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Transcript
MATH 1000 /11 Chapter 1 1.2 Symbols and Set of Numbers 1.3 Fractions Sets A set is collection of objects, each of which is called a member or element of the set. Notation: A pair of brace symbols { } encloses the list of elements is a the set of elements containing that set of elements. MATH100/05/ Dr. H. Melikyan Sets of numbers • • • • Natural numbers – {1, 2, 3, 4, 5, 6 . . .} Whole numbers – {0, 1, 2, 3, 4 . . .} Integers – {. . . –3, -2, -1, 0, 1, 2, 3 . . .} Rational numbers – the set of all numbers that can be expressed as a quotient of integers, with denominator 0 • Irrational numbers – the set of all numbers that can NOT be expressed as a quotient of integers • Real numbers – the set of all rational and irrational numbers combined • A Number Line used to represent ordered real numbers has negative numbers to the left of 0 and positive numbers to the right of 0. • Order Property for Real Numbers indicates how to use inequality signs. If a and b are real numbers, a < b means a is to the left of b on a number line. a > b means a is to the right of b on a number line. • Absolute value of a number is the distance of that number away from 0. a 0, since distances are non-negative. | -7 | = 7, |- 0.13| = 0.13, | 3| = 3. Section 1.3 Fraction is a quotient of two numbers. • Numerator is the top number. • Denominator is the bottom number. Simplifying fractions (lowest terms) • Involves factoring numerator and denominator into prime numbers (natural numbers other than 1 whose only factors are 1 and itself : 2, 3, 5, 7, 13, 37 …). • A natural number bigger 1 , that is not prime is called Composite number ( 15, 21, 1222) MATH100/05/ Dr. H. Melikyan Fundamental Principle of Fractions • Can cancel common factors in numerator and denominator. • If a, b, c are real numbers such that b and c 0. a c a bc b MATH100/05/ Dr. H. Melikyan Example Simplify the following fractions. 30 2 35 5 5 48 2 2 2 2 3 2 2 2 8 22 2 11 45 3 3 5 Since there are no common terms, the fraction is already simplified. 12 2 23 1 60 2 2 3 5 5 MATH100/05/ Dr. H. Melikyan Multiplying fractions • Multiply numerators and denominators. a c ac b d bd b, d 0 Dividing fractions • Invert the divisor fraction (the reciprocal of divisor ). • Then multiply the fractions. a c a d ad b d b c bc MATH100/05/ Dr. H. Melikyan b, d, c 0 Adding and subtracting fractions • Required to have the same denominator. • Have to change fractions to equivalent ones until they have same denominator. • Then combine the numerators, denominator will be the common denominator. a c ac b b b a c ac b b b b0 b0 MATH100/05/ Dr. H. Melikyan Example Add the following fractions. 3 9 2 23 12 3 20 20 20 2 25 5 Subtract the following fractions. 7 5 7 6 5 5 42 25 17 10 12 10 6 12 5 60 60 60 MATH100/05/ Dr. H. Melikyan MATH100/05/ Dr. H. Melikyan