native Americans. These are the people who already don’t see themselves at the table and are therefore most likely to deal with stereotyping and subtle discouragement.</p>
<p>I suspect this because within my own social network, the article was being shared mostly by women. My social network is certainly not a representative sample of the U.S. at large—as a woman in math myself, I happen to meet a lot of other women mathematicians—so I am not claiming it was shared more among women in general. But it made me wonder who gets what messages about becoming a mathematician.</p>
<p><a></a>People who are traditionally underrepresented in STEM fields will take the message of how difficult this career path is more literally. Those from underrepresented groups already have to fight against stereotype threat and <a href="http://www.slate.com/articles/business/the_ladder/2016/04/is_impostor_syndrome_real_and_does_it_affect_women_more_than_men.html">impostor syndrome</a>: They’re more likely to interpret an assessment of job prospects as <em>do you think you’re really good enough, </em>or<em> should you quit before you even start? </em><a href="http://www.sciencemag.org/careers/2016/07/low-math-confidence-discourages-female-students-pursuing-stem-disciplines">A recent study of college calculus students published in <em>Plos One </em>and reported by <em>Science</em></a> found that women are more likely to leave STEM majors after taking calculus than men are, even after controlling for factors including academic preparedness and career plans.<a href="http://www.slate.com/articles/health_and_science/science/2016/08/discouraging_people_from_academia_could_have_a_devastating_effect_on_women.html#correction">*</a></p>
<p>Mathematics can’t afford to lose more women and minorities. Or to put it another way, mathematics can’t afford <em>to not recruit</em> more women and minorities. In 2014, the most recent year for which data is available, the <a href="http://www.ams.org/profession/data/annual-survey/docsgrtd">annual survey of new doctoral recipients in the mathematical sciences</a> reported that a little under one-third of newly minted math Ph.D.s are women; only 2.5 percent are black, 3.5 percent are Hispanic or Latino, and less than 1 percent are Native American. The numbers get worse as we move up the career ladder. According to the most recent report by the <a href="http://www.ams.org/profession/data/cbms-survey/cbms2010">Conference Board for the Mathematical Sciences</a>, 23 percent of tenured or tenure-track faculty are women. The numbers for other underrepresented groups are mostly too low to register as percentages. Mathematics suffers from its lack of diversity. In a 2014 <a href="http://twas.org/article/maths-also-women">interview with the World Academy of Sciences, Ingrid Daubechies</a>, past president of the International Mathematical Union, said: “Many mathematicians believe that mathematical talent is distributed more or less uniformly around the globe.” When we don’t include all of that talent, math suffers.</p>
<p>Young people from marginalized groups don’t need more realism, they need to know it’s OK to dream. In 2013, I <a href="http://blogs.scientificamerican.com/roots-of-unity/mathematics-live-demarco-wilkinson/">interviewed mathematicians Laura DeMarco and Amie Wilkinson</a> as part of a series for the <a href="https://sites.google.com/site/awmmath/home">Association for Women in Mathematics</a>. They talked about their career paths, struggles they had faced as women in the field, and advice for young people who think they might want to be mathematicians. Both of them mentioned their lack of confidence as undergraduates and how important it was for people to encourage them at that time. Wilkinson ended the interview by urging people just to go for it if they might be interested in pursuing mathematics at a more advanced level. “If it doesn’t work out, big deal, it’s a year of your life. I just think more people should try,” she said.</p>
<p>The post was shared on University of Wisconsin mathematician <a href="https://quomodocumque.wordpress.com/2013/06/11/interview-with-demarco-and-wilkinson/">Jordan Ellenberg</a><a href="https://quomodocumque.wordpress.com/2013/06/11/interview-with-demarco-and-wilkinson/">’s blog</a> and received this comment:</p>
<blockquote>
I have to say I am somewhere between annoyed and disturbed by how completely encouraging and positive the published interview was. ... I don’t want someone going to graduate school picturing he or she will become the next Laura or Amie when such an outcome is only a somewhat unlikely possibility.
