Naked Science Forum
On the Lighter Side => Famous Scientists, Doctors and Inventors => Topic started by: DoctorBeaver on 19/03/2008 08:01:39
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I assume you are referring to decimal fractions rather than simply base 10 counting numbers (i.e. the extension of positional notation of base 10 number to non-integer numbers).
http://en.wikipedia.org/wiki/Al-Uqlidisi
Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi was an Arab mathematician, possibly from Damascus. He wrote the earliest surviving book on the positional use of the Hindu-Arabic numerals, around 952. It is especially notable for its treatment of decimal fractions, and that it showed how to carry out calculations without deletions.
Ofcourse, the use of positional notation preceded that (the Babylonians used a positional notation, but using a sexagesimal counting system, but it was extended for the use of fractions, as this is a natural extension for a positional numerical notation).
http://en.wikipedia.org/wiki/Kerala_School
The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559-1632). In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c.1500-c.1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.
Their discovery of these three important series expansions of calculus—several centuries before calculus was developed in Europe by Leibniz and Newton—was a landmark achievement in mathematics. However, the Kerala School cannot be said to have invented calculus, because, while they were able to develop Taylor series expansions for the important trigonometric functions, they developed neither a comprehensive theory of differentiation or integration, nor the fundamental theorem of calculus.
In the fields of geometry, arithmetic, and algebra, the Kerala school discovered a formula for the ecliptic, Lhuilier's formula for the circumradius of a cyclic quadrilateral by Parameshvara, decimal floating point numbers, the secant method and iterative methods for solution of non-linear equations by Parameshvara, and the Newton-Gauss interpolation formula by Govindaswami
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So we can blame the Arabs or the Indians.
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I think we should go back to Roman Numerals.
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I think we should go back to Roman Numerals.
And Egyptian fractions?
Incidentally, what do you suggest about removing the antiquated Babylonian sexagesimal system we still use for representing time (OK, so we use Indian/Arabic or Roman numerals, but we still use the same underlying number base)?
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The Babylonian system was superb. 12 was a very important number to them and has the advantage of being divisible by 2,3 & 4, whereas 10 is only divisible by 2 & 5. 12 is much more versatile.
They used the segments of their fingers to count on, touching them with their thumbs. This meant they could count to 24 using just their fingers (3 segments per finger, 4 fingers per hand).
The only reason some people have difficulty with it is that we were all raised on base 10 numbering. Computer programmers often use binary, octal or hexadecimal numbering (base 2, base 8 or base 16) and they don't seem to have too much trouble. I certainly got the hang of hex easily enough and could do basic hex arithmetic in my head.
The problem arises when you try to apply different base numbering to the decimal system. So, 2 x base 12 = 24 whereas in base 12 numbering it would be 20 (with 2 extra symbols for 10 & 11).
Were we all to have been raised using the Babylonian numbering system, counting & simple arithmetic would be no more difficult than with the base 10 that we use.
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Actually, the biggest problem with base 60 is, if one is not using base 10 notation in base 60, the large number of symbols you would have to maintain (would make pocket calculators a bit unwieldy). Actually, not even the Babylonians used 60 discrete symbols.
Many people have long argued in favour of duodecimal (and it is not difficult to scale duodecimal to sexagesimal).
You are ofcourse correct that duodecimal has 3 different factors in its base, on the other hand, it just about balances out when you look at the base number, and those adjacent either side of it (it is fairly easy to determine if something is divisible by 3 or 9 in a decimal system because 3 is a factor of 9, and 9 is one below 10 - the comparable numbers for the duodecimal system are 11 and 13, both of which are prime).
The other advantage often quoted for the duodecimal system is in packaging. We still mostly prefer to buy eggs by the dozen and half dozen. We can now buy eggs by the 10, but we do not package eggs by 5. In many other contexts, it is often very convenient to package things in an array of 4 x 3, where it is far less convenient to package things in the more elongated arrays of 2 x 5. And, ofcourse, we still use duodecimal in many aspects of measuring time (duodecimal is a factor of the sexagesimal system we use, but is also a factor in the base 24 we measure hours in, and we still have 12 months in a year, albeit they are not equal length months).
Ofcourse, as you point out, there is increasingly a practical advantage in using hexadecimal as a base. Binary is very unwieldy (which is why octal and hexadecimal is much preferred as an appropriate balance between using a limited number of symbols while maintaining a more manageable length of number for most common uses).
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You are ofcourse correct that duodecimal has 3 different factors in its base, on the other hand, it just about balances out when you look at the base number, and those adjacent either side of it (it is fairly easy to determine if something is divisible by 3 or 9 in a decimal system because 3 is a factor of 9, and 9 is one below 10 - the comparable numbers for the duodecimal system are 11 and 13, both of which are prime).
