Chapter 5

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Department
Mathematics & Statistics (Sci)
Course
MATH 111
Professor
Nabil Kahouadji
Semester
Winter

Description
MATH 111 – Winter 2012 5: RATIONAL NUMBERS AS FRACTIONS 5.1 The Set or Rational Numbers rational numbers: Q = {a/b | a and b are integers, and b ≠ 0}; ie/ fractions proper fraction: a/b where 0 ≤ a < b; if not, it is an improper fraction 5.1.1 Equivalent or Equal Fractions fundamental law of fractions: let a/b be any fraction and n a nonzero integer. then a/b = an/bn 5.1.2 Simplifying Fractions simplest form: a rational number a/b is in the simplest form if b > 0 and GCD (a,b) = 1; that is, if a and b have no common factor greater than 1 and b > 0 5.1.3 Equality of Fractions equality of fractions: two fractions a/b and c/d are equal if, and only is, ad = bc 5.1.4 Ordering Rational Numbers - if a, b and c are integers and b > 0, then a/b > c/d if, and only if, a > c - if a, b, c and d are integers with b > 0 and d > 0, then a/b > c/d and only if, ad > bc 5.1.5 Denseness of Rational Numbers denseness property for rational numbers: given two different rational numbers a/b and c/d there is another rational number between these two numbers - let a/b and c/d be any rational number with positive denominators, where a/b < c/d then a/b < a+c / b + d < c/d 5.2 Addition, Subtraction, and Estimation with Rational Numbers 5.2.1 Addition of Rational Numbers additional of rational numbers with like denominators: if a/b and c/b are rational numbers, then a/b _ c/b = (a + c) / b - if a/b and c/d are two rational numbers, than a/b + c/d = (ad + bc) / bd 5.2.2 Mixed Numbers mixed number: number made up of an integer and a proper fraction - sometimes inferred 2 ¾ means 2 times ¾ since xy means x times y, but this is not correct - -2 ¾ means – (2 + ¾) or equivalently -2 – ¾ but NOT -2 + ¾ MATH 111 – Winter 2012 5.2.3 Properties of Addition of Rational Numbers additive inverse property of rational numbers: for any rational number a/b, there exists a unique rational number – a/b, the additive inverse of a/b such that: a/b + (- a/b) = 0 = (- a/b) + a/b addition property of equality: if a/b and c/d are any rational numbers such that a/b = c/d, and if e/f is any rational number, then: a/b + e/f = c/d + e/f 5.2.4 Subtraction of Rational Numbers subtraction of rational numbers in terms of addition: if a/b and c/d are any rational numbers, then a/b – c/d is the unique rational number e/f such that a/b = c/d + e/f subtraction of rational numbers with like denominators: if a/b and c/b are rational numbers, then: a/b – c/b = (a – c)/b - if a/b and c/d are any rational numbers, then a/b – c/d = (ad – bc)/bd 5.2.5 Definition of “Greater Than” and “Less Than” in Terms of Subtraction less then and greater than for rational numbers: if a/b and c/d are any rational numbers, then a/b < c/d if c/d – a/b > 0, and c/d > a/b if, and only if a/b < c
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