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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