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 X 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.

The following are the principal and most useful arithmetical signs:

22. The Sign of Addition is a perpendicular cross, +. It is called Plus, and indicates that the numbers between which it is placed are to be added; 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, 83, 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, 79, indicates that the value of 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, 20 ÷ 5, 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.