HIGHER ARITHMETIC.

DEFINITIONS.

1. Quantity is any thing that can be increased, diminished, or measured; as distance, space, weight, motion, time.

2. A Unit is one, a single thing, or a definite quantity. 3. A Number is a unit, or a collection of units.

4. The Unit of a Number is one of the collection constituting the number. Thus, the unit of 34 is 1; of 34 days is 1 day. 5. An Abstract Number is a number used without reference to any particular thing or quantity; as 3, 24, 756.

6. A Concrete Number is a number used with reference to some particular thing or quantity; as 21 hours, 4 cents, 230 miles. 7. Unity is the unit of an abstract number.

8. The Denomination is the name of the unit of a concrete number.

9. A Simple Number is either an abstract number, or a concrete number of but one denomination; as 48, 52 pounds, 36 days. 10. A Compound Number is a concrete number expressed in two or more denominations; as, 4 bushels 3 pecks, 8 rods 4 yards 2 feet 3 inches.

11. An Integral Number, or Integer, is a number which expresses whole things; as 5, 12 dollars, 17 men.

12. A Fractional Number, or Fraction, is a number which expresses equal parts of a whole thing or quantity; as, of a pound, of a bushel.

13. Like Numbers have the same kind of unit, or express the same kind of quantity. Thus, 74 and 16 are like numbers; so are 74 pounds, 16 pounds, and 12 pounds; also, 4 weeks 3 days, and 16 minutes 20 seconds, both being used to express units of time. 14. Unlike Numbers have different kinds of units, or are used

to express different kinds of quantity. Thus, 36 miles, and 15 days; 5 hours 36 minutes, and 7 bushels 3 pecks.

15. A Power is the product arising from multiplying a number by itself, or repeating it any number of times as a factor.

16. A Root is a factor repeated to produce a power.

17. A Scale is the order of progression on which any system. of notation is founded. Scales are uniform and varying.

18. A Uniform Scale is one in which the order of progression is the same throughout the entire succession of units.

19. A Varying Scale is one in which the order of progression is not the same throughout the entire succession of units.

20. A Decimal Scale is one in which the order of progression is uniformly ten.

21. Mathematics is the science of quantity.

The two fundamental branches of Mathematics are Geometry and Arithmetic. Geometry considers quantity with reference to positions, form, and extension. Arithmetic considers quantity as an assemblage of definite portions, and treats only of those conditions and attributes which may be investigated and expressed by numbers. Hence,

22. Arithmetic is the Science of numbers, and the Art of computation.

NOTE. When Arithmetic treats of operations on abstract numbers it is a science, and is then called Pure Arithmetic. When it treats of operations on concrete numbers it is an art, and is then called Applied Arithmetic. Pure and Applied Arithmetic are also called Theoretical and Practical Arithmetic.

23. A Demonstration is a process of reasoning by which a truth or principle is established.

24. An Operation is a process in which figures are employed to make a computation, or obtain some arithmetical result.

25. A Problem is a question requiring an operation.

26. A Rule is a prescribed method of performing an operation. 27. Analysis, in arithmetic, is the process of investigating principles, and solving problems, independently of set rules.

28. The Five Fundamental Operations of Arithmetic are, Notation and Numeration, Addition, Subtraction, Multiplication, and Division.

SIGNS.

29. A Sign is a character indicating the relation of numbers, or an operation to be performed.

30. The Sign of Numeration is the comma (,). It indicates that the figures set off by it express units of the same general name, and are to be read together, as thousands, millions, billions, etc.

31. The Decimal Sign is the period (.). It indicates that the number after it is a decimal.

32. The Sign of Addition is the perpendicular cross, +, called plus. It indicates that the numbers connected by it are to be added; as 3+ 5+ 7, read 3 plus 5 plus 7.

33. The Sign of Subtraction is a short horizontal line, called minus. It indicates that the number after it is to be subtracted from the number before it; as 12-7, read 12 minus 7.

34. The Sign of Multiplication is the oblique cross, X. It indicates that the numbers connected by it are to be multiplied together; as 5 × 3 × 9, read 5 multiplied by 3 multiplied by 9.

