Place Value

Laplace

“The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.”  (Laplace 1814, quoted in O’Conner et al)

Number Systems

A number system is a method for writing numbers, using numerals or other symbols in a consistent manner.

The Roman number system is illustrated below. In its development it demonstrates a number of different types of number system. In its simplest form it is an additive system. That is the values of the basic Roman numerals are simply added together, so MLXVI (1000+50+10+5+1) represents the Hindu-Arabic number 1066. In principle, with additive number systems the order of the numerals is not important, but in the Roman system numbers are usually written with largest and smallest numerals going from left to right.

/Users/grahamshawcross/Documents/blog_drafts/places_values/Roman

When subtraction is optionally used, order does become important, so VI is 6 but IV is 4, XL is 40, XC is 90, CM 900 etc. This convention does not form part of the Roman method of calculation but is an aid to a more compact written representation. See Five Finger Exercises.

The subtraction pairs CM (1000-100) and XL (50-10) boxed above, are in effect surrogate numerals that need to be evaluated before being added to the other numerals.

The supplementary nature of subtraction can be seen in the simple Python program below that, without any checks, takes a number represented by Roman numerals and converts it to the equivalent Hindu-Arabic representation. It first adds the values of all the numerals together and then deducts any subtractions twice over to prevent double counting.

arabic

As shown in the diagram above multiplication is used to represent larger numbers. In some periods a horizontal bar above a group of numerals is used to multiply the bracketed numerals by 1,000. At various other times vertical bars are used as brackets. So |V| is 5,000, |X| 10,000, |XIV| 14,000 and |XL| is 40,000. Sometimes both horizontal and vertical bars are used to multiply by 1,000 x 1,000 or 1,000,000.

Placeholders

The Babylonian Number System is base 60 or sexagesimal. To avoid having separate numeral symbols for the numbers 1-59, it uses a base 10 system to generate these numerals. As shown below the 1-9 numerals use the appropriate number of copies of the units glyth. The 10, 20, 30, 40 and 50 numerals use the appropriate number of copies of the tens glyth. All the other numerals are made up from pairs of units and tens numerals, so for instance the 42 numeral consists of the 40 numeral (4 tens glyths) paired with the 2 numeral (2 units glyths). Although the numerals have a strong and memorable pattern on average they have 7 glyths each and at worst 14. See Subitising for relevance of this.

/Users/grahamshawcross/Documents/blog_drafts/places_values/dwgs/

Numbers below the base 60, are simply represented by the 1-59 numerals themselves. Numbers above 59 are represented by using the 1-59 numerals to indicate the quantity of 60s. This numeral is placed to the left of the units numeral. Similarly for numbers above 359 a numeral representing the quantity of 36os (60×60) is placed to the left of the 60s numeral.

Given that 60 to the power zero (600 ) is 1, the numeral in the first right-most place or position represents the quantity of units. The numeral in the second place, to the left of the first place, represents the quantities of 6os or 601. The next place to the left represents the quantities of 360s or 60 x 60 that is 602 .

So each place, starting from zero on the right and moving to the left, represents increasing powers of the base of the number system in the Babylonian system, base 60.

Document1

In some cases as in the Babylonian version of 3609 (1 x 602 )+(9 x 60), illustrated above, there are places without a value, the second (60) place in this case. Initially this was dealt with, as in grid counting systems, by just leaving a space. This was a source of error and possible fraud so later a placeholder symbol was put in places where there were no values to be recorded.

However in the Babylonian number system the placeholder was never used in the first place or position, as it would be in the Hindu-Arabic decimal number system to differentiate say 7, 70 and 700 where zero is being used as a placeholder in the first place or position.

So as in the examples in the diagram it is difficult to differentiate (1 x 602 )+(9 x 60) from (1 x 603)+(9 x601) but they represent greatly different values, 3,609 and  24,840 respectively. It is probably this difference in value that  obviated the need to use placeholders in the first position.

Place Value Systems

The migration of the placeholder symbol to a true zero numeral that can be used in all places represents the arrival of place value systems. Our Hindu-Arabic base 10 number system is an example of such a place value system.

/Users/grahamshawcross/Documents/blog_drafts/places_values/dwgs/

The base of a number system determines how many numerals that system requires. So a base 10 place value system requires ten numerals as illustrated above and a base 2 or binary system requires just two, 0 and 1. As shown earlier the Babylonian base 60 system has sixty numerals, if the placeholder symbol is included.

A place value system can be used with any base, the Mayan number system for instance is base 20 and has a genuine zero symbol. As shown below the Mayans wrote their numbers vertically with the first place at the bottom and succeeding places moving upwards.

/Users/grahamshawcross/Documents/blog_drafts/places_values/Roman

Using a dot to represent units and a bar to represents fives, means that for each numeral there are never more than two sets of four glyths to be subitised.

Discussion

As Laplace thought, place value number systems are difficult to explain, yet culturally extremely important; perhaps as important as the invention of the alphabet.

At the heart of the problem might be the difficulty in understanding that any number to the power zero is 1. This allows numerals in the first place to represent numbers up to the base of the number system. So there are 2 numerals in a binary (base 2) system, 10 in a decimal (base 10) system, 20 in the Mayan (base 20) system and 60 in the Babylonian (base 60) system if one counts their placeholder as a numeral.

Calculating systems such as Chinese rod counting, Roman grid calculating and the abacus have no problem representing zero without placeholders or a symbol for zero and use position to indicate powers of the base being used including, in the base 10 example below, negative powers for tenths, hundredths etc. Here vertical rods are units and a horizontal rod five units, again reducing the number of glyths required to five or less, that is within the subitising range.

/Users/grahamshawcross/Documents/blog_drafts/subitising/Subitisi

Spoken number words, that use a hybrid system of addition and multiplication, also have no need of placeholders or zeroes. See Five Finger Exercises

/Users/grahamshawcross/Documents/blog_drafts/subitising/SubitisiAfter Dehaene (1992) Varieties of Numerical Abilities Cognition, 44 1-42

The need for placeholders, zeroes and place value number systems arrises from the need to record in writing the results of calculations for administrative and trade purposes, be it on clay tablets, papyrus or paper.

 

Bibliography

Dehaene, S. (1992) Varieties of Numerical Abilities Cognition, 44 1-42

Knuth, D. (1997) The Art of Computer Programming. Volume 2, 3rd Ed. Addison–Wesley. pp. 194–213, “Positional Number Systems”.

Lam, L.Y. (1996) The Development of Hindu-Arabic and Traditional Chinese Arithmetic Chinese Science 13: 35–54

O’Connor, John J.; Robertson, Edmund F., “Pierre-Simon Laplace”, MacTutor History of Mathematics archive, University of St Andrews., accessed 16 September 2015

 

About Graham Shawcross

Architect PhD student at Edinburgh University Interested in order, rhythm and pattern in Architectural Design
This entry was posted in Architecture, Classification, Enumeration, Logic and tagged , , , . Bookmark the permalink.

One Response to Place Value

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