decimal

1.2.5

The Advent of Modern Decimal Notation

Introduction

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ORIGINAL TEXT

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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