## How Dirac Made Every Number

A couple weeks back I offered a challenge taken from Graham Farmelo’s biography (**The Strangest Man**) of the physicist Paul Dirac. The physicist had been invited into a game to create whole numbers by using exactly four 2’s and the normal arithmetic operations, for example:

While four 2’s have to be used, and not any other numerals, it’s permitted to use the 2’s stupidly, as every one of my examples here does. Dirac went off and worked out a scheme for producing *any* positive integer from them. Now, if all goes well, Dirac’s answer should be behind this cut and it hasn’t been spoiled in the reader or the mails sent out to people reading it.

Dirac’s formula, according to Farmelo, is this:

You should have n square roots in that innermost logarithm; that is, for example,

This is a magnificent solution, obviously, and it’s a good problem for your pre-algebra class to show *why* it works, at least for some particular number. It does make a slightly stupid use of one of the 2’s, since there’s no discernable difference between and , but that’s within the rules.

The system can be extended to other whole powers of two, so if you want to build any integer using exactly four 4’s, or 16’s, or 1024’s, you’re in luck, but I admit I don’t see how this gets you to building any number out of a string of four 3’s. The square root symbol can be traced back to the early 16th century (and a forerunner, RR, to Leonardo da Pisa in about 1220), and extensions of it to take cube or fifth or other roots have always been available, but by writing the number of the root somewhere around the base symbol. Dirac’s scheme works for 2 (and powers of 2) because a square root symbol by itself carries that 2 implicitly; cube roots require the 3 be written explicitly.

## delarsea 9:29 pm

onFriday, 18 April, 2014 Permalink |Reblogged this on How and Why?.

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## nebusresearch | The Math Blog Statistics, April 2014 1:35 pm

onThursday, 1 May, 2014 Permalink |[…] How Dirac Made Every Number, the answer to that puzzle of how to construct any counting number using precisely four 2′s and ordinary operations (it’s a forehead slapper once you’ve seen it) […]

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## Reading the Comics, March 31, 2015: Closing Out March Edition | nebusresearch 6:44 pm

onFriday, 3 April, 2015 Permalink |[…] and John Newcombe’s Zack Hill (March 29) passes on one of the famous quotes attributed to Paul Dirac, that if there is a god he must be a mathematician. Wikipedia attributes this to a May 1963 […]

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## abyssbrain 2:54 am

onFriday, 10 April, 2015 Permalink |Nice post. I’m not familiar with the history of this problem and that it’s attributed the Dirac. Though I have seen the version of using the 4’s to produce every number.

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## Joseph Nebus 3:58 am

onFriday, 17 April, 2015 Permalink |I don’t know that the problem traces to Dirac, but at least a biography of him did credit him with a solution using exactly four 2’s. Using 4’s to make numbers is another wonderful puzzle, though.

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## Any Requests? | nebusresearch 3:00 pm

onSaturday, 30 January, 2016 Permalink |[…] this brings things back around to how Paul Dirac worked out a way to produce any whole number using exactly four 2’s and the normal arithmetic operations anybody […]

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