</blockquote>
<p>The ensuing discussion got passionate and occasionally heated. Some people, including Wilkinson, wondered if an interview with two men would have received the same criticism. In general, do we expect successful people to go around telling everyone else they probably won’t be as successful? Ellenberg asked whether interviews with, say, NBA players should also follow this advice. Should LeBron James tell kids how few of them will grow up to be in the NBA? Arguably, his duty to do that should be even greater than DeMarco’s and Wilkinson’s, as the NBA employs far fewer athletes than the nation’s math departments do mathematicians. Should Hillary Clinton not tell little girls they could be president someday? Should her speeches emphasize the fact that only five of the 320 million people in this country are current or former presidents, so little girls, or any children for that matter, who look up to her should not even dream about seeing themselves in the White House?</p>
<p>Being a math professor is not nearly as remote a possibility as being a professional athlete or president of the United States, but nonetheless not everyone who wants to be will end up tenured at a prestigious research university. The odds are long, but there’s a reason we read profiles and interviews of the people who beat them: We want to see how the few who make it get there, and we want to dream about how we might make it, too.</p>
<p>I think Wilkinson said it perfectly in a comment on Ellenberg’s blog: “[T]he world is filed with gifted women. Just as many as gifted men. They do not go into math. Math suffers for it. I’m not going to be the one to tell them not to try. I’m going to be the one to say, go for it. You belong. Period.” That discussion was specifically about women in math, but the sentiment applies to other underrepresented groups as well and other STEM fields.</p>
<p>Our dreams are not going to magically land us tenure-track jobs at top research institutions: The job crisis is real. But there are other things to resolve before we resort to limiting our applicant pool. We need to address funding problems and the fact that we rely on underpaid adjuncts and graduate students to teach large numbers of classes. We need to address other barriers to graduate school, too: the lack of financial security; the cruel way academic and biological clocks often interfere with each other for people who want to have children; and racist and sexist admissions and hiring committees, whether that racism is conscious or unconscious. But these are problems for the larger community to address, not to throw at the feet of undergraduates before they’ve even decided where to apply for grad school.</p>
<p>So what <em>should</em> advisers discuss with their students who are thinking about graduate school? By all means, they should encourage students to think about their future careers and the realities of the job market, but they should also remind them that people like them have successful careers both in and out of academia. If all prospective grad students received more information about more prospective careers, perhaps there would be less pressure to succeed in academia alone—which would help everyone.</p>
<p>People from underrepresented groups already get plenty of messages telling them that <em>maybe</em> they’re not really good enough, <em>maybe</em> they don’t really belong. They need to know there is a place for them. Talk to them about finding role models and mentors. <a></a>Talk to them about programs like <a href="http://www.edgeforwomen.org/">Enhancing Diversity in Graduate Education</a> that will give them a leg up, as well as the beginnings of a professional and social support network when they enter graduate school. Talk to them about professional organizations that support students. Tell them you will be there to listen to and believe in and support them if they are faced with harassment, sexism, racism, or other prejudice. Tell them it’s OK to try, even if they aren’t sure they’re the best. That there’s no shame in trying, failing, and learning from the experience, or in trying, succeeding, and finding out you’d rather do something else with your life.</p>
<p>Everybody should be allowed to explore and find themselves, not just the privileged. It’s OK if they don’t all end up as professors—but we’d be better off if more of them did.</p>
