The Naked Scientists

The Naked Scientists Forum

Author Topic: Where did the numbers 0 and infinity come from?  (Read 5674 times)

Heronumber0

  • Guest
Where did the numbers 0 and infinity come from?
« on: 03/08/2007 12:29:07 »
Being Heronumber0, I know about next to nothing. So, could you guys please tell me why 0 is a number at all and why infinity exists at all in a rational person's Universe? It doesn't make sense to my simple insouciant mind. [:-[]


 

another_someone

  • Guest
Where did the numbers 0 and infinity come from?
« Reply #1 on: 03/08/2007 13:54:50 »
Zero has at least two purposes.

At its simplest, it is a placemarker for the absence of anything else, and it is the identity for addition.

If you use a positional numbering system (as the Babylonians did, and as we do, but unlike the Romans or Hebrews), then you need to have a value for the position where there is nothing.

In our decimal numbering system, we have numerals 1 to 9 for positive numbers, but then if we wish to express the number 12, we place a 1 in the tens column, and a 2 in the units column; but if we wish to express the number ten, then we have 1 in the tens column, but we need to place a symbol in the units column to denote there are no units over the 10.  Similarly, if we write one hundred and one, then we would write a 1 in the hundreds column, and a 1 in the units column, but we have to now have a symbol in the tens column to say there are no tens, and we use the symbol '0'.

The Romans did things differently.  The Romans had different symbols for their units, their hundreds, and their tens; so it was clear if one group was missing without having to fill the space explicitly with a symbol (the Romans did not use a zero).  In Roman numbers, one was 'I', but ten was 'X', so if you wanted to write 12 you would write 'XII', but if you just saw an 'X' you did not need to see that there were no units over the ten because you had no need to identify the position of the numeral 'X' (in the decimal system, if you see '1' without a zero following it, you think it a '1', but if you see an 'X' without anything following it, you still know it to be a ten).  Similarly, for Romans, the number one hundred was denoted by the letter 'C', so they could unambiguously write one hundred and one as 'CI', and one hundred and eleven as 'CXI', and nowhere did they need to have a placemarker for a missing digit, whereas we need to write '101' otherwise if we just saw '11' we would read that as eleven rather than one hundred and one.

The other function of zero is as an identity for addition.  Although the conceptual idea of zero must be as old as trade, where the idea that if you had a debt of $100, and paid $100, the remaining debt would be zero; but much of early abstract mathematics dealt more with geometry than than with algebra, and so they did not really deal with negative numbers, and so had little need for zero.  It was only with the adoption of Algebra (from the Arabic al-jabr, and it was from the Arabs that the Europeans learnt the modern numbering system which included the zero, although the Arabs themselves learnt it from the Indians) that we started to include the idea of algebraic addition and subtraction, and so had to include the number zero as the value when you subtract a number from itself (or, the equivalent, to add a number to its own negative number).

Infinity is another number that has long been part of the philosophy of humanity, the idea that something goes on forever without end.  The word itself is from the Latin Latin 'infinitas' meaning 'unboundedness'.  The concept is relevant both to geometry (the meeting point of two parallel lines) as well as to algebra, and is key to calculus (but is still a branch of mathematics that continues to develop, particularly in set theory and hyperreal numbers).  The use of the symbol '∞' to denote infinity seems to be credited to John Wallis some time in the late 17th century.
« Last Edit: 03/08/2007 13:58:19 by another_someone »
 

Offline dentstudent

  • Neilep Level Member
  • ******
  • Posts: 3146
  • FOGger to the unsuspecting
    • View Profile
Where did the numbers 0 and infinity come from?
« Reply #2 on: 03/08/2007 14:28:15 »
How did the Romans do mathematics with a letter based numeric system? How did they know that X * X = C? And I'm guessing that they didn't have fractions either, or something analogous to decimals...."I've just cut this gladiator in I/II!"
 

another_someone

  • Guest
Where did the numbers 0 and infinity come from?
« Reply #3 on: 03/08/2007 14:43:42 »
How did the Romans do mathematics with a letter based numeric system? How did they know that X * X = C? And I'm guessing that they didn't have fractions either, or something analogous to decimals...."I've just cut this gladiator in I/II!"

