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5. A bar placed over a letter increases its value a

thousand times; as,
V=5000; L=50000; M=1000000.

ARABIC NOTATION

18. The Arabic Notation is now more widely used than any other method. It is the simplest and clearest system of notation known to civilization. It received its name from the Arabs, who introduced the system into Christian Europe about the year 1200. The symbols, except the zero, originated in India before the Christian era; therefore, the term Hindu is sometimes used instead of Arabic. These symbols were not generally taught and used until the fifteenth century. In 1478 the first printed arithmetic appeared at Treviso, and in 1482 the first German arithmetic, at Bamberg, and these explained the system. In its present state of perfection, ten figures are employed in expressing numbers.

Figures: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Names: zero, one, two, three, four, five, six, seven, eight, nine.

The superiority of this system is due to the use of the zero, which renders possible the distinctive feature known as local, or place value. Each of the nine digits has two values—a simple value and a local value. The simple value of a figure is the value indicated by its name. A figure always has its simple value when standing alone or in units order. The local value of a figure is its value when placed in any order except units.

Thus, in 703, the simple value of “7” is seven, but the local value of “7is seven hundred.

19. The principles involved in writing numbers are: 1. Ten units of any order make one of the next higher

order. 2. Moving a significant figure one place to the left in

creases its value tenfold; one place to the right

decreases its value tenfold. 3. Vacant orders in a number are filled with zeros.

20. Index notation may also be mentioned as a special form of the above system. Scientists often find it convenient in approximations to introduce powers of ten.

Thus, 246000000000 may be written 246 X 10', or 24.6 X 1010.

Since 10-1=0.1, and 10-2=0.01, etc., a decimal fraction may be expressed by using 10 with a negative exponent.

E.g., 0.0000000375 may be written 37.5X10-9. These numbers may be multiplied thus,

246 X 37.5 X 109-9=246 X 37.5=9225.

COMMON OR FRENCH METHOD OF NUMERATION

21. The Common or French Method of Numeration is the method used by the people of the United States and of France. The figures are considered as far as possible in groups of three figures each, commencing at the right hand. The groups are called periods. The first twelve periods are as follows:

1. units 5. trillions 9. septillions
2. thousands 6. quadrillions 10. octillions
3. millions 7. quintillions 11. nonillions
4. billions 8. sextillions 12. decillions.

ENGLISH METHOD OF NUMERATION

22. The English people and people of many other European nations consider a period as made up of six figures. This method of grouping the figures is called the English Method. The first six periods according to this method are as follows:

1. units 3. billions 5. quadrillions
2. millions 4. trillions 6. quintillions

By the English Method, the number 234179386564 is read two hundred thirty-four thousand one hundred seventynine million, three hundred eighty-six thousand, five hundred sixty-four. By the French Method, it would read two hundred thirty-four billion, one hundred seventy-nine million, three hundred eighty-six thousand, five hundred sixty-four.

SCALES OF NOTATION

23. The scale (Latin scala, a ladder), in any system of notation, is the succession of ascending and descending order values.

24. The radix of a scale (Latin radix, root) is the number of units which it takes of one order to make one unit of the next higher order. The radix is also called the base-number. In the Arabic Notation the radix is ten and is uniform. In any scale the number of characters, including 0, is the same as the number of units in the radix of that scale. The scale gets its name from the number of units in its radix. To express numbers in scales higher than the deci

mal, new characters must be employed. Thus, a may be used to represent ten, b, eleven, c, twelve, etc.

Scale.
Characters Used.

Radix.
Duodecimal 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, 0 twelve, expressed 10
Undenary 1, 2, 3, 4, 5, 6, 7, 8, 9, a, 0... eleven,

10 Decimal 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. ten,

10 Nonary 1, 2, 3, 4, 5, 6, 7, 8, 0..

nine,

10 Octary 1, 2, 3, 4, 5, 6, 7, 0.

eight,

10 Septenary 1, 2, 3, 4, 5, 6, 0.

seven,

10 Senary 1, 2, 3, 4, 5, 0..

six,

10 Quinary 1, 2, 3, 4, 0.

five,

10 Quaternary 1, 2, 3, 0.

four,

10 Ternary 1, 2, 0..

three,

10 Binary

two,

10

1, 0..

25. Inasmuch as the names of the orders used in expressing numbers are adapted to the decimal scale, numbers expressed in other scales should be read by naming the number of units in each order. For example, the number 256 in the octary scale should be read: octary scale, 2 units of the third order, 5 of the second, and 6 of the first. Usually a small subscript is used to indicate the radix. Thus, 423, means 423 in the quinary scale.

EXAMPLES

26. Write in the senary scale the numbers corresponding to the numbers from 1 to 15 in the decimal scale.

EXPLANATION.—The radix is 6; therefore, the figures used are 1, 2, 3, 4, 5, 0. It must be remembered that when the number of units in any order reaches six, it makes one of the next higher order.

Radix ten : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.
Radix six : 1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23.

In like manner, write in other scales the numbers corresponding to the numbers from 1 to 35 in the decimal scale. 27. To change from any scale to the decimal scale:

Example.—Express 3432g in the decimal scale.

SOLUTION: 3 x 6' + 4 x 6' + 3 6' + 2 = = 648 + 144 + 18 + 2 = 812.

.:. 34326 = 81210.

This example may be treated as one in compound numbers. Thus,

OPERATION.

3432
6
22

6
135

6 812

EXPLANATION.—Since each higher unit is equal to six of the next lower order, 3 units of the fourth order are equal to 18 of the third. By adding 4, the number of the third order given, we obtain 22, the number of the third order. Proceeding in the same manner, until the number of units of the first order is obtained, the number in the decimal scale is 812.

28. To change from the decimal to another scale: Example. Change 5287 from the decimal to the qui

nary scale.

OPERATION.

5 | 5287
5 1057, rem. 2
5 211, rem. 2

42, rem. 1
5 8, rem. 2

1, rem. 3

EXPLANATION.-By dividing by 5, we obtain the number of units of the second order and the number of units of the first order remaining. By continuing to divide by 5, the number of units in the successive orders is obtained, and the number of units remaining after division. It is thus found that 528710 = 132122s.

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