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Base ten makes a lot of sense as a numbering system, given the number of digits humans typically have on their hands.

That said, some older money systems weren't based on the number of fingers we have, e.g. pounds, shillings, and pence. Similarly, a day is divided into 24 hours, not ten. Both of these suggest perhaps some cultures may have used alternate numbering systems in the past.

Have any non decimal numbering systems been used frequently in the past? If so, what were they?

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  • $\begingroup$ Could you clarify "mathematical". Eg: My example only definitively refers to sheep counting. $\endgroup$ – winwaed Oct 29 '14 at 20:05
  • $\begingroup$ I've removed the word. Counting is counting... $\endgroup$ – yochannah Oct 29 '14 at 20:13
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    $\begingroup$ Possibly of interest: The Number Concept: Its Origin and Development by Levi Leonard Conant, describes the counting systems used by an enormous number of cultures and languages. $\endgroup$ – MJD Oct 30 '14 at 17:04
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    $\begingroup$ There is a meta discussion about whether this is on topic. $\endgroup$ – BMS Oct 30 '14 at 22:26
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To this day many people use various systems besides the decimal one. I was surprised to read that "Old Babylonians used the system based on 60":-) Do not we all use their system today? Not only for time but for angle measurement?

When the French revolutionary government introduced the decimal system as a standard, they also tried to reform the angle and time measurement. 10 hours in a day and 10 in a night, 100 minutes in an hour, 100 second in a minute. For the angles: 100 degrees in a right angle, 400 in a circle, each degree is 100 minutes of arc, etc. This is why kilometer was defined as 1/400000 of the meridian. It is nothing but the "decimal minute". (The nautical mile is one "Babylonian minute" of the meridian). Clocks, watches and angle measuring instruments with decimal scales can be found in museums.

But this did not work. So we still use the Babylonian system.

Now, how many inches are there in a foot? How many ounces in a pound ? Not everyone uses decimal system today.

An interesting system is used for angle measuring in artillery. The circle is divided into 6000 parts. This is convenient for quick estimation of the distance to an object of known size, using the crude approximation pi=3.

There was an attempt to introduce the system based on 8 for everyday use. (Charles XII of Sweden). This did not work.

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  • $\begingroup$ Of course we still use base 60 for time and angle measurements. What I wanted to say by that statement was that it comes from Babylonia, so what I in fact meant was 'already old Babylonians used base 60' $\endgroup$ – Ondřej Černotík Oct 30 '14 at 7:50
  • $\begingroup$ I wanted to emphasize that not only in time and angles. Simple (not decimal) fractions are used in screw sizes, and in stock price quotations on NY Stocks exchange. And English measurement system is certainly not based on the decimal system:-) $\endgroup$ – Alexandre Eremenko Oct 30 '14 at 18:35
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Old Babylonians used base 60 for calculations. (This is also where 60 seconds in a minute and 60 minutes in an hour come from.) According to Wikipedia, the main advantage of this was that it made practical calculations rather easy due to the number 60 having many divisors. Their mathematics was generally developed for the time but that would make up for a completely different question and answer.

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The Celts (ie. Iron Age Britons and their descendents) are reputed to have used a base 20 system. This is meant to be the origins of the Yan Tan Tethera counting system used by many upland sheep farmers well into the 19th Century and surviving today in a few places such as Swaledale. See http://en.wikipedia.org/wiki/Yan_tan_tethera for a list of the variations.

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The French language still has the residue of a base-twenty system. Their word for 80 is 'quatre-vingts', which literally translates to 'four twenties'.

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    $\begingroup$ English (and German even more) still has the residue of a base twelve system. Think at eleven, twelve and elf, zwölf which are specific constructions. $\endgroup$ – mau Oct 29 '14 at 20:32
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    $\begingroup$ Not to forget the dozen (and their equivalents in other languages, see link). $\endgroup$ – Wrzlprmft Oct 29 '14 at 22:15
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Maya used a vigesimal system too. Note that many decimal systems use a quinary subsystem: think at the Latin numerals, but also at the Japanese abacus.

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  • $\begingroup$ by quinary subsystem, you mean that 5 is significant too? $\endgroup$ – yochannah Oct 29 '14 at 21:28
  • $\begingroup$ Yes, in a certain sense it is an auxiliary basis $\endgroup$ – mau Oct 29 '14 at 22:02
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The binary system is widely used today, and hexadecimal and octals numbers have gained a lot of attention due to their proximity to the binaries. This is due to computers working mainly with boolean values, which makes it natural to count in base 2 when dealing with such machines.

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    $\begingroup$ It used to be the case for Octal as well. $\endgroup$ – Ali Caglayan Oct 29 '14 at 20:23

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