1.2.3 - Early Numeration

1.2.3

Early Numeration

"The Ancient world rarely had need or employed the use of numbers larger than a million"

-- Sbiis Saibian

Previous article: 1.2.1 - Carving the Notch : Dawn of Mathematics

EGYPTIAN NUMBER GLYPHS

The Egyptians had a remarkably straight forward way of notating numbers that is easy for the modern person to understand. They had special "glyphs" (symbols) assigned exclusively for the purpose of numbers.

The Egyptians used a total of 7 glyphs for their notation ...

A Number was simply written out by writing a string of such symbols in succession.

The first glyph , a simple line , represented a single tally mark ...

The second glyph , a bow , represented "ten" tally marks ...

The third glyph , a curly staff , represented ten bows ...

The fourth glyph , a lotus plant , represented ten curly staffs ...

The fifth glyph , a bent finger , represented ten lotus plants ...

The sixth glyph , a falcon , represented ten bent fingers ...

and the seventh glyph , a man with arms raised in astonishment , represented ten falcons or a great multitude ...

In other words, the symbols represented the denominations of ones, tens, hundreds, thousands, ten thousands , hundred thousands , and millions.

In order to represent any number, one would simply write out the symbols whose sum was the appropriate number. The symbols were usually written in descending order starting with the largest from left to right. It was also intended that the least number of glyphs be used in a representation, so one would never actually write out 10 of any single type of glyph, because it could always be replaced by one copy of the next higher glyph.

For convenience, allow me to establish the following ascii form. Let "|" represent a tally mark, "n" represent a bow, "@" represent a curly staff, "<" represent a lotus, "i" represent a bent finger, "&" represent a falcon, and "V" represent a man with arms raised up in astonishment.

So for example, if an Egyptian scribe wanted to write out 523,406 he would have to write out ...

&&&&& ii <<< @@@@ ||||| |

While this system is certainly a big improvement over a simple "tally system", it does have some major draw backs. When denominations have a large number of units, the scribe has to individually write out each unit. This makes some numbers very long to write. For example 999,999 would require 54 symbols:

VVVVV VVVV &&&&& &&&& iiiii iiii <<<<< <<<< @@@@@ @@@@ nnnnn nnnn ||||| ||||

Also this system is limited in the numbers it can express in practice. There is no "ten million" symbol, so in order to write out ten million, you would have to write out ten of the seventh glyph ( Man with arms raised in astonishment) . A large enough number would require more copies of the seventh glyph than a scribe could reasonably write out

VVVVVVVVVV ... ... VVVVVVVVVV (with V Vs) = million million

From the Egyptians standpoint however this was not an important draw back, because numbers in the millions were rare. So rare in fact that the seventh glyph was sometimes used to represent a very large unknown quantity; in other words, "many".

Later civilizations would go along similar lines as the Egyptians in their numeration systems. The popular Roman numerals provide a few interesting innovations on the basic scheme laid out by the Egyptians.

ROMAN NUMERALS

Like the Egyptians the Romans used a set number of symbols to represent various denominations.

They used some of the letters from their own alphabet. The Seven letters they used were ...

I V X L C D M

Again the " I " represented a single tally mark.

The V was not "ten" but "five" tally marks ...

V = IIIII

The X was "two" V's or "ten" tallies ...

X = VV = IIIII IIIII

The L was five X's

L = XXXXX

The C was two L's , or ten X's

C = LL = XXXXX XXXXX

The D was five C's

D = CCCCC

and the M was two D's, or ten C's

M = DD = CCCCC CCCCC

In other words, the Romans had symbols for the denominations of one , five , ten , fifty , one hundred , five hundred , and a thousand. These denominations are still in use today on printed money, which often includes denominations such as five, twenty , and fifty.

Again symbols were combined additively to form a sum representing a number. The use of "half-way" symbols such as V , L , and D helped to shorten and simplify the cases. For example 888 would be ...

DCCCLXXXVIII

X is equivalent to the bow, C to the curly staff, and M to the lotus plant.

Originally the system only worked additively, so that numbers such as 44 would be expressed as ...

XXXXIIII

later in the systems development though, there was a further truncation. If 2 consecutive symbols were written in ascending order instead of the usual descending, then you would subtract the smaller unit from the larger. For example, instead of writing IIII for four, one would write IV . This is what most people are familiar with.

Oddly this rule only applies under specific set-ups. So something like IC which we could interpret as 99 inductively, would not be a legal construction.

Instead only these combination's were used ... IV IX XL , XC , CD , CM

No other ascending combination's were allowed. So the numerals would usually go this way ...

I , II , III , IV , V , VI , VII , VIII , IX , X , XI , XII , XIII , XIV , XV , XVI , XVII , XVIII , XIX , XX , XXI , etc.

using these rules greatly shortened the expressions cutting them in half since each denomination would never require more than 4 symbols.

Still one might complain that they only got as far as the lotus (thousands).

It turns out that they also implemented a way to notate larger numbers much later.

To form new denominations after M, one would write a bar over a symbol to multiply it by a factor of M . So for example V with a bar on top would be MMMMM or five M's. X bar would be ten M's, and so on. This means that M bar would be the same as the seventh glyph (millions). It was also possible to continue "indefinitely" by adding another bar and another as necessary to form newer and newer denominations.

Because I can't use bars I will instead use the slash ( / ) before a symbol to signify multiplying it by a factor of M (thousand).

To illustrate how this could be used to describe some really big numbers here's an example ...

