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The IRS's Favorite Mathematical Law

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When it comes to catching tax cheats, the IRS has more than just federal law on its side. The agency’s arsenal also includes a mathematical truth known as Benford’s law. Armed with this law, the IRS can sniff out falsified returns just by looking at the first digit of numbers on taxpayers’ forms.

While most Americans wouldn’t put it past the IRS to use black magic, the truth behind Benford’s law is far from mystical. In 1938, GE physicist Frank Benford undertook a comprehensive study of numbers and how they occur. His findings mirrored the discovery of American astronomer Simon Newcomb, who had undertaken similar research in 1881. Benford found that when it came to naturally or socially generated data, the distribution of the first digit in a series of numbers is not uniform.

In analyzing 20,000 sets of numbers from a variety of sources—numbers from a newspaper, population figures, American League baseball stats—Benford found that a whopping 30 percent of the numbers in his sample had one as their first digit. The numeral two turned up in the first position 18 percent of the time, and three occurred 12 percent.

There’s a simple explanation for what Benford observed. In the number set 0 to 99, 11 percent of the numbers start with 1, and 11 percent start with each digit from 2 to 9. In the number set 0 to 199, over half of the numbers start with 1, and less than 6 percent start with 2 to 9. In the number set 0 to 299, 37 percent start with 1 and 37 percent start with 2, and the numbers 3 through 9 start 3.7 percent each. This situation goes on forever, so over a large enough data set, the distribution of leading digits follows a predictable pattern. The bigger the integer, the less likely it is to be the first digit in a data set.

How does the IRS use this distribution? Many folks who happen to fudge a bit on their tax returns or expense reports call attention to their creativity by using too many dollar amounts that start with an eight or nine (the least common integers found in the first position) and not enough that start with numeral one. Savvy CPAs know what to look for, and many computer systems that tabulate figures are also programmed to catch any suspicious strings of numbers.

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History
When Math Discoveries Led to Banned Numbers
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The literature world has seen more than its share of controversy. The best stories tend to provoke the strongest reactions—both positive and negative—in readers, which is why so many classic books have been banned at one point or another. But even a more objective field like math isn’t immune to conflict. In its new video, TED-Ed rounds up the numbers that caused such a stir when they were introduced that they were banned in math circles.

One of the earliest examples comes from ancient Greece. A mathematician named Hippasus was having trouble solving certain equations with fractions and whole numbers alone, so he came up with irrational numbers to make these values easier to express. The ruling school of thought at the time dictated that everything in nature could be explained elegantly with the numbers that already existed. Threatened by Hippasus’s new notion, his fellow mathematicians rejected the irrational numbers and had him exiled.

Other numbers have been banned for legal reasons. When Arab traders brought their positional number system, which included zero, to Italy in the Middle Ages, Florence banned it from record-keeping fearing that they would be easier to forge than Roman numerals. The Arabic way of counting also led to the rise of negative numbers, which were regarded with disdain by many experts into the 19th century. For more banned numbers, including some that are prohibited today, check out the full story below.

[h/t TED-Ed]

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Live Smarter
An Ex-Google Engineer Just Reinvented the Measuring Cup
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Euclid

Recognizing a problem most people didn’t even know they had, former Google and Facebook software engineer Joshua Redstone has made a bold claim for his recent Kickstarter venture: He’s developed a better measuring cup.

According to the Boston Business Journal, Redstone spent four years tinkering with a solution to something that had long annoyed him as an amateur chef: Traditional measuring cups, which are stocky and not very well tapered, don’t do a great job of accurately measuring their own contents. Redstone believes the shape of a cup determines its success, particularly when a cook overfills a liquid or solid by a tiny amount. The smaller the volume, the more the problem is magnified.

Euclid

Redstone’s cup, Euclid, resolves the issue. According to the Kickstarter page: “With traditional measuring cups, the smaller the amount, the harder it is to measure accurately. The culprit? The shape. Straight sides magnify errors when measuring lower down in the cup. Some have tried to solve this problem with conical measuring cups, but their results fall short of Euclid’s by up to 60 percent. Euclid is the only measuring cup with a mathematically optimal, tapered design for consistent accuracy across amounts.”

Euclid is just about ready to overshoot its $30,000 Kickstarter goal. Backers can pay $24 for the cup now, or wait until it’s available at retail for a slightly higher price to be determined. The cup is scheduled for release in May 2018.

[h/t Boston Business Journal]

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