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Its worth noting that this asymmetry between addition and multiplication happens because our usual representations of numbers (i.e. on a piece of paper or in a
by eigenket 3y ago
Its worth noting that this asymmetry between addition and multiplication happens because our usual representations of numbers (i.e. on a piece of paper or in a a computer) favour addition over multiplication. You can come up with different representations of rationals, for example you could store prime exponents as integers so the number 20 gets stored as (2,0,1,0...) because its 2^2 × 3^0 × 5^1, or 5/6 would be (-1, -1, 1,...).
With a representation like this, or just by taking logs if you like, you can come up with a representation where multiplication and division are very cheap but addition and subtraction are more expensive.
- vikingerik 3y agoNitpick, 50 is 2^1 × 5^2, not the other way around. But anyway, thanks a ton for this post, I felt my brain shift at a new way of thinking about numbers, seriously. I have to wonder what serious and useful implementations of this method might have been done somewhere.
- eigenket 3y agoThanks :) I edited my comment so the arithmetic is (hopefully) right now
- eigenket 3y agoBy the way, if you're wondering if anyone has done anything useful with the method of describing numbers as prime exponents I mentioned the answer is yes, depending on your definition of "useful". The encoding in terms of prime exponents is known as the Godel encoding, and its a fairly important step in proving things like Godel's incompleteness theorems and the undecidability of the Halting problem. Essentially it's a one-to-one map between natural numbers and finite sequences of natural numbers (since you can encode (a,b,c,d...) as 2^a × 3^b × 5^c × 7^d etc), which turns out to be mathematically a convenient thing to have.
- xpe 3y agoI was going to add a less eloquent comment along these lines, but you got here first!
- the-printer 3y agoPlease share it if you can recall. As someone who just figured out what Base-2 and Base-10 meant, I’m eager for all sorts of Mathematica.
- xpe 3y agoStated very broadly, I think of the mathematical “design space” as the set {operations, representations}. Classic examples include: - time domain / frequency domain - lookup table / functional form - traditional binary representation / gray codes
- foota 3y agoMakes me wonder... Is it known whether there is some intermediate representation that makes both fast?
- psychphysic 3y agoYeah you store the log of every number.
- foota 3y agoCould you explain a little more of post a reference?
- psychphysic 3y agoIf A × B = C then log A + log B = log C And if D^E = F then E × log D = log F Mostly a tongue in cheek comment but if you don't know log rules at all check out log tables [0]. And you'll see that if you use log values then multiplication and division become addition and subtraction and equally simply. Relies on you either having stored the needed data or efficiently calculating logirithms and antilogirithims. https://www.cuemath.com/algebra/log-table/ https://www.cuemath.com/algebra/log-table/
- voldacar 3y agoBefore mechanical calculators, large tables of logarithms were computed by hand and used for exactly this purpose.