COMMERCIAL ARITHMETIC.

DEFINITIONS.

1. Arithmetic is the Science of Numbers and the Art of Computation.

2. A Unit is a single thing.

3. A Number is a unit or a collection of units.

4. The Unit of a number is one of the collection of units forming the number; thus, the unit of 5 is 1; of 17 dollars, 1 dollar; of 30 pupils, 1 pupil. 5. An Integer is a whole or entire number.

6. An Even Number is one that can be exactly divided by 2; as, 6, 8, 44. 7. An Odd Number is one that cannot be exactly divided by 2; as, 5, 9, 23. 8. A Composite Number is one that can be resolved or separated into factors; as, 4 = 2 × 2; 12 = 3 × 2 × 2.

9. A Prime Number is one that cannot be resolved or separated into factors, being divisible only by itself and unity; as, 1, 2, 3, 5, 7, 19, 83.

10. An Abstract Number is one used without reference to any particular thing or quantity; as, 3, 11, 24.

11. A Concrete Number is one used with reference to some particular thing or quantity; as, 3 dollars, 11 men, 24 cords of wood.

12. A Compound Denominate, or Compound Number, is a concrete number expressed by two or more orders of units; as, 3 dollars and 11 cents; 5 pounds, 2 ounces and 15 pennyweights.

13. Like Numbers are such as have the same unit value; as, 5, 14, 37; or, 5 men, 14 men, 37 men; or, if denominate, the same kind of quantity; as, 5 hours 14 minutes 37 seconds.

14. Unlike Numbers are such as have different unit values; as, 11, 16 days, 265 dollars, 5 pounds, 4 yards.

15. Ratio is the comparison of magnitudes. It is of two kinds; arithmetical and geometrical.

16. Arithmetical Ratio expresses a difference.

17. Geometrical Ratio expresses a quotient.

18. A Problem in Arithmetic is a question to be solved; its analysis, the logical statement of its conditions and of the steps required for its solution.

19. The Conclusion of the analysis is called the answer, or result. 20. A Rule is an outline of the steps to be taken in a solution.

SIGNS.

21. A Sign is a character used to express a relation of terms or to indicate an operation to be performed.

It is called Plus, are to be added;

The following are the principal and most useful arithmetical signs: 22. The Sign of Addition is a perpendicular cross, +. and indicates that the numbers between which it is placed thus, 54 indicates that 4 is to be added to 5.

23. The Sign of Subtraction is a short horizontal line, It is called Minus, and indicates, when placed between two numbers, that the value of the number on its right is to be taken from the value of the number on its left; thus, 8 3, indicates that 3 is to be subtracted from 8.

24. The Sign of Multiplication is an oblique cross, X. It indicates that the numbers between which it is placed are to be multiplied together; thus, 7 × 9, indicates that the value of 7 is to be taken 9 times.

25. The Common Sign of Division is a short horizontal line with a point above and one below, . It indicates a comparison of numbers to determine a quotient, it being understood that the number at the left of the sign is to be divided by the one at its right; thus, 205, indicates that 20 is to be divided by 5.

26. The Sign of Ratio is the colon, : ; it also indicates division.

27. The Sign of Equality is two short horizontal lines,=. It is read equals, or, is equal to, and indicates that the numbers, or expressions, between which it is placed are equal to each other; thus, 2 + 2 = 4.

28. The Signs of Aggregation are the parenthesis, (), brackets, [ ], brace,, and vinculum, ———. They indicate that the quantities included within, or connected by them, are to be taken together and subjected to the same operation.

29. The Index, or Power Sign, is a small figure placed at the right of and above another figure. It indicates that the number over which it is placed is to be taken as a factor a number of times equal to the numerical value of the index. Thus 42 indicates that 4 is to be taken twice as a factor, or multiplied by itself once; 43 indicates that 4 is to be used three times as a factor. 42 is read 4 squared; 43 is read 4 cubed; also, the second power of 4; the third power of 4. 30. The Root, or Radical Sign, is the character, √; it is the opposite of the index, or power sign. When there is no figure in the opening, it indicates that the quantity over which the sign is placed is to be separated into two equal factors, or its square root taken. A figure placed in the opening indicates the number of equal factors required, or the root to be extracted; as, 27, 16.

31. The Dollar Sign is the character, $.

32. The Cent Sign is the character, f.

33. The Decimal Point is the period, . ; when employed to separate dollars from cents it is called a Separatrix.

Fractional parts of a dollar are expressed only as hundredths; thus $14.53 is read 14 dollars and 53 cents, or 14 and 53 hundredths dollars.

ABBREVIATIONS AND CONTRACTIONS.

34. The following are some of the principal abbreviations and contractions in

NOTE. Other and more complete lists of abbreviations and contractions, together with illustrations of their uses, will be given in the advanced part of this work..

NOTATION AND NUMERATION.

35. Notation is the method of expressing numbers.

There are three ways of expressing numbers.

I. By Words; as one, two, three.

II. By Figures, called the Arabic, or, more properly, the Indian Notation; this notation employs the nine digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, and the naught, 0, which is also called zero, and cipher.

By this method a number is written and read with direct reference to its successive periods, commencing with the highest.

III. By Letters, called the Roman Notation; this notation employs the seven capital letters; I for one, V for five, X for ten, L for fifty, C for one hundred, D for five hundred, and M for one thousand.

By this method a number is written and read with direct reference to its successive orders, and multiplication by one thousand is indicated by over-scoring the letter whose value is to be so increased; as, V for five, V for five thousand; M for one thousand, M for a thousand thousand, or a million.

36. Numeration is the method of reading numbers expressed by words, figures, or letters.

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ARABIC METHOD OF NOTATION.

37. By the Arabic Method the value of numbers increases from right to left, and decreases from left to right in a ten-fold ratio; the successive figures from right to left or from left to right are called orders of units, the value of one of any order being ten times the value of one of the order next to its right, and only one-tenth the value of one of the order next to its left; for example, in the number one hundred and eleven, expressed 111, the second 1 is equal in value to ten times the first 1, but to only one-tenth the value of the third 1.

The succession of the orders of units in writing numbers by this method, establishes a decimal system in which the numbers are divided for convenience into periods of three figures, or places, each. Numbers so written are read or enumerated from right to left to ascertain their value, and from left to right to announce their value. The naught, or cipher, is always read as of the order of the place it occupies.

For example, in reading to ascertain the value of the expression 265017, we begin at the right and name the successive orders of units: units, tens, hundreds, thousands, tens of thousands, hundreds of thousands. Having now determined the names of the given units, we read from the left, and announce the number as two hundred sixty-five thousand seventeen.

FRENCH SYSTEM OF NUMERATION.

38. The separating of written numbers into uniform periods of three figures, or places, as explained above, is known by its origin and use as the French system of numeration. This is the system invariably used in the United States.

39. The Periods take their names from the Latin numerals, with certain established variations, and numbers are divided into orders of units and into periods, and are read as shown by the following