Machinery's Handbook, 31st Edition
REAL NUMBERS
3
REAL NUMBERS AND THEIR OPERATIONS Real Numbers
Most mathematical computation is performed in the real number system . The universal set of the “reals” includes the subsets: naturals, whole numbers, integers, rationals, and irrationals . The naturals (also called counting numbers ): {1, 2, 3, . . .} are included in the whole numbers: {0, 1, 2, 3, . . .}, which are included in the integers (or signed whole num- bers): {. . . ,– 2, – 1, 0, 1, 2, . . .}. And all of these subsets are included in the rationals. Rational numbers, including integers, can be written in fraction form. Since all frac- tions can be divided numerator by denominator, their decimal form either terminates or repeats. Examples of rational numbers: – 4/1, 3/5 = 0.6, 1/3 = 0.333. . . . The only set in the real numbers larger than the naturals that does not contain any of the other sets is the irrationals. These are not expressible as ratios. An irrational number’s decimal representation does not terminate and it has no pattern of repetition. Examples of irrational numbers are roots that cannot be simplified, such as 6 70 3 and , as well as quantities like π and the natural log base e . The entire real number set is the union of the rationals and the irrationals. Properties of Real Numbers.— Though often obvious and followed almost automati- cally, the properties of real numbers are critical to mathematical reasoning. These prop- erties justify various steps in solving algebraic problems, such as those in this Handbook. Equivalence properties (symmetry, reflexivity, transitivity) and operational properties of numbers are summarized here. Equivalence Properties: The properties of equivalence relations are the basis of equa- tion solving. Reflexive: a = a . Symmetric: If a = b , then b = a . Transitive: If a = b and b = c , then a = c . Substitution: If a = b , then a may be replaced by b in any equation or expression. Operational Properties: These concern addition, subtraction, multiplication, and divi- sion, as summarized in the table below. Property Addition Multiplication Commutative: a + b = b + a a × b = b × a Associative: ( a + b ) + c = a + ( b + c ) ( a × b ) × c = a × ( b × c ) Identity: a + 0 = 0 + a = a 1 × a = a × 1 = a Inverse: a + (–a ) = 0 a × 1/ a = 1 Other Properties: Distributive of multiplication over addition: a × ( b + c ) = ( a × b ) + ( a × c ) ( a + b ) × c = ( a × c ) + ( b × c ) Zero property of multiplication: If a × b = 0, then either a = 0 or b = 0 Zero property of division: If a / b = 0, then a = 0 ( b ≠ 0) Integers (Signed Numbers).— Positive whole numbers extend to the right of zero on the number line. Negative whole numbers extend to the left of zero. Together with zero, these make up the integers (sometimes called signed numbers ): {. . . ,– 2, – 1, 0, 1, 2, . . .}. The sciences (as well as economics and other fields) deal with negative as well as non- negative quantities. Temperature is an obvious example; so is land altitude, which can be
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