Another possible answer is: in base 60 you would have 60 digits (from 0 to 59), and you would have to créate a symbol for each of them... Not very practical, is it? 

Arunagirinathan,

>Why we use base 10 almost everywhere 

I look at my hands, and I wonder....

>than base 60 which was first invented method?

I don't think that that is provably true.

Egyptian records suggest a base 10 system in use ~3000 BCE.

Earlier systems may have existed, but I know of no such records.

>Suppose we change time and angles from base 60 to base 10 in all places

>then what will happen?

A lot of programmers and clock makers will become very rich.

But for what point? A needless coherence with other systems? Ask the British how that has worked for miles and pints.

   Another possible answer is: in base 60 you would have 60 digits (from 0 to 59), and you would have to créate a symbol for each of them... Not very practical, is it? 

Raul,

Indeed - calculators would need to be rather large. And numeric keypads, and telephone button arrays, etc.

But decimal time and angles wouldn't be so hard. After all, we already have the gradian.

If we would have 2x6=12 (=60/5) fingers instead of 2x5=10 our base system would had probably arrived at base 12 instead of base 10. Decompostion of 12 into primes leads to 2 and 3, whereas decompostion of 10 leads to 2 and 5. Concerning mental arithmic base 12 might be favourable. So a next question to ask would be why do human beeings unfortunatly have 2x5=10 fingers instead of 2x6=12 fingers.

And if we had 8 digits, 4 on each hand, like cartoon characters, 3 fingers and a thumb, we would likely use some form of Hexadecimal notation, making understanding of binary number schemes, powers of 2, practically instinctive. This would make the learning curve for machine language programming easier. But alas, we have 10 digits on our hands.

Why we do not go back to base 60 has been largely answered. The point that seems unanswerable is the large number of digits (note, by the way, that Babylonian ``digits'' were, in fact, composite, being made up of elements indicating units and groups of five).

But the other half of the question could be: why do we not decimalise everything, including time and angles? One answer: it has been tried. In the French Revolution, a day of ten hours, with an hour of a hundred minutes and a minute with a hundred seconds were introduced.

I guess the main reason this failed was the introduction of a week with ten days: man needs more rest than such  decimal system allows for.

Two funny mystic games with physical constants and physical theories:

Consider some of the most important physical constants and calculate the digits in some unusual bases and wonder about regularities and irregulaties of the sequences of digits.

Consider the question wether there exists something in (theoretical) physics which is associated with a SU(n) symmetry , n=2,3,4,5,6,...,10,11,12,... .

This might give a hint to the question of what bases could be considered as meaningful (but unhandy) with respect to physics.

Dear Karl G. Kreuzer, please give more details for your answer.....

Hi   Arunagirinathan ,

SU(n) groups (and other groups) and their representation by complex-valued matrices (or by operators in Hilbert spaces) are important in the theory of classical fields and quantized fields which try to describe the physical behaviour and properties of elementary particles in scattering experiments and which try to find some kind of 'unification' of the different interactions ('forces') between elem. particles.

Maybe

https://en.wikipedia.org/wiki/Special_unitary_group  

and

https://en.wikipedia.org/wiki/Quark_model

can help you to find an appropriate introductory textbook about this specific area of mathematics and mathematical physics. 

An excellent book for those who want to become an expert in this specific area: "Theory of Group Representations and Applications" by A.O.Barut and R.Raczka (1977,1980,?) 

The ones who invented the base 60, actually used a base 5 system, for notational purpose.  That has survived in the use of dozens, which is a fifth of 60.  The question could be as well about the binary system.  In fact, it has been shown that the most efficient system is base 3, taking into account the number of symbols and the number of digits.  Some others think that the better system is base 12 because this number has many divisors, but that's useful rather for manual calculus.  It was probably one of the reason of the base 60 system: 3*4*5 = 60. One other is that there are 6x60 days in a year.

Claude.

You have mixed Babylonian base 60 and Egyptian base 5 incorrectly.  Sumer used base 10 before Babylonians created a cursive base 60 that rounded off 1/91 to 1/90. Upper Egypt used base 5 before merging with Lower Egypt base 2 to form the Old Kingdom to a cursive base 10. Sub-Saharan Africa used a fractal form of base 2 in ways that infinitely extended two parent families into spiritual home villages.

After 2000 BCE Egypt replaced its cursive base 10 that rounded off,  1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 in weights and measures by throwing away a 1/64 unit, to a finite base 10 number theory base arithmetic, algebra, geometric plus arithmetic and geometric progressions finite system used by Greeks and Hellenes until 800 AD.