<p><em><strong>Correction, Aug. 8, 2016:</strong> Due to a production error, an earlier version of this article misstated that a study on college calculus students had been published in Science. It was published in Plos One and reported on in Science. (<a href="http://www.slate.com/articles/health_and_science/science/2016/08/discouraging_people_from_academia_could_have_a_devastating_effect_on_women.html#return">Return</a>.)</em></p>Mon, 08 Aug 2016 15:16:47 GMThttp://www.slate.com/articles/health_and_science/science/2016/08/discouraging_people_from_academia_could_have_a_devastating_effect_on_women.htmlEvelyn Lamb2016-08-08T15:16:47ZBut discouraging people from pursuing the career could have a devastating effect on underrepresented groups.Health and ScienceAcademia Is Hard to Get Into. Everyone—Especially Women and Minorities—Should Still Try.100160808007academiastemjobsEvelyn LambSciencehttp://www.slate.com/articles/health_and_science/science/2016/08/discouraging_people_from_academia_could_have_a_devastating_effect_on_women.htmlfalsefalsefalseDiscouraging people from academia could have a devastating effect on women, minorities:If we tell people to reconsider entering a field, we need to be aware of who will listen. Often, it’s the people we need the most.monkeybusinessimages/ThinkstockMathematics can’t afford to lose more women and minorities.It Doesn’t Add Up
http://www.slate.com/articles/health_and_science/education/2016/03/andrew_hacker_s_the_math_myth_is_a_great_example_of_mathematics_illiteracy.html
<p>In his new book <a href="http://www.amazon.com/dp/1620970686/?tag=slatmaga-20"><em>The Math Myth: And Other STEM Delusions</em></a><em>,</em> Andrew Hacker lays out a bold case for substantially changing the instruction of mathematics. Hacker’s thesis is that too many students drop out of high school and college because they fail math classes, a problem he would solve by removing such obstructions from high schools’ required curricula and providing students with more options for math classes he deems relevant. These would include courses that emphasize numeracy—a facility with numbers and arithmetic as they generally show up in everyday life—and “citizen statistics,” an ability to understand and contextualize figures that appear in media and politics.</p>
<p>Hacker’s conclusions aren’t entirely without merit. Math education fails many students for a variety of reasons: high-stakes testing under time pressure, focus on rote calculation rather than imagination and reasoning, <a href="http://mrhonner.com/archives/14673">inane word problems</a> that fail to show the relevance of mathematics to the real world, and, yes, a failure to expose students to statistics earlier and more effectively. Like Hacker, I am skeptical of doom-and-gloom warnings about the U.S. running out of skilled engineers and scientists, and I think it would be wonderful if we <a href="http://news.nationalgeographic.com/2016/03/160324-math-stem-education-hacker-opinion/">stopped making high school math a straight line to calculus</a>. And who can argue with taking a closer look at whether the mathematics requirements to get into medical or actuarial school are reasonable?</p>
<p>But Hacker’s book has so many problems in both substance and form that it’s hard to take his thesis seriously. I kept track of errors, unreferenced claims, and misleading arguments as I read <em>The Math Myth</em>, and I found so many that I’m halfway tempted to publish an annotated edition of the book. (I’m not the only one frustrated by Hacker’s many specious arguments. Mathematician <a href="http://devlinsangle.blogspot.com/2016/03/the-math-myth-that-permeates-math-myth.html">Keith Devlin</a> and math and statistics teacher <a href="http://alittlestats.blogspot.com/2016/02/the-wrong-way-to-target-math-part-i.html">Amy Hogan</a>, among others, have written posts pointing out other flaws in his arguments and conclusions.) Ironically, many of his arguments would make excellent examples for a <a href="http://www.amazon.com/dp/0809058405/?tag=slatmaga-20">John Allen Paulos–style book about mathematical illiteracy.</a></p>
<p>I am a mathematician, thinking about math brings me great joy, and I want more people to have joyful experiences with mathematics. <em>Of course</em> I think many of Hacker’s conclusions are incorrect. Most troubling to me is the idea that mathematics is important only insofar as we use it in our careers, and therefore anyone whose job path doesn’t involve math shouldn’t have to take math classes beyond basic numeracy. Education isn’t valuable simply because we use it in our jobs. Literature, music, and art enrich our lives and nourish our spirits. History and political science can make us more informed citizens. Science can help us understand why <a href="http://fivethirtyeight.com/features/failure-is-moving-science-forward/">research is rarely conclusive</a>. I reject Hacker’s idea that mathematics doesn’t <a href="http://www.slate.com/articles/life/education/2016/03/algebra_ii_has_to_go.html">help us understand other areas of life and enhance our experience of the world</a>. In her recent <strong><em>Slate </em></strong>piece on Hacker’s book, Dana Goldstein described how her husband sees concepts such as derivatives as connecting the concrete to the abstract, of helping us understand the world. He’s right.</p>