I'm not sure what point you are trying to make. 11 is also adjacent to 10. But what difference does it make if the numbers either side of the base are prime?
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You are ofcourse correct that duodecimal has 3 different factors in its base, on the other hand, it just about balances out when you look at the base number, and those adjacent either side of it (it is fairly easy to determine if something is divisible by 3 or 9 in a decimal system because 3 is a factor of 9, and 9 is one below 10 - the comparable numbers for the duodecimal system are 11 and 13, both of which are prime).
I'm not sure what point you are trying to make. 11 is also adjacent to 10. But what difference does it make if the numbers either side of the base are prime?
The point is that if you are regarding it useful to have a greater number of factors in your base because you can easily work out if a given number is divisible by that factor; it is also easily possible to tell if a number is divisible by a factor one greater or less than the base (which is why you can quickly work out if a number is divisible by 3 or 9 using base 10, simply by adding the digits of the number together and determining if the sum of the digits are themselves divisible by 3 or 9).
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I wouldn't have thought that's something an ordinary person would do every day, so it isn't really of any practical benefit.
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I wouldn't have thought that's something an ordinary person would do every day, so it isn't really of any practical benefit.
What 'ordinary' people do every day, who can say.
I have known people who cannot tell you if a number was divisible by 10, and even though they can easily say whether a number was odd or even, they could not tell you if it was divisible by 2.
Most 'ordinary' people are very good at doing what they need to do daily; and most 'ordinary' people were quite happy for centuries just using Roman numerals, or using pounds, shillings, and pence. Those things that they did daily, they did not need to think about, and had no need to generalise. It was the accountants, and later the mathematicians, who did not like the Roman numeral system; and it was the computer industry which supposedly did not like LSD and so forced decimalisation (although one wonders if it was not more political - being only a couple of years before we entered the EEC, and about 8 months before Parliament actually voted the terms for entry to the EEC).
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Although your reply is accurate, it is somewhat tangential to my point.
I can't recall ever needing to know if a number is divisible by 9 and I can't think of any instances where the average person in the street would need to know either - unless they've got 9 kids to share a packet of sweets among.
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Although your reply is accurate, it is somewhat tangential to my point.
I can't recall ever needing to know if a number is divisible by 9 and I can't think of any instances where the average person in the street would need to know either - unless they've got 9 kids to share a packet of sweets among.
It works for 3, not only for 9 (since 3 is a factor of 9).
If you don't need to know this, then why the supposed benefit in a duodecimal system which has 3 as a factor of the base number?
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Dividing by 2, 3 and 4 is fairly commonplace. I was querying division by 9. In any case, in all but the simplest instances, dividing a number by 3 is no slower than summing the digits and then dividing.
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In any case, in all but the simplest instances, dividing a number by 3 is no slower than summing the digits and then dividing.
On the contrary, without using electronic aids, would you find it easier to divide 178232 by 3, than simply to sum 1+7+8+2x2+3 = 2x8+4+3 = 23, which at first glance is not divisible by 3.
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In any case, in all but the simplest instances, dividing a number by 3 is no slower than summing the digits and then dividing.
On the contrary, without using electronic aids, would you find it easier to divide 178232 by 3, than simply to sum 1+7+8+2x2+3 = 2x8+4+3 = 23, which at first glance is not divisible by 3.
It took me less than 3 seconds to do the original division.
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It is called the Dewey Decimal system. It was invented by a guy named Dewey pronounced DOO WEE. Decimals themselves were invented by a woman. She was getting even for the fur coat her husband missed on the last hunting trip.
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It is called the Dewey Decimal system. It was invented by a guy named Dewey pronounced DOO WEE. Decimals themselves were invented by a woman. She was getting even for the fur coat her husband missed on the last hunting trip.
Like this Wiki site explains??
http://en.wikipedia.org/wiki/Dewey_Decimal_Classification
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Nobody ever talks about the number 30 as a base. Its the product of the first three primes and is divisible by 2,3,5,10 and 15. The digits can be expressed easily with the digits we use now. I played with this years ago and used a dot over the digit for the numbers 10-19, and bar over the digit for numbers from 20 to 29. I can't do that here, but using apostrophes we get for example: 0'(10),7'(17),0"(20),5"(25), 10 (30). Its much easier to express fractions: .0'(1/3); .0"(2/3) etc. Quiz: what's 10'4"? (This is a three place number.)
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Of course to us (in the UK - more so in Europe) the imperial system of measurement inches etc is outdated, with 12 inches to the foot, 36 inches to the yard... but traditional engineers do fractions of inches in 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128 etc. So binary fractions of inches!
To go off on a tangent: printing is based on the "point" which is 1/72nd inch. So 12-pt text = 12/72inch = 1/6th inch linespacing.
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I'd like to bring back an old unit of measurement, the barleycorn. It was 1/3 of an inch.