Milo,

I do not know about the rest, but the Greeks definitely did not use base 10: their system was, for everyday use, similar to Roman numerals: they used letters with different values, which were concatenated without using a place value system.

For astronomical purposes, on the other hand, as in Ptolemy's trigonometric tables, the base sixty system was fully used, with positional value and even a well defined zero.

because we have 10 fingers :)  It's much more practical. Also mathematically.

Hi,

Civilisations seldom use a unique base for their calculations. We all use base 60 and 10. When I was a boy I had to learn base 12 for counting money when going to England.

>Suppose we change time and angles from base 60 to base 10 in all places then what will happen?

It works. French people and others in Europe lived with a full decimal system at the time of the first Republic 1792-1804. One day was 2 times ten hours! One week was 10 days! All this stopped for political reasons, not for practical ones.

The metric system systematized the use of decimal system but it did not succeeded in changing the way we specify calendar days or the time of the day or the standard ratios in lenght in the British Imperial Units.

''the Chinese as early as 1400BC introduced a bamboo stick decimal numbering system and developed the abacus based on this system. Later these designations were replaced in the west by the Hindu-Arab symbols for one through nine with which we are all familiar today. Also the concept of zero represented by the Chinese as a blank space was replaced by the clearer representation of 0 in the Hindu-Arab system.''

http://www2.mae.ufl.edu/~uhk/DECIMAL-SYSTEM.pdf

''For zero to fulfil its potential in mathematics, it is necessary for each number up to the base figure to have its own symbol. This seems to have been achieved first in India. The digits now used internationally make their appearance gradually from about the 3rd century BC, when some of them feature in the inscriptions of Asoka.

 The Indians use a dot or small circle when the place in a number has no value, and they give this dot a Sanskrit name - sunya, meaning 'empty'. The system has fully evolved by about AD 800, when it is adopted also in Baghdad. The Arabs use the same 'empty' symbol of dot or circle, and they give it the equivalent Arabic name, sifr.''

http://www.historyworld.net/wrldhis/PlainTextHistories.asp?historyid=ab34#ixzz43CLm7MYN

Greeks did use base 10 scaled rational numbers that closely followed Egyptian multiplication scales rational numbers such that

n/p times m/m = mn/mp, with the best divisors of mp that optimally summed to mn in a unit fraction series, written from left to right in ciphered Ionian or Dorian alphabets, with only minor changes made to the traditional Egyptian hieratic notation.

For example, the 300 BCE Hibeh papyrus scaled units of one day in n/45 tables, with 2/2 and 4/4  being the only scaling factors. At other times, the Akhmim Papyrus, about 400 AD Coptics wrote n/2 to n/32 tables that followed Ahmes style of writing 2/n tables with n = 2, ..., 101, that accepted small and large m/m scaling factors . 

After 800 AD ciphered numerals ended, with subtraction substituted for multiplication in the Arab and medival texts written by Fibonacci and others., by seven methods, distinctions per LE Sigler. That is,

(n/p - 1/m) = (mn - p)/mp set (mn - p) = 1 as often as possible to define 2-term 

unit fraction series. When impossible, ie 4/13 selected 1/4 and 1/18 that reached

4/13 = 1/4 + 1/18 + 1/468, 

written from rom right to left in the Greek and Egyptian manner.

Hi,

The Babylonians, who were, by the way, quite competent in manipulating numbers, in geometry (they came very close to calculate pi) and more, used base 60, which is, mathematically speaking, more powerful than base 10 system.  Based 10 was introduced solely due to an error by nature that evolved us into animals with a total of 10 fingers in our hands.  There is no other reason.  It is among the poorest choices for a base.  Even 12 is a lot more powerful base than 10.  But we are stuck with it.

Base 2 is more powerful than base 60. Ask any computer engineer that is a member of IEEE. Babylonian base 60 was used by Ptolemy and the Almagest, set the locations of stars, planets, and our place on our planet. The 24 length of the day came from Egyptian sideral time developed around 1100 BCE. Mixed numeration systems seem to have marked the development of science for 5,000 years.

Hello,

Base 2 is used  in computer because of Boolean algebra,  not because it is powerful.  In fact, many theoretical researchers have been trying to come up with non-binary logic that could be used in a different generation of computers. So far, the effort has not been very successful.  (There are, to be sure, non-binary logic, but they are not amenable to be used on the existing hardware technology.)

