THE

NORMAL

HIGHER ARITHMETIC.

INTRODUCTION.

NATURE OF ARITHMETIC.

1. Mathematics is the science of quantity. It treats of the properties and relations of quantity.

2. Quantity is anything that can be measured. It is of two kinds, Number and Extension.

3. Arithmetic is the science of Number; Geometry is the science of Extension.

4. Arithmetic embraces ideas and truths. The ideas give rise to Definitions: and the truths, to Principles and Problems.

5. A Definition is a concise statement of the distinctive qualities of anything.

6. A Principle is a truth of science. Principles may be in the form of Axioms or Theorems.

7. An Axiom is a self-evident truth. laws which control the reasoning processes.

Axioms are the

8. A Theorem is a truth which becomes evident by a process of reasoning called a Demonstration.

9. A Demonstration is a process of reasoning by which the truth of a theorem is proved.

10. A Problem is a question requiring some unknown result from that which is known.

11. The Conditions of a Problem are the known truths that are given.

12. A Solution of a Problem is a process of obtaining the desired result.

13. A Mental Solution is one in which the operations are performed without the aid of written characters.

14. A Written Solution is one in which the operations are performed by the aid of written characters.

15. A Rule is a statement of the method of solving a problem.

16. Arithmetical Analysis is the process of solving problems by a comparison of their elements.

17. Arithmetic is the science of numbers and the art of computing with them.

18. A Number is a unit or a collection of units. Numbers are Concrete and Abstract.

19. A Unit is a single thing or one. A single thing is a concrete unit; one is an abstract unit.

20. A Concrete Number is one applied to some particular object: as, two yards, five books, etc.

21. An Abstract Number is one not applied to any particular object; as, two, five, etc.

22. Similar Numbers are those in which the units are alike; as, two boys and four boys.

23. Dissimilar Numbers are those in which the units are unlike; as, two boys and four books.

24. The General Classes of Numbers treated of in Arithmetic are Integers, Fractions, and Denominate Numbers. 25. An Integer is a number of integral units; as, four, five, etc.

26. A Fraction is a number of the equal divisions of a unit; as, two-thirds, three-fourths, etc.

27. A Denominate Number is a number in which the unit is a measure of continuous quantity; as, three yards, four pounds.

28. These Three Classes of numbers admit of the same general processes, and, as subject matter, give rise to a triune division of Arithmetic.

29. The Processes of Arithmetic may all be embraced under the three heads, Synthesis, Analysis, and Comparison. 30. The fundamental idea of Arithmetic is the Unit or The synthesis of units gives rise to Numbers. Numbers may be subjected to the operations of synthesis, analysis and comparison, and out of these processes arise all the subjects of arithmetic.

one.

31. Fundamental Processes.-A general synthesis is called Addition. A special case of Addition, in which the numbers united are all the same, is called Multiplication.

A general analysis is called Subtraction. A special case of Subtraction, in which the object is to find how many times one number contains another, is called Division.

These four processes are called the Fundamental Operations of Arithmetic. From these four processes arise others which we may call Derivative Processes.

32. Derivative Processes.-A general synthesis of factors to form composite numbers may be called Composition. A synthesis of equal factors is Involution. A synthesis of factors to find a number which is one or more times several numbers, is called Common Multiple.

An analysis of a number into its factors is called Factoring. An analysis into equal factors is called Evolution. The finding of a common factor of several numbers is called Common Divisor.

These divisions have their origin in synthesis and analysis, and grow out of them. There are several other divisions which have their origin in and grow out of Comparison.

33. Comparison.-The comparison of two numbers gives rise to Ratio. The comparison of equal ratios gives

rise to Proportion. The comparison of several numbers differing by a common ratio gives rise to Progression.

In comparing numbers, we see that we can often change a number from one class of units to another, which gives rise to Reduction. In comparing numbers, we may assume some number as a basis of reference, and develop their relation in regard to this basis; when this basis is a hundred, we have Percentage. Numbers may be compared and their properties investigated, which gives rise to the Properties of Numbers. 34. We thus have a complete outline of Arithmetic. It is considered, first, as treating of three classes of numbers, Integers, Fractions and Denominate Numbers. Its processes are also three-föld, Synthesis, Analysis and Comparison. The whole science of Arithmetic is an outgrowth of this triune basis.

NOTE. For a fuller discussion of this subject, see Brooks's Philosophy of Arithmetic.

SECTION I.

ARITHMETICAL LANGUAGE.

35. Arithmetical Language is the method of expressing numbers.

36. Arithmetical Language is of two kinds, Oral and Written. The former is called Numeration and the latter Notation.

NUMERATION.

37. Numeration is the method of naming numbers, and of reading them when expressed by characters. It is the oral expression of numbers.

38. Since it would require too many words to give each number a separate name, numbers are named according to the following simple principle:

Principle. We name a few of the first numbers, and then form groups or collections, name these groups, and use the names of the first few numbers to number these groups.

39. A single thing is named one; one and one more are named two; two and one more, three; three and one more, four; and thus we obtain the simple names,

One, two, three, four, five, six, seven, eight, nine, ten.

40. Regarding the collection ten as a single thing, we might count one and ten, two and ten, etc., as far as ten and ten, or two tens, which modified by use would give the following numbers:

Eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty.

41. Proceeding in the same way, we would have two tens and one, two tens and two, two tens and three, etc., which modified by use would give the following numbers:

Twenty-one, twenty-two, twenty-three, twenty-four, twentyfive, twenty-six, twenty-seven, twenty-eight, twenty-nine.

42. Continuing in the same manner, we would have three