Doing complex arithmetic using Roman numerals is not easy, which is one good reason that once algebra and abstract arithmetic was introduced from the Arabs, the Roman numerals rapidly went into decline amongst the mathematical fraternity.

As far as I am aware, the Romans did not have fractions, but they did use the precursor of fractions, the notion of ratios (i.e. they would not conceive of the of the unitary value of 2/3rds, but they could conceive of the ratio 2:3).  After all, the Romans knew how to decimate things (take 1 part in 10 of them).

The Roman system was quite adequate, and very suitable, for expressing numbers when calculated on an abacus, but their competence in arithmetic would not go beyond that.

Aside from the fact that the Roman system made multiplication more complex, the really big problem with the Roman numbering system is that it is not open ended.  The Romans used the letter M to stand for mille, meaning 'one thousand', but beyond that, the numbering system would start to break down, and the idea of expression trillions and above would simply break the system totally (but then, how many cases would the average Roman need to speak of trillions?).
« Last Edit: 03/08/2007 14:50:04 by another_someone »
 

another_someone

  • Guest
Where did the numbers 0 and infinity come from?
« Reply #4 on: 03/08/2007 15:32:50 »
Incidentally, the issues regarding the move from Roman numerals to decimal number representations were not unlike the arguments put forward when Britain moved from the old £sd system of currency to the decimal system of currency in February 1971 - it was to make computation of money easier with the upcoming computer technology (no need to mix base 12 with base 20 - just do everything in base 10).
 

Heronumber0

  • Guest
Where did the numbers 0 and infinity come from?
« Reply #5 on: 03/08/2007 17:09:44 »
Thanks for that reply George and Stuart.  It just puzzled me and I had asked Mathematicians about it but had not received a suitable reply. So these numbers have been used for reasons of practicality.

Does that mean that they are imaginary numbers then because they don't appear to have real correlates. I had one ice cream C**netto today but I cannot count zero ice creams or an infinity of them...   ???
 

another_someone

  • Guest
Where did the numbers 0 and infinity come from?
« Reply #6 on: 03/08/2007 18:48:15 »
Does that mean that they are imaginary numbers then because they don't appear to have real correlates. I had one ice cream C**netto today but I cannot count zero ice creams or an infinity of them...   ???

They are not what would be formally known as "imaginary" numbers, since that term has a very precise meaning to mathematicians, referring only to number that are multiples of the square root of minus one (√-1).

There seem to be some disagreement as to whether the number zero should be included in the set of natural numbers (some groups regard natural numbers as all positive integers excluding zero, while others regard it as positive integers plus zero).

It is certainly true that zero, like negative numbers, and like non-integer values, are abstract concepts that cannot be used to count things, but because there is a very strict formal definition of what is an "imaginary" number, one should not use that term to refer to either fractions, transcendental numbers, negative numbers, or zero.
 

Offline DoctorBeaver

  • Naked Science Forum GOD!
  • *******
  • Posts: 12656
  • Thanked: 3 times
  • A stitch in time would have confused Einstein.
    • View Profile
Where did the numbers 0 and infinity come from?
« Reply #7 on: 06/08/2007 15:13:49 »
The Romans did actually use fractions; but they used words to represent them. For instance, 3/8 would have been "tres octavae".

They also had the "uncio" (the root of the word "ounce") which was equivalent to 1/12 of an as - a unit of weight. Dividing something into twelfths means you can have quarters, thirds & halves - which makes life a lot easier.

There were symbols for amounts of unciae but they were not standardised and many Western scholars use other symbols instead.
 

paul.fr

  • Guest
Where did the numbers 0 and infinity come from?
« Reply #8 on: 06/08/2007 18:58:12 »
Slightly off track, but i have always wondered why kids are tought their Zero times table.
 

Offline DoctorBeaver

  • Naked Science Forum GOD!
  • *******
  • Posts: 12656
  • Thanked: 3 times
  • A stitch in time would have confused Einstein.
    • View Profile
Where did the numbers 0 and infinity come from?
« Reply #9 on: 06/08/2007 19:19:02 »
Slightly off track, but i have always wondered why kids are tought their Zero times table.

 

The Naked Scientists Forum

Where did the numbers 0 and infinity come from?
« Reply #9 on: 06/08/2007 19:19:02 »

 

SMF 2.0.10 | SMF © 2015, Simple Machines
SMFAds for Free Forums