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Zziggy
1141751.  Fri Jul 17, 2015 4:41 am Reply with quote

Natural numbers are counting numbers: 1, 2, 3, etc. The earliest example we have of natural numbers is found on a 30,000-year-old Czechoslovakian wolf bone. It appears to be a quinary (base-5) system, since the 55 notches thereon are grouped into 11 groups of five.

Counting was important to the Ancient Egyptians unlike the Greeks, who paid their way to the underworld, the Egyptians needed to prove they could count their fingers. The Egyptians, though, had improved on that wolf bone by inventing symbols to stand for various values: an arch represented 10, a finger ten thousand, a pharaoh one million, for instance. They were also notable for extending the number system from the naturals to the rationals, though, as they invented fractions.

Although special fractions were represented by particular symbols individual sections of the Eye of Horus Egyptian fractions are most notable these days by their construction, since only fractions whose numerator was one, and denominators distinct, were allowed: for instance, '5/6' would be written as '1/2 + 1/3'. The cumbersome nature of this system was probably the reason that as early as the second century Ptolemy was recommending the use of the Babylonian positional system, even though it was base-60 (indeed, astronomers used sexagesimal notation until the sixteenth century because of this).


Fractions of the Eye of Horus

The decimal point was invented in 1585 by Simon Stevin, private tutor to the son of William the Silent, Inspector of Dykes, Quatermaster-General of the Army and Minister of Finance of the Netherlands. Inspired by the Ancient Babylonians and his distaste for fractions, he published a new system in which each number was followed immediately by its decimal exponent within a circle; e.g. 4.537 would be written

4⓪5①3②7③.

As people got used to this system, they stopped writing the circled numbers, until only the ⓪ remained. This got smaller and smaller, until it became the decimal point we know and love today.

In first millenium Chinese mathematics a system of counting rods was used for arithmetic. When solving an equation, these rods would be arranged in a certain way, with red rods used to symbolise addition and black rods symbolising subtraction. This led to the concept of negative numbers: cheng fu shu, and a negative number would be represented the same way as its positive equivalent, with the addition of another rod placed diagonally over the top. Hindu mathematicians accepted negative numbers, but weren't happy about it the twelfth-century mathematician Bhaskara remarked that a negative solution is "not to be taken; people do not approve of [them]". Western mathematicians simply tried to ignore them, however, with even Descartes refusing to accept negative numbers as roots of equations, calling them "false roots" - indeed, he never even extended his eponymous coordinate system below zero.

On the other hand, perhaps Decartes is not a deserving benchmark it is thanks to his scorn at the idea of a square root of a negative number that we have the term "imaginary numbers". Leibniz, however, was more happy with the notion: he likened i to the Holy Spirit, with their ethereal existence.

If complex numbers are the two-dimensional analogue of real numbers, then quaternions are the four-dimensional analogue of complex numbers. There is also an eight-dimensional analogue Cayley numbers but sixteen-dimensional numbers have been proved impossible.


References:
Zero: The Biography of a Dangerous Idea, Charles Seife*
Taming the Infinite: The Story of Mathematics, Ian Stewart
50 Mathematical Ideas You Really Need to Know, Tony Crilly
From Eudoxus To Einstein: A History of Mathematical Astronomy, C. M. Linton
https://en.wikipedia.org/wiki/Egyptian_fraction


*I purposefully didn't mention zero in this post, because it really merits one of its own!

 
CharliesDragon
1141776.  Fri Jul 17, 2015 7:09 am Reply with quote

In Norway we use a comma and not a full stop as a decimal point.

Actually, the world seems to be quite evenly divided in relation to what decimal mark they prefer.

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

 
Zziggy
1141777.  Fri Jul 17, 2015 7:15 am Reply with quote

Zero - or nought if you prefer.

People were scared of zero for a very long time. Zero represented nothingness, the void, chaos: it was the very definition of evil for mediaeval scholars, who reasoned: God is omnipotent; there is nothing he can't do. But God is the ultimate good, and cannot do evil. Therefore, evil is nothing. This fear came partly from Western society's basis on Greek philosophy zero was anathema to the Greeks.

Zero violates the Axiom of Archimedes, which states that if you add any number to itself enough times it will eventually exceed any other number. Zero makes no geometric sense you can't take a physical ratio of zero to some number and all mathematics was geometry to the Greeks. Most importantly, zero violated one of the most fundamental tenets of Greek wisdom: there is no void. The only time you will ever see zero in ancient Greek texts is within the realm of astronomy, where Babylonian notation was used but they used it as seldom as possible, and neglected it when the results were translated back into Greek.

The Greek rejection of zero lasted two millennia. It was challenged by Zeno's paradox an attempt to prove that the universe was changeless and immobile but, although the atomists came up with a solution which embraced the void, it was eventually Aristotle who won out with his philosophy of the finite universe and prime mover. Zero had become heresy.

In contrast, by the ninth century, India had a symbol for zero, and gave it a place on the number line. They nevertheless still saw the connection to the supernatural: Bhaskara wrote, [the] fraction of which the denominator is a cipher [zero], is termed an infinite quantity [adding a number gives] no alteration as no change takes place in the infinite and immutable God. Nearly a thousand years later, in September 1997, the billion-dollar missile cruiser USS Yorktown was crippled by this piece of philosophy, when its computer system tried to divide by zero and it was left dead in the water.