35. The Sign of Division is a short horizontal line, with a point above and one below, ÷. It indicates that the number before it is to be divided by the number after it; as 18 ÷ 6, read 18 divided by 6.

Division is also expressed by writing the dividend above, and the divisor below, a short horizontal line. Thus, 18, read 18 divided by 6.

36. The Sign of Equality is two short, parallel, horizontal lines,=. It indicates that the numbers, or combinations of numbers, connected by it are equal; as 4+ 8 = 15—3, read 4 plus 8 is equal to 15 minus 3. Expressions connected by the sign of equality are called equations.

37. The Sign of Aggregation is a parenthesis, (). It indicates that the numbers included within it are to be considered together, and subjected to the same operation. Thus, (8 + 4) × 5 indicates that both 8 and 4, or their sum, is to be multiplied by 5. A vinculum or bar, has the same signification. Thus,

7 x 93 = 21.

38. The Sign of Ratio is two points, :. Thus, 7:4 is read, the ratio of 7 to 4.

39. The Sign of Proportion is four points, 3: 6:48, is read, 3 is to 6 as 4 is to 8.

Thus,

40. The Sign of Involution is a number written above, and a little to the right, of another number. It indicates the power to which the latter is to be raised. Thus, 128 indicates that 12 is to be taken 3 times as a factor; the expression is equivalent to 12 x 12 x 12. The number expressing the sign of involution is called the Index or Exponent.

41. The Sign of Evolution, ✔, is a modification of the letter r. It indicates that some root of the number after it is to be extracted. Thus, ✔25 indicates that the square root of 25 is to be extracted; 364 indicates that the cube root of 64 is to be extracted.

AXIOMS.

42. An Axiom is a self-evident truth. The principal axioms required in arithmetical investigations are the following:

1. If the same quantity or equal quantities be added to equal quantities, the sums will be equal.

2. If the same quantity or equal quantities be subtracted from equal quantities, the remainders will be equal.

3. If equal quantities be multiplied by the same number, the products will be equal.

4. If equal quantities be divided by the same number, the quotients will be equal.

5. If the same number be added to a quantity and subtracted from the sum, the remainder will be that quantity.

6. If a quantity be multiplied by a number and the product divided by the same number, the quotient will be that quantity. 7. Quantities which are respectively equal to any other quantity are equal to each other.

8. Like powers or like roots of equal quantities are equal.

9 The whole of any quantity is greater than any of its parts. 10. The whole of any quantity is equal to the sum of all its parts.

NOTATION AND NUMERATION.

43. Notation is a system of writing or expressing numbers by characters; and,

44. Numeration is a method of reading numbers expressed by characters.

45. Two systems of notation are in general use -the Roman and the Arabic.

NOTE. The Roman Notation is supposed to have been first used by the Romans: hence its name. The Arabic Notation was first introduced into Europe by the Moors or Arabs, who conquered and held possession of Spain during the 11th century. It received the attention of scientific men in Italy at the beginning of the 13th century, and was soon afterward adopted in most European countries. Formerly it was supposed to be an invention of the Arabs; but investigations have shown that the Arabs adopted it from the Hindoos, among whom it has been in use more than 2000 years. From this undoubted origin it is sometimes called the Indian Notation.

47. The Roman notation is founded upon five principles, as follows:

1st. Repeating a letter repeats its value. Thus, II represents two, XX twenty, CCC three hundred.

2d. If a letter of any value be placed after one of greater value, its value is to be united to that of the greater. Thus, XI represents eleven, LX sixty, DC six hundred.

3d. If a letter of any value be placed before one of greater value, its value is to be taken from that of the greater. Thus, IX represents nine, XL forty, CD four hundred.

4th. If a letter of any value be placed between two letters, each of greater value, its value is to be taken from the united value of the other two. Thus, XIV represents fourteen, XXIX twentynine, XCIV ninety-four.

5th. A bar or dash placed over a letter increases its value one thousand fold. Thus, V signifies five, and V five thousand; L fifty, and L fifty thousand.