1-1 Alternate Methods for Representing Numbers. Origins of Representation of numbers

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1 Learning nit : - Handout - Alternate Methods for Representing Numbers Origins of Representation of numbers As twentieth century Americans, we take so much for granted. We are so accustomed to writing our numerals that we may not even question their origin. There are three theories which have been advanced which may explain the development of our numerals. We know that quantities were represented in ancient times by collections of pebbles. That is, three pebbles represented three of whatever object was being recorded. If numerals began when people tried to draw the pebbles, they would have drawn dots. The ot Theory says the numerals came from recording an appropriate number of dots, but when the process was hurried the dots lost their distinction and our numerals resulted. For example: If 3 were represented with three dots, and written in a hurry, the writer may have smeared the writing. 3 The ash Theory is similar, only supposes that tallies (dashes) were recorded originally instead of dots. We know, too, that ancient people tallied quantities by carving notches in bones and sticks. So numerals might have developed this way: 3 The Angle Theory opines that our numerals originally contained the appropriate number of angles in them. If the angle theory is right, it could look like this: 3 PASE: Experiment with each of the three theories. See if you can guess how other numerals were developed from dots, and from dashes, and from angles. Remember the appropriate number of dots, dashes, and angles has to be present in your guess. Experiment especially with the numerals: 2, 4, 6, and 8. Remember, these are theories; there isn t going to be a wrong guess. Page of 7

2 Learning nit : - Handout Verbal Representation of Numbers Around the world, words for numbers are quite revealing. For example, when missionaries translated John : for a Papaun tribe in New Guinea it was rendered, A man lay ill one man, both hands, five and three years. Without looking in your Bibles, how long was the man ill? (Hint: New Guinea is tropical. People don t wear shoes, so toes are just as available for counting as fingers are.) In British New Guinea the number 99 comes out as, four men die, two hands come to an end, one foot ends, and four. Why so morbid? (In our culture we count the fingers that are held up on a hand. Some cultures count them down, so when all five fingers have been counted down, the hand comes to an end. ) In Western Sudan the word dibi is used for 40. ibi literally means a mattress. What's the connection? The Mandingo tribe of West Africa uses kononto for 9. That word literally means to the one in the belly. Can you explain this? PASE: How would you answer the questions above? Here are some optional ways to represent numbers. Though these methods were physical in nature (i.e. not meant as written notation), we will write them with pictures. When using these ideas with young students, plan to use manipulative objects (pebbles, checkers, etc.) Physical Representation of numbers The Roman Calculator was a board or flat stone with groves and holes in it. The indentations were given the values as shown below:. Roman Calculator Pebbles in grooves & holes to represent values. M C L X V I Placing a pebble in the indentation recorded that value. The values of the pebbles were then added together to obtain the quantity represented. Roman numerals were a natural result of the calculator. For example to write 400, they used four C s because there would be four pebbles in the C groove. Page 2 of 7

3 Learning nit : - Handout Here s an example:. Roman Calculator Pebbles in grooves & holes to represent values. M C L X V I Illustrated above: = 926 Two simple rules governed the use of the calculator. No more than four pebbles were to appear in a groove, and no more than one pebble was allowed in a hole. (By the way, our word calculator comes from the Latin word calculus meaning pebble.) raw a calculator that shows the number 487. The calculator (as the name implies) was actually used to perform calculations. When adding, the first number was represented by pebbles in the grooves and holes. Pebbles for the second number would be places above the calculator in the appropriate locations. Beginning at the right, the pebbles above the calculator were added into the grooves and holes. The two basic rules were followed and the sum obtained. We will look at a very similar method in the Alternate Methods for Addition section in a little while, so don t fret over doing calculations with this tool. PASE: You need to know how to represent a given value on a Roman Calculator that you draw, and you need to be able to read a value represented on a Roman Calculator that you see. 2. Japanese Abacus There are actually three styles of abaci. The Russian abacus has ten beads on each rod and no center divider. The Chinese abacus (suan-pan) has seven beads on each rod with a divider bar to separate the one s from the five s. The Japanese Abacus (soroban) has five beads on each rod. It is explained here: On the Japanese Abacus there are five beads on each rod. The beads below the center divider represent,, 0, etc. The beads above the divider represent five times the lower quantity:, 0, Page 3 of 7

4 Learning nit : - Handout 00, etc. Beads away from the divider are in storage and not counted. Beads placed next to the divider are to be counted. 0,000, ,000, Could you read the number illustrated above (without the decoding numerals at the right)? Where would you place the beads to represent the number 36,07? raw an abacus that illustrates 36,07 below. on t forget to draw the beads in storage, too! 0,000, How will you represent the 0 in the hundred s place?,000,000 0 PASE: You need to know how to represent a given value on a Japanese Abacus that you draw, and you need to be able to read a value represented on a Japanese Abacus that you see. 3. Korean Chisanbop In this method the fingers and thumbs are used to report quantities. In this method, the fingers on the right hand have a value of and the thumb is a. The fingers on the left hand have a value of and that thumb has a value of 0. If the finger (or thumb) is placed down on the tabletop, it is counted. Fingers that are not to be counted are held up in the air. (This method even provides calisthenics for your fingers!) Page 4 of 7

5 Learning nit : - Handout Values: Example: 0 * values are given to fingers: * fingers down on tabletop are counted, those up in air are not = 73 raw some hands that illustrate the number 4 using and as in the example. 4. The Ciphers of the Monks A Forgotten Number-Notation of the Middle Ages by avid A King (Hardcover. ISBN ) is a comprehensive study of an ingenious number-notation from the Middle Ages that was devised by monks and mainly used in monasteries. With this system any number up to 9999 can be represented by a single drawing. Locate the book from your library if you d like more information. PASE: Study the key and examples below. Can you understand the code? What is in common for all the 00 s, for all the 0 s, for all the s and for all the s? Page of 7

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7 Learning nit : - Handout ANSWERS:. Roman Calculator Pebbles in grooves & holes to represent values. M C L X V I Illustrated above: = Japanese Abacus 0,000, Illustrated: 36,07,000, Korean Chisanbop Illustrated: 4 iane Mason and Indian Hills Community College Page 7 of 7