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ADVANCED

ARITHMETIC

NOTATION AND NUMERATION

1. Our remote ancestors doubtless did their counting by the aid of the ten fingers. Hence, in numeration it became natural to divide numbers into groups of tens. This accounts for the almost universal adoption of the decimal scale of notation.

2. It is uncertain what the first number symbols were. They were, probably, fingers held up, groups of pebbles, notches on a stick, etc. Quite early, however, groups of strokes I, II, III, IIII, ..., were used to represent numbers.

3. The earliest written symbols of the Babylonians were cuneiform or wedge-shaped symbols. The vertical wedge (1) was used to represent unity, the horizontal wedge (-) to represent ten, and the two together (Y=) to represent one hundred. Other numbers were formed from these symbols by writing them adjacent to each other. Thus,

YYY=1+1+1=3,

<<M = 10 + 10 +1+1=22,

-Y= 10 x 100=1000,

WW=5 x 100 + 10 + 2 = 512.

To form numbers less than 100 the symbols were placed adjacent to each other and the numbers they represented were added. To form numbers greater than 100 the symbols representing the number of hundreds were placed at the left of the symbol for one hundred and used as a multiplier.

4. The Egyptians used hieroglyphics, pictures of objects, or animals that in some way suggested the idea of the number they wished to represent. Thus, one was represented by a vertical staff (1), ten by a symbol shaped like a horseshoe (n), one hundred by a short spiral (e), one hundred thousand by the picture of a frog, and one million by the picture of a man with outstretched hands in the attitude of astonishment. They placed the symbols adjacent to each other and added their values to form other numbers. Thus, eni= 100+ 10+1=111.

The Egyptians had other symbols also.

5. The Greeks used the letters of their alphabet for number symbols, and to form other numbers combined their symbols much as the Babylonians did their wedgeshaped symbols.

6. The Romans used letters for number symbols, as follows:

1 5 10 50 100 500 1000
I V V X L с

D

M Numbers are represented by combinations of these symbols according to the following principles :

(1) The repetition of a symbol repeats the value of the number represented by that symbol; as, III=3, XX=20.

(2) The value of a number is diminished by placing a symbol of less value before one of greater value; as,

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