D 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 (11) 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, (84) × 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, 7x9÷3= 21. |