Zero was finally introduced to the West in Liber Abaci by Fibonacci, who had learned Arabic numerals from Muslim scholars. Zero and infinity started to take hold in Western thought; Aristotle's philosophy was losing its authority, and eventually the church would face the rise of Protestantism.

The Catholic counter-reformation reaffirmed that zero was heresy, but even devout scholars had a hard time accepting this. Descartes, a Jesuit, started his coordinate system at (0,0) but insisted to his death that the void does not exist.

The acceptance of zero and infinity means that today we can talk about black holes in physics and limits in algebra, we can draw paintings with perspective and play with quantum mechanics. The idea of extraterrestrial life is open to us, we can draw fractals ... and a sprinter can overtake a tortoise.


Reference: Zero: The Biography of a Dangerous Idea, Charles Seife

 
Zziggy
1141778.  Fri Jul 17, 2015 7:16 am Reply with quote

(P.S. I'm not sure if I should have put these in the 500 Word Essay sections ... it was full of 'M' topics so I put them here.)

 
Zziggy
1141780.  Fri Jul 17, 2015 7:25 am Reply with quote

Hindu-Arabic numerals were introduced into Europe in 1202 (as noted above, by Fibonacci); in 1299 they were banned in Florence - governments were much more averse to their use than the local merchants.

However, their advantages (particularly the inclusion of zero) were too numerous, and the local merchants continued to use them. As well as business transactions, the number system started to be used for sending secret codes, which led to the word cipher - the name of the number so inextricably tied to the new system - coming to mean code.

Reference: Zero: The Biography of a Dangerous Idea, Charles Seife

 
Zziggy
1141781.  Fri Jul 17, 2015 7:37 am Reply with quote

The golden ratio, or φ, is exactly equal to (1 + √5)/2 and roughly equal to 1.618. It can be written as the continued fraction [1;1,1,1,...]:



It was probably the first irrational number ever discovered, by the Pythagorean Hippasus of Metapontum, since it is found within the pentagram which was considered holy by that group. For the heresy of positing the existence of irrational numbers, the legend goes, he was drowned at sea.

φ* is also the most irrational number. All numbers have a Markov constant , which measures the smallest interval about the number which contains infinitely many rational numbers. The Markov constant of φ is 1/√5, which can be proven to be the largest of any number.


*and its equivalent fractions. One continued fraction is said to be equivalent to another if they eventually agree. So

[1;2,3,6,1,6,4,4,4,4,4,4,...]

is equivalent to

[3;4,4,4,4,4,...],

because eventually they are both just 4 repeating.

 
Zziggy
1141782.  Fri Jul 17, 2015 8:07 am Reply with quote

CharliesDragon wrote:
In Norway we use a comma and not a full stop as a decimal point.

Actually, the world seems to be quite evenly divided in relation to what decimal mark they prefer.

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

Should I infer from that map a teeny tiny possible link to the British Empire maybe ...?

 
suze
1141789.  Fri Jul 17, 2015 8:49 am Reply with quote

Very much so, although it is my belief that - contrary to what is stated in that Wiki article - Latvia uses the point rather than the comma. No idea why, and it's a relatively minor country whose little peculiarity should not be taken as strong evidence against the hypothesis.

There's an International Standard (ISO 4217) which specifies that the decimal mark should always be the comma (and that the thousands separator should always be the point). But the City of London and Wall Street have no intention of changing the way they've always written sums of money, and so global finance has tended not to follow ISO 4217.

It is the official policy of the Republic of South Africa that the decimal mark is the comma, and so a price should be quoted as R 12,34. But AIUI, most English speakers in South Africa use the point and utterly ignore the official policy. Alfred and frantic, how was it for you?

 
Alfred E Neuman
1141792.  Fri Jul 17, 2015 8:54 am Reply with quote

CharliesDragon wrote:
In Norway we use a comma and not a full stop as a decimal point.

Actually, the world seems to be quite evenly divided in relation to what decimal mark they prefer.

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


Technically we also use a comma, but since the advent of spreadsheets, I don't know of many people who actually do use a decimal comma. When talking I'll say five comma three as easily as five point three, but if writing it down, I'll always use a point, even though I used comma at school. I have no idea what they teach in school these days.

 
crissdee
1141797.  Fri Jul 17, 2015 10:34 am Reply with quote

Zziggy wrote:
William the Silent, Inspector of Dykes.....


You surely didn't expect that to slip by unoticed, did you?????

It's a tough job but somebody has to do it.

 
Guest
1141798.  Fri Jul 17, 2015 10:44 am Reply with quote

Oh dear. To clarify, it was Simon Stevin who inspected the dykes.

But no, I definitely didn't expect it to go unnoticed in a place like QI :P

 
Zziggy
1141799.  Fri Jul 17, 2015 10:45 am Reply with quote

Anonymous wrote:
Oh dear. To clarify, it was Simon Stevin who inspected the dykes.

But no, I definitely didn't expect it to go unnoticed in a place like QI :P

^ that was me

 
tetsabb
1141808.  Fri Jul 17, 2015 12:06 pm Reply with quote

crissdee wrote:
Zziggy wrote:
William the Silent, Inspector of Dykes.....


You surely didn't expect that to slip by unoticed, did you?????

It's a tough job but somebody has to do it.


Thinking of applying for the post?

 
Jenny
1141814.  Fri Jul 17, 2015 12:39 pm Reply with quote

Wow those are great posts Zziggy! Very much in the spirit of QI.

 
Zziggy
1141818.  Fri Jul 17, 2015 1:03 pm Reply with quote

:D thanks!

 

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