<p>But even <em>if</em> math is only useful for our jobs, few high schoolers know what job they will eventually have. (I wanted to be a herpetologist. And president. And a professional musician. And a biomedical researcher. And a priest.) How can they know exactly what math topics to learn? Fourteen-year-olds who choose not to take algebra II are limiting their future career options, or at least making it much more difficult to catch up if they decide in college that they want to be engineers. It’s impossible for every student to take every class that might help them in the future, but foundational math classes keep doors open.</p>
<p>I’m also nervous that letting children opt out of the more abstract math classes so early will reinforce society’s <a href="http://blogs.scientificamerican.com/roots-of-unity/the-media-and-the-genius-myth/">biases about who can do math</a>, which already do a lot of damage by keeping some groups of people underrepresented in mathematics. If math is seen as something <a href="http://www.slate.com/blogs/future_tense/2012/06/14/stem_gender_gap_research_on_telling_girls_they_re_bad_at_math_.html">boys are better at than girls</a> (despite <a href="https://www.quantamagazine.org/20140812-a-tenacious-explorer-of-abstract-surfaces/">evidence to the contrary</a>), I worry that some girls will pre-emptively self-select out of classes like algebra II, making it less likely that we’ll see parity in STEM careers anytime soon.</p>
<p>As much as the content of his conclusions, though, the arguments Hacker uses to reach them are disingenuous. Over and over again, he relies on the reader’s ignorance or fear of mathematics to make mathematics education sound scarier than it is. These repeated misunderstandings and misrepresentations undermine his credibility. I know much more about math than I do about pedagogy, policy, and other topics he addresses. If a huge amount of what he says about math is incorrect or misleading, why should I trust him on the other subjects?</p>
<p>Throughout the book, Hacker uses jargon to make math topics sound more intimidating. Students, he laments, are asked to master “associative properties.” It sends shivers down the spine … unless you know that the associative property of addition is the one that says 5+1+3=6+3=5+4—that is, it doesn’t matter whether you add the first two numbers together and then the third or the last two and then the first. This is a basic property of addition that most students should learn in elementary school.</p>
<p>Then there’s the word <em>linear</em>. A linear, or straight-line, relationship is practically the simplest relationship two variables can have. A change in one variable leads to a proportional change in another. <em>Linear</em> is basically math-speak for <em>simple and easy to solve</em>.</p>
<p>Early in the book, to show us that he really does think students should learn some basic math, Hacker gives an example of where he would use algebra in the real world: “If a twenty-muffin recipe specifies thirty-five ounces of flour and we want only thirteen muffins, how much flour (<em>x</em>) do we need? The equation we jot down—20 is to 13 as 35 is to x—is elementary algebra, or simply ‘solving for <em>x</em>.’ True, this involves only multiplication and division. Still, every teenager and adult should have this skill and understand the reasoning behind it.”</p>
<p>Later, he uses the word <em>linear</em> to scare us: “a course in linear and quadratic equations” and “linear inequalities” are supposed to demonstrate how out-of-touch math education is with real-world requirements. The thing is, the reasoning behind the multiplication and division he uses in his muffin recipe <em>is</em> a linear equation. The equation 20x=35y, or x=35y/20, might look intimidating, but it’s an algebraic way to represent the amount of flour <em>(x)</em> needed to make a certain number <em>(y)</em> of muffins. When he wants to use the idea of a linear relationship as an example of the kind of numeracy that is necessary for everyone, it is simply basic algebra; when it is an example of out-of-control math education, it is a linear equation. Hacker is pandering to many people’s aversion to mathematics, and it’s dishonest and insulting.</p>
<p>Then there are the outright errors, as in this problem Hacker uses as an example of a question he has given to his numeracy classes. Yes, it’s a word problem. Bear with me.</p>
<blockquote>
A rectangular-shaped fuel tank measures 27 1/2 inches in length, 3/4 of a foot in width, and 8 1/4 inches in depth. How many gallons will the tank contain? (231 cubic inches=1 gallon.)
