29. Example.-Change .324, to the decimal scale.

30. Example.-Change .71210 to a decimal in the scale of 5.

31. The operations, addition, subtraction, multiplication, and division, may be performed with numbers in any scale. The processes are the same as in the decimal scale. The student must bear in mind every time the number of units in each order required to make one unit of the next higher order.

The first fifteen numbers in the following scales are:

6, 7, 8, 9, a, b, 10, 11, 12, 13.

6,

7, 8, 9,

Duodecimal, 1, 2, 3,

4,

5,

Undenary, 1, 2, 3,

4,

5,

6,

Decimal, 1, 2, 3,

4,

5,

6,

7,

Nonary, 1, 2, 3,

Octary,

1, 2, 3,

6,

a, 10, 11, 12, 13, 14. 8, 9, 10, 11, 12, 13, 14, 15. 7, 8, 10, 11, 12, 13, 14, 15, 16. 7, 10, 11, 12, 13, 14, 15, 16, 17. 10, 11, 12, 13, 14, 15, 16, 20, 21. 11, 12, 13, 14, 15, 20, 21, 22, 23. 12, 13, 14, 20, 21, 22, 23, 24, 30. Quaternary, 1, 2, 3, 10, 11, 13, 20, 21, 22, 23, 30, 31, 32, 33. Ternary, 1, 2, 10, 11, 12, 20, 21, 22, etc. Binary, 1, 10, 11, 100, 101, 110, 111, etc.

Quinary 1, 2, 3, 4,

Exercise I

1. Define arithmetic, number, unit, principle, and axiom. What purpose do the axioms serve? In what year was the first arithmetic printed?

2. What is notation? What two systems of notation are now in common use? State the five principles of the Roman notation; the three of the Arabic notation.

3. Express 23479, 5087, and 355 by the Roman notation. 4. Express XXIX, XCIX, CXIV by the Arabic notation.

5. Express 24000 and 0.0000017 by the index notation. To what kind of numbers is the index notation adapted? 6. Read by the French method 342378921476; read the same number by the English method.

7. Define scale and radix. From what does the scale get its name?

8. Write the first sixty numbers in each scale. 9. Change the following to the decimal scale: (1) 58679; (2) 231234; (3) 231a511.

10. Change 58375 from the decimal (1) to the octary scale, (2) to the ternary, (3) to the duodecimal, (4) to the senary.

11. Add 31235, 4124, 32435, 42335.

12. Subtract 34562, from 624567.

13. Multiply 3424, by 2345.

14. Divide 2034431, by 34245.

CHAPTER II

THE FOUR FUNDAMENTAL OPERATIONS

I. ADDITION

32. Addition (Latin addere, to increase) is the process of finding the sum of two or more numbers. It is a short method of counting.

33. The numbers to be added are called addends.

34. The result of addition is called the sum or amount. 35. The sign of addition (+) is read plus, meaning more. 36. The sign of equality (=) is read equals.

NOTE. The signs (+) and (−) were used by Johannes Widman in his Mercantile Arithmetic, published in Leipzig in 1489. He used them, however, to denote excess and deficiency. These symbols were first used to denote addition and subtraction by a German mathematician (Michael Stifel) in his work published in 1544.

The sign of equality (=) was first used by the English mathematician, Robert Recorde, in an algebra called the Whetstone of Witte, published in 1557.

37. Fundamental principles used in addition:

1. Only similar addends can be added; therefore, the addends must have the same unit.

2. The sum is similar to the addends, and contains all the units of all the addends.

3. The sum is the same regardless of the order of grouping the addends.

4. The sum is the same regardless of the order of performing the operation.

NOTE.-Principle 3 is called the associative law of addition; principle 4, the commutative law of addition.

38. To test the accuracy of addition:

1. Review the work.

2. Add the columns in reverse order.

3. Cast out the 9's.

39. RULE: To cast the 9's out of any number:

Divide the sum of the digits by 9, and find the excess.

Thus, to find the excess of 9's in 23784, begin at the left: 2+3+7=12; cast out the 9 and say 3+8=11; cast out the 9 and say 2+4-6; the excess is 6. Or, 2+3+7 +8+4=24; 24÷9=2, with a remainder 6.

The test of accuracy by casting out the 9's is based upon the principle that the excess of 9's in any number equals the excess of 9's in the sum of its digits. This may be shown by the following

.. 4352 the above multiples of 9 + 14

Observe that the number has been separated into multiples of 9, and the sum of the digits of the number. There can be no excess when the multiples are divided by 9, so the only excess must be in the sum of the digits.