321,875,686 would be ...

//C //C //C //X //X /M /D /C /C /C /L /X /X /V D C L X X X V I

The bar notation is actually a big improvement over the Egyptian glyphs, because it allows one to generate new denominations, where as the Egyptian system doesn't have any convenient way to notate higher denominations than /M ( millions ). Ultimately though, the slashes would themselves become cumbersome when dealing with very large numbers:

/////////////////////////////////////////////////////////////////////////////////////////////////////M

However, most romans had no use for such ridiculously large numbers anyway.

ARABIC DECIMAL NOTATION

Our modern notation for writing numbers is sometimes referred to as Arabic decimal notation on account that it has it's roots in India.

The major draw back of both the Egyptian and roman numerals is that in order to form new denominations new symbols need to be employed.

With the Egyptian notation this would mean inventing new glyphs, which would be a problem. The infrequent use of the more esoteric glyphs, would mean that no standard set would likely be universally accepted. Also the invention of a continuous set of glyphs would eventually exhaust human imagination and ingenuity.

The modern Roman system fairs a little better because it at least provides a tool to create "new" denomination symbols. Still, eventually these "symbols" become cumbersome, such as //////////M which would be a thousand thousand thousand thousand thousand thousand thousand thousand thousand thousand thousand.

Note that neither system has any symbol to represent a denomination of zero. For example 304 in glyphs would be ...

@@@||||

The empty denomination, zero tens, is simply represented by the absence of the ten marks ( bows ).

Likewise Roman numerals would write 304 as CCCIV , and the lack of tens means that there will be no X's or L's appearing in the numeral.

Both of these systems make no use of a zero symbol, or any kind of place holder.

It was the way that Indian mathematicians made use of 'zero' as both a 'number' and a 'place holder' in the sixth century AD that lead to the development of our modern day numerals.

The earliest known text to include a decimal place value notation was in a Jain text in india dated 458 AD. In 628 AD Brahmagupta's book 'Brahmmasputha Siddhanta' he introduces properties of the number zero. The earliest use of unique glyphs for this decimal notation occur in 876 AD.

Arabic numerals were eventually adopted by the Europeans and gradually replaced the use of roman numerals which had been common.

The glyphs we use today, called "digits", namely 0,1,2,3,4,5,6,7,8,9 were a later European development. The original glyphs used evolved over time, and branched out across regions, but ultimately the digits we use today have become the most widely used symbols for numerals in the world.

I'll will now leave the story of it development, and switch to discussing how the current system works.

Our modern decimal system makes use of only ten symbols, called digits. Each digit is assigned a base value.

The chart below displays the digit in the first column, the name of the digit/number in the second column, and in the third column the number of tally marks represents the "base value" of each digit respectively.

DIGIT NAME TALLY REPRESENTATION

0 Zero

1 One |

2 Two | |

3 Three | | |

4 Four | | | |

5 Five | | | | |

6 Six | | | | | |

7 Seven | | | | | | |

8 Eight | | | | | | | |

9 Nine | | | | | | | | |

Note: the digits are listed in order such that each one represents one more than the preceding one. The Zero represents the absence of objects to count, literally nothing, so it is shown in tally representation as the absence of any tally marks.

In addition "Ten" represents one more than nine ( namely | | | | | | | | | | , or the number of fingers on both hands ). Ten is the "base" of the decimal notation. Ten isn't represented by a single unique digit, but is instead thought of as "one group of ten".

An Arabic numeral is represented as a sequence of digits:

ie. 4637890 ... 14595142

The position of each digit determines its value. The positions start with the highest value on the far left and then decrease in value until you reach the far right.

The digit furthest to the right represents the values shown in the table above. The next digit to the left (we'll call it 2nd position ) represents "groups of tens".

So for example 23 means " 2 groups of ten and 3 ones" . Zero comes in handy when their is none left over. For example if you have 3 groups of ten but nothing else it is written as 30 meaning " 3 groups of tens and 0 ones ". The zero in this case is being used as a place holder and helps distinguish it from 3.

The 3rd position represents the next denomination, "groups of ten tens". Ten tens is better known as a hundred.

In general, each position is groups ten times the size as those in the previous position. These positions are usually referred to in sequence as ones (1st position ) , tens ( 2nd position ) , hundreds ( 3rd position ) , thousands ( 4th position ) , ten thousands ( 5th position ) , hundred thousands ( 6th position ) , and millions ( 7th position ).

You might remember the tedious task of decimal expansion taught at school. The exercise is meant to emphasize the positional nature of decimal notation. For example you can expand...

12345

as ...

1x10000 + 2x1000 + 3x100 + 4x10 + 5x1

Reading this out we can say it is " 1 group of ten thousand , 2 groups of thousands , 3 groups of hundreds , 4 groups of tens , and 5 groups of ones "

Arabic decimal notation is highly compact and the size of the numbers increases rapidly with the addition of more digits. In the sequence 1 , 10 , 100 , 1000 remember that the value is being multiplied ten fold with every zero added.

The following link shows the tally representations of various modern numerals to help you better understand the scale of these numbers, even the smallish large ones ...

Direct Representation

The modern numerals however are not without their drawbacks. Firstly, because a digits value depends on its position, in order to express large numbers it is necessary to specify the digit in every position prior to the first non-zero digit. Furthermore their are instances where numbers are expressed more compactly in either egyptian glyphs or roman numerals. Look at the table for example ...

10 = = X

100 = = C

1000 = = M

10000 = = /X