</blockquote>
<blockquote>
(a) 7.366 gallons (b) 8.839 gallons (c) 170,156 gallons.
</blockquote>
<p>Hacker writes, “The correct answer, 8.839 gallons, requires converting one of the measurements from feet to inches. Too speedy a scanning of the question caused students to overlook that mismatch. Without thinking twice, they might simply multiply 3/4 (or .75) by 27.5 inches and then by 8.25 inches to obtain the tanks cubic volume. If they do, they’ll get the erroneous 7.366 gallons, which the test makers obligingly provide as an answer.”</p>
<p>I puzzled over this paragraph when I first read it. I wasn’t using a calculator, and I’m not terribly swift at mental arithmetic, but I knew something was off. Forgetting to convert feet to inches should lead to an answer that is off by a factor of 12, but 7.366 is quite close to 8.839. The decoy answer should be 0.7366 gallons, not 7.366. It’s a simple case of a missing decimal point. It’s an easy mistake to make, and in Hacker’s defense, it seems the original question also listed the “wrong” incorrect answer. Still, his number sense should have been tingling. After all, one of the goals of the numeracy education he advocates is to be able to eyeball a number and see if it seems reasonable.</p>
<p>Am I nitpicking? Yes—and I normally wouldn’t jump on someone for making minor arithmetic mistakes. (I make plenty myself.) But it’s fair game for an author who holds out his approach to numeracy as an example other schools should adopt. He should have picked up on the error.</p>
<p>More nefariously, Hacker tries to pull some statistical shenanigans in a chapter about whether mathematics enhances our minds. He says 36 percent of people who scored a 700 or higher on the math portion also scored a 700 higher on the critical-reading portion, whereas 44 percent of people who scored a 700 higher or on the critical-reading portion also scored a 700 or higher on the math portion. (Though those numbers are only 8 percentage points apart, he describes 36 percent as “only somewhat over a third” and 44 percent as “fully 44 percent.”)</p>
<p>Hacker writes, “these SAT figures suggest that literary proficiency is more likely to be accompanied by mathematics achievement than the other way around.” Sure, but only insofar as they “suggest” that more people score a 700-plus on the math section than on the critical-reading section. He’s putting the same numerator—people who got a 700 or higher on both sections—on top of two different denominators—people who got a 700-plus on math and people who got a 700-plus on critical reasoning—and acting like it is meaningful that he gets two different numbers.</p>
<p>Let’s say very few people took the SAT this year, and only 100 people got a 700-plus on the math portion. Based on Hacker’s numbers, 36 of them also got a 700-plus on the critical-reading portion. I can use a linear equation if I want to be fancy (basic multiplication and division if I want to be down-to-earth) to see that there must have been 82 people who got a 700-plus on the critical-reading section. This surely says something about the SAT—namely, the math portion of the test is easier than the critical-reading portion for many students—but it’s a silly way to try to make a point about whether mathematics helps students understand other subjects. And if Hacker actually understands anything about statistics, it’s deliberately misleading.</p>
<p>One of Hacker’s soapboxes is that even jobs that use math really don’t use as much as we think. For example, he writes that the mathematical requirements on the actuarial exam “aren’t on the tests to ensure actuarial competence. Rather, they’re included as part of an effort to raise the profession’s stature.” The topics he cites as evidence are Gaussian distributions, Markov chains, Brownian motion, and Chapman-Kolmogorov equations. That sure sounds like a scary pile of terminology, but what does it mean?</p>
<p>The Gaussian distribution he invokes is also called a normal distribution, or a bell curve. You’ve probably seen a picture of it—it’s the natural curve many variables (height, weight, IQ, etc.) take when you plot them. Someone who thinks “citizen statistics” is important should want everyone to be familiar with this curve, whether they call it the Gaussian distribution or not. Actuaries, who deal with statistics in large populations on a regular basis, certainly use this topic in their work.</p>
