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IV = 4, XL=40, XC = 90. The less number is subtracted from the greater.

(3) The value of a number is increased by placing a symbol of less value after one of greater value, as XI=11, CX 110. The less number is added to the greater number.

(4) The value of a number is multiplied by 1000 by placing a bar over it, as ī= 100,000, X = 10,000.

=

7. Among the ancients we do not find the characteristic features of the Arabic, or Hindu system where each symbol has two values, its intrinsic value and its local value, ¿.e. the value due to the position it occupies. Thus, in the number 513 the intrinsic value of the symbol 5 is five, its local value is five hundred. Written in Roman notation 513

= DXIII. In the Roman notation each symbol has its intrinsic value only.

8. The ancients lacked also the symbol for zero, or the absence of quantity.

The introduction of this symbol made place value possible.

9. With such cumbersome symbols of notation the ancients found arithmetical computation very difficult. Indeed, their symbols were of little use except to record numbers. The Roman symbols are still used to number the chapters of books, on clock faces, etc.

10. The Arabs brought the present system, including the symbol for zero and place value, to Europe soon after the conquest of Spain. This is the reason that the numerals used to-day are called the Arabic numerals. The Arabs, however, did not invent the system. They received it and its figures from the Hindus,

11. The origin of each of the number symbols 4, 5, 6, 7, 9, and probably 8 is, according to Ball, the initial letter of the corresponding numeral word in the Indo-Bactrian alphabet in use in the north of India about 150 B.C. 2 and 3 were formed by two and three parallel strokes written cursively, and 1 by a single stroke. Just when the zero was introduced is uncertain, but it probably appeared about the close of the fifth century A.D. The Arabs called the sign 0, sifr (sifra = empty). This became the English cipher (Cajori, “History of Elementary Mathematics").

12. The Hindu system of notation is capable of unlimited extension, but it is rarely necessary to use numbers greater than billions.

13. In the development of any series of number symbols into a complete system, it is necessary to select some number to serve as a base. In the Arabic, or Hindu system ten is used as a base ; i.e. numbers are written up to 10, then to 20, then to 30, and so on. In this system 9 digits and 0 are necessary. If five is selected as the base, but 4 digits and 0 are necessary. If twelve is selected, 11 digits and 0 are necessary.

The following table shows the relations of numbers in the scales of 10, 5, and 12. (t and e are taken to represent ten and eleven in the scale of 12.)

BASE

10

3

5

6 7

8

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48 100

1 2 3 4 5 6 7 8 9 10 11 12 21 1 2 3 4 10 11 12 13 14 20 21 22 41 143 400

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Ex. 1. Reduce 431, to the decimal scale.
Note. 431, means 431 in the scale of 5.
Solution. 4 represents 4 x 5 x 5 = 100

3 represents 3 x 5
1 represents 1

= 1
.. 431,

= 11610 Ex. 2. Reduce 463270 to the scale of 8.

, Solution. 8 4632

579 0 = 579 units of the second order and none of the first order. 72 3= 72 units of the third order and 3 of the second order. 90= 9 units of the fourth order and none of the third order. 1 1 1 unit of the fifth order and 1 of the fourth order.

:. 463210 = 11030g

=

EXERCISE 1

1. What number symbols are needed for the scale of 2 ? of 8? of 6 ? of 11 ? Write 12 and 20 in the scale of 2.

2. Reduce 234; and 546, to the decimal scale. 3. Reduce 764910 to the scale of 4. 7.5

4. Compare the local values of the two 9's in 78,940,590,634. What is the use of the zero? Why is the number grouped into periods of three figures each ? Read it.

5. If 4 is annexed to the right of 376, how is the value of each of the digits 3, 7, 6 affected ? if 4 is annexed to the left ? if 4 is inserted between 3 and 7?

6. What is the local value of each figure in 76,345 ? What would be the local value of the next figure to the right of 5 ? of the next figure to the right of this?

7. For what purpose is the decimal point used ? 8. Read 100.004 and 0.104; 0.0002; 0.0125 and 100.0025. ADDITION

14. If the arrangement is left to the computer, numbers to be added should be written in columns with units of like order under one another.

15. In adding a column of given numbers, the computer should think of results and not of the numbers.

He should not say three and two are five and one are six 329 and four are ten and nine are nineteen, but simply five, six, 764 ten, nineteen, writing down the 9 as he names the last num- 221 ber. The remaining columns should be added as follows: 9642 three, seven, nine, fifteen, seventeen, writing down the 7; 7823 nine, fifteen, seventeen, twenty-four, twenty-seven, writing 18779 down the 7; nine, eighteen, writing down the 18. Time in 211 looking for errors may be saved by writing the numbers to be carried underneath the sum as in the exercise.

16. Checks. If the columns of figures have been added upward, check by adding downward. If the two results agree, the work is probably correct.

Another good check for adding, often used by accountants, is to add beginning with the left-hand column.

16000 16 Thus, the sum of the thousands is 16 thou- 2600

26 sands, of the hundreds 26 hundreds, of the tens

160

16 16 tens, and of the units 19 units.

19

19 18779 18779

or

EXERCISE 2

Define

1. What is meant by the order of a digit? addend, sum.

2. Why should digits of like order be placed in the same column ? State the general principle involved.

3. Why should the columns be added from right to left ? Could the columns be added from left to right and a correct result be secured? What is the advantage in beginning at the right ?

4. In the above exercise, why is 1 added (“carried ”) to the second column ? 1 to the third column ? 2 to the fourth column ?

17. Accuracy and rapidity in computing should be required from the first. Accuracy can be attained by acquiring the habit of always checking results. Rapidity comes with much practice.

18. The 45 simple combinations formed by adding consecutively each of the numbers less than 10 to itself and to every other number less than 10 should be practiced till the student can announce the sum at sight. These combinations should be arranged for practice in irregular order similar to the following: 1 2 2 5 9 8 1 4 5 7 6 4 2 3 4 1 2 3 1 3 8 9 7 6 8 6

4 6 4

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19. Rapid counting by ones, twos, threes, etc., up to nines is very helpful in securing both accuracy and rapidity.

Ex. Begin with 4 and add 6's till the result equals 100. Add rapidly, and say simply 4, 10, 16, 22, ..., 94, 100. ,

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