The machine language code I learned in the early 1960's used zero and one (0 and 1) with no Boolean logic. Fixed and floating points were awkward during the early years, easily overflowing computer memory. Only with hard work has the binart on and off logic improved by the IEEE community, Joe Ascoly of Aberdeen proving grounds being one. With the introduction of parallel computers, the Chudnovsky brothers from Columbia, working with IBM s think tank was speed up 1,000 times by considering a little Fermat 2^n the relationship... No Boolean logic there either...

0 and 1 IS Boolean logic.  Does it matter whether you call it 0 and 1, or true and false, or white and black.  Floating points were stored in a table.  That is why you cannot compare a floating number against 0.  Furthermore, as far as I know, during Manhattan project computers were used in the mid forties  to aid solving very large number of equations.

By the way, it is a bit awkward to understand you.  Are you sure you worked on computers in USA?

Arsalan,

Thank you for your challenges. The logic of any given program is contained in the external structure of solving specific problems, be the language Fortran IV, equation based, COBOL, business and language based, or modern JAVA or HTML. 

Having taken the MIT computer memory and computer construction course, online, internal algorithms must work in both directions. For  example studying ogami, the Japanese folding math; reverse engineerings particular shape can be easier than that algorithm needed to point out folds that otherwise are irrational that are hard to see, are more important today than the Boolean internal logic of early computers.

Look up "Little Fermat", and Greory and Albert  Chudovsky, brothers from Columbia with math PHDs that sped up supercomputers 1000 times, and you may find the 1987 innovation improved the internal operating system in an unexpected manner.

Best Regards,

Milo (mi espousa es Hispanic, so Hola)

It's simply a matter of convention-there's nothing particularly profound about any base. 

Most of the above answers are good but it seems clear to me that these simple questions are not being posed to students at a young enough age. There is much learning that can be achieved at an earlier age if such questions are considered. I agree there is particularly nothing profound about any base. However, the point is that a base is both a linguistic and logical decision used to optimize both processing and communication of information within a given context.

As a side note, the prime factors of a chosen base determine which rational values have either one or two place-based representations. If two then one is with finitely many non-zero characters and one is with infinitely many occurring in a repeated pattern. If only one then it is of the latter form. I believe such observations have educational value at a younger age which could improve both linguistic and logical skills later in life. If that is not a fundamental goal of education, I don't know what is.

Actually base 10 recursive was used before base recursive in Babylon. Egyptian finite base 10 unit fraction numeration systems were continuously used from 2050 bce until base 10 decimals emerged as a reaction to the Ottomans ended the Byzantine Empire in 1454 AD. When developed base 10 decimals merged finite and infinite approximations .

That is, number theory was known in Egypt by its abstract nature. Greek, Arab and medieval finite unit fraction math followed Egyptian abstract math, primarily in mental ways, leaving hard to read scribal shorthand notes that scholars too often skipped over or grossly muddled and under valued by suggesting only additive and practical transliterations.

To parse the actual abstract nature of 22/7, and it's arithmetic and algebra used by Ahmes and how these abstractions were first calculated by Egyptians and improved upon by Greeks like Archimedes inverse proportions and the binomial theorem must be mastered per

https://www.academia.edu/10608644/Archimedes_Square_Root_of_3_5_6_7_and_29_and_Sulba-sutra_Square_Root_of_2

Egyptian rational numbers stated in finite unit fraction series were exact a fact muddled by Babylonians that rounded off 1/91 to 1/90 . For example, Ahmes selected LCM 70 to scale 2/91 per

2/91 (70/70) = 140/((70)(91)) = (91 + 49)/((70)(91)) = 1/70 + 1/130

Meant,

1/91 = 1/2(1/70 + 1/130) = 1/140 + 1/260

exactness that Babylonian arithmetic and algebra never attained.

Summing up, there were many irrational numbers that were approximated by the 4,000 year old inverse proportion square root method, a method needed to solve algebra''s fundamental theorem, nth degree equations have n roots, as Gauss proved in 1796 , listing all 2nd degree equation roots by adding complex numbers to our world, while earning his PhD.

Today we oddly merge rational and irrational numbers into a thoughtless base 10 decimal system, a clone of Babylonian thinking built upon the mapping of the binomial theorem and coefficients from Pascal's Triangle by an algorithm , facts exposed by Oystein Ore. Number Theory and it's history a must read.

MIlo Gardner