<p><a href="https://en.wikipedia.org/wiki/Markov_chain">Markov chains</a> are more complicated, but the bottom line is that they model random processes like fluctuations in the stock market or the <a href="http://boingboing.net/2013/06/22/solving-monopoly-with-markov-c.html">game of Monopoly</a>. Markov chains are not used by every actuary every day, but a cursory Google search gives many <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.55.3521&rep=rep1&type=pdf">examples</a> of <a href="https://www.actuaries.org/Boston2008/Papers/IPW3_Spivak.pdf">Markov chains</a> in actuarial science. (The other two topics Hacker mentions, Brownian motion and the Chapman-Kolmogorov equation, are both related to Markov chains, so he’s complaining about Markov chains three times.)</p>
<p>Hacker is either clueless about how actuaries use math, in which case he should not be using them as an example in his book, or he is once again deliberately using people’s discomfort with technical terms to make mathematics requirements sound unreasonable.</p>
<p>Taken individually, each of these examples might seem like an insignificant misstep, but the book is littered with them. I almost hope they’re Easter eggs for numerate people and that Hacker has a secret agenda of improving math education to the point that everyone can recognize that his arguments are full of crap.</p>
<p>Where does that leave us? Few mathematicians or educators would argue that <a href="http://mathwithbaddrawings.com/2016/03/02/why-the-math-curriculum-makes-no-sense/">the math curriculum is perfect or perfectly taught</a>. Hacker is not the first to recognize or call attention to the problem; there are thousands of talented and passionate math teachers working to address the math phobia that permeates our culture and gets <a href="https://student.societyforscience.org/article/parents%25E2%2580%2599-math-anxiety-can-%25E2%2580%2598infect%25E2%2580%2599-kids">handed down from generation to generation</a>, teachers working to make their classrooms places where students will see the utility, beauty, and fun of doing mathematics. Of course we should work to make mathematics education better. But while we consider the options, we shouldn’t let our emotional reactions to math terminology lead us to accept shoddy arguments from Hacker or anyone else.</p>Tue, 29 Mar 2016 09:45:00 GMThttp://www.slate.com/articles/health_and_science/education/2016/03/andrew_hacker_s_the_math_myth_is_a_great_example_of_mathematics_illiteracy.htmlEvelyn Lamb2016-03-29T09:45:00ZAndrew Hacker argues that abstract math is scary, damaging, and should be optional in American education. He should check his calculations.Health and ScienceThis Book Takes a Swipe at the State of Math Education. It Also Gets a Lot of Math Wrong.100160329001educationmathematicsEvelyn LambEducationhttp://www.slate.com/articles/health_and_science/education/2016/03/andrew_hacker_s_the_math_myth_is_a_great_example_of_mathematics_illiteracy.htmlfalsefalsefalseThis book takes a swipe at the state of math education—and gets a lot of math wrong!It doesn’t add up.1519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t478153673800124340692670011519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t478153673800124340692670011519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t478153673800124340692670011519028539001AQ~~,AAAAAASoY90~,_gW1ZHvKG_2pKN0AJTySft1Irx-gT62t47815367380012434069267001Photo illustration by Sofya Levina. Images by Picsfive/Thinkstock and Sonya_illustration/ThinkstockYour Life in Pi
http://www.slate.com/articles/technology/technology/2013/04/pi_meme_on_reddit_and_george_takei_your_life_really_is_encoded_in_its_digits.html
<p><em>"Pi is an infinite, nonrepeating decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever love, the date, time, and manner of your death, and the answers to all the great questions of the universe. Converted into a bitmap, somewhere in that infinite string of digits is a pixel-perfect representation of the first thing you saw on this earth, the last thing you will see before your life leaves you, and all the moments, momentous and mundane, that will occur between those two points.</em></p>
<p><em>All information that has ever existed or will ever exist, the DNA of every being in the universe, EVERYTHING: all contained in the ratio of a circumference and a diameter."</em></p>
<p>From what I can tell, this meme comes from Redditor <a href="http://www.reddit.com/r/AskReddit/comments/zm0ac/reddit_what_is_the_most_mindblowing_sentence_you/c65xvns">kenfoldsfive's answer</a> to the question, "<a href="http://www.reddit.com/r/AskReddit/comments/zm0ac/reddit_what_is_the_most_mindblowing_sentence_you/">Reddit, what's the most mind-blowing sentence you can think of?</a>" It's been bouncing around the Internet for several months now and got a little boost recently when George Takei's Facebook page shared it with nearly 4 million followers earlier this month. It certainly is a mind-blowing idea! Infinity is always hard for our puny finite brains to handle, and I admit that the vastness of irrational numbers blows my mind whenever I accidentally think about it for too long. We're talking about a number that encodes not only my life story, but also a version in which my fly wasn't down the first time I taught a class.</p>
<p>The only problem is, it isn't true. Or, it's probably true, but we don't know for sure, and get off my lawn. At least that's what the <em><a href="http://www.huffingtonpost.com/2013/04/12/pi-meme-misleading-mathematical-constant_n_3056299.html">Huffington Post</a></em> said last week. (On Tuesday it posted a more <a href="http://www.huffingtonpost.com/david-h-bailey/are-the-digits-of-pi-random_b_3085725.html">cheerful story</a> about pi with some interesting illustrations.) The article makes some good points, but I think it misses the forest for the trees. If you look at it the right way, pi really does have it all.</p>
<p>The sticking point is the first assertion of the meme: Does pi contain every possible finite combination of digits? All irrational numbers, including pi, have infinite, nonrepeating decimal representations, but this is not enough to ensure that they include all strings. For example, 0.1010010001 … is a perfectly acceptable irrational number, and it never even includes the digit 2.</p>
<p>In an edit to the original post, kenfoldsfive notes that the statement is true if pi is a "normal number," meaning that every finite string occurs with exactly the frequency you'd expect if the digits were random. For example, 10 percent of digits are 1s, 10 percent are 2s, and so on. (This is for numbers written in base 10. Normality can be defined for binary, hexadecimal, or any other base.) In 1909, mathematician Émile Borel proved that "almost every" real number is normal. The mathematical meaning of "almost every" is more extreme than the typical English meaning. Borel showed that there is basically a 0 percent chance that if you pick a real number truly randomly, you'll get one that isn't normal.</p>
<p>For this reason and analysis of the first few trillion digits of pi, most people who care about such things believe that pi is indeed normal. There's no reason to think it isn't, except that no one has proved it yet. There's a lot we don't know about pi. After all, we only know <a href="http://www.numberworld.org/misc_runs/pi-10t/details.html">10 trillion digits of it</a>, a mere speck in the grand scheme of things. We don't even know whether every digit appears infinitely often. Maybe there are only 10<sup>1,000</sup> 7s.</p>
<p>But the focus on whether or not pi is normal misses an interesting question: Exactly how would we translate an irrational number into a bunch of text? At its heart, the meme is saying that there is something essentially infinite about irrationality that can be used to represent everything contained in our finite world. And that's right, if you choose your "code" correctly.</p>
<p>The meme suggests ASCII, a method of rendering characters using either seven or eight binary digits. (There is a decimal version as well.) Ignoring some technical details about how ASCII is really implemented, let's pretend that every two-digit combination from 00 to 99 encodes a different letter, number, space, or punctuation mark. Then we just go through the digits of pi two at a time and get some string of symbols out. If pi is normal, your life story is in there somewhere.</p>
<p>But there's nothing essential about this method of coding. My friend David Ralston at the mathematics department of SUNY Old Westbury, told me about a different way to extract text from numbers.</p>
<p>Let's say you want to find your life story in pi. We'll assume your life story isn't going to take up more room than the Bible, around 3.5 million characters in English, according to <a href="http://www.missionalchallenge.com/2008/07/how-many-letters-in-bible.html">this guy's grandfather</a>. Close enough for me. (You can pick any upper limit to the number of characters you think you'll need; the process can handle any number.) Now you need to make a list of all possible words that are no more than 3.5 million characters long. I'd recommend writing all the one-letter words first, followed by all the two-letter words, and so on, using alphabetical order at each step. This gives you some huge but still finite list of possible words—let's say K is the number of them.</p>
<p>What we need now is a way to assign each word on this list to a unique chunk of information that appears in pi. In other words, we need to find at least K distinct "things" in the number pi. To do this, we exploit the fact that for any irrational number, there are at least K+1 distinct strings that are K digits long. For example, if you look for strings that are two digits long in the irrational number 0.101001000100001 … , you will find three strings: 10, 01, and 00. For strings of three digits, you will find at least four examples.</p>
<p>Using this fact, we look at the digits of pi in chunks of length K. To the first chunk, assign the first word on your list. To the second distinct chunk (which may overlap with the first; that's OK), assign the second word, and so on. We "only" have K words on our list, and we have at least K+1 distinct blocks of length K, so we'll run out of words before we run out of blocks. If we have more distinct blocks than we have words on our list (we will), we can just start over at the beginning of our list of words. There's nothing wrong with encoding the same word twice. That will happen anyway because some K-digit blocks will show up multiple (and even infinitely many) times.</p>
<p>This process is not as straightforward as ASCII coding, but for any given irrational number, it gives us all possible strings up to some finite length. It might feel like cheating because we've specified what we want to find. But I think it's like finding a needle in a haystack: It's not going to happen if you're not looking for needles. And there is no upper limit to how long the things on your list can be, Ralston wrote in an email. It could be, "for example, all strings whose length is not larger than the number of elementary particles in the observable universe, which is (I think) a reasonable restriction to place on 'all possible words,' and includes the binary code for a very high-resolution JPEG of that awkward moment from your senior prom."</p>
<p>We don't know for sure that pi contains all possible strings of decimal digits, but on a deeper level, the meme is right. And if it gets you pondering the mysteries of infinity, so much the better. You won't be alone. The idea that monkeys sitting at a typewriter would eventually produce the complete works of Shakespeare has been around for decades. To continue bending your mind, I recommend these variations on the theme. Jorge Luis Borges explores the darker side of the concept in his short story <a href="http://jubal.westnet.com/hyperdiscordia/library_of_babel.html">The Library of Babel</a>, and the always entertaining Vi Hart's video for Pi Day 2012 tackles the question, "<a href="http://www.youtube.com/watch?v=uXoh6vi6J5U">Are Shakespeare's Plays Encoded in Pi?</a>" Enjoy!</p>Wed, 17 Apr 2013 20:43:00 GMThttp://www.slate.com/articles/technology/technology/2013/04/pi_meme_on_reddit_and_george_takei_your_life_really_is_encoded_in_its_digits.htmlEvelyn Lamb2013-04-17T20:43:00ZEverything in your past—and future—is encoded in the digits of pi.TechnologyIs Your Entire Life History Really Encoded in the Digits of Pi?100130417011mathmemesEvelyn LambTechnologyhttp://www.slate.com/articles/technology/technology/2013/04/pi_meme_on_reddit_and_george_takei_your_life_really_is_encoded_in_its_digits.htmlfalsefalsefalseIs Your Entire Life History Really Encoded in the Digits of Pi?Is Your Entire Life History Really Encoded in the Digits of Pi?Illustration by Robert Neubecker