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Chapter 3
Elementary Number Theory and
Methods of Proof
3.2
Direct Proof and Counterexample 2
Rational Numbers
Rational Numbers
• Definition
– A real number r is rational if, and only if, it can be
expressed as a quotient of tow integers with a
nonzero denominator. A real number that is not
rational is irrational.
– r is a rational ⇔∃integers a and b such that r = a/b
and b ≠ 0.
– (informal) quotient of integers are rational numbers.
– (informal) irrational numbers are real numbers that
are not a quotient of integers.
Example
• Is 10/3 a rational number?
– Yes 10 and 3 are integers and 10/3 is a quotient of integers.
• Is –(5/39) a rational number?
– Yes –(5/39) = -5/39 which is a quotient of integers.
• Is 0.281 rational?
– Yes, 281/1000
• Is 2/0 an irrational number?
– No, division by 0 is not a number of any kind.
• Is 0.12121212… irrational?
– No, 0.12121212… = 12/99
• If m and n are integers and neither m nore n is zero, is (m+n)/mn a
rational number?
– Yes, m+n is integer and mn is integer and non-zero, hence rational.
Generalizing from the Generic
Particular
• Generalizing from the particular can be used to
prove that “every integer is a rational number”
1.
2.
3.
–
arbitrarily select an integer x
show that it is a rational number
repeat until tired
Example:
•
7/1, -9/1, 0/1, 12345/1, -8342/1, …
• Theorem 3.2.1
–
Every integer is a rational number.
Proving Properties of Rational
Numbers
• Sum of rational is rational
– Prove that the sum of any two rationals is rational.
– (formal)∀real numbers r and s, if r and s are
rational then r + s is rational.
– Starting Point: suppose r and s are rational
numbers.
– To Show: r + s is rational
Proving Properties of Rational
Numbers
– r = a/b, s = c/d , for some integers a,b,c,d where b
≠ 0 and d≠0
– it follows that r + s = a/b + c/d
– a/b + c/d = (ad + bc)/bd
– the fraction is a ratio of integers since bd ≠ 0
– ad + bc = p (integer) and bd = q (integer)
– therefore, r + s = p/q is rational by the definition.
• Theorem 3.2.2
– The sum of any two rational numbers is rational.
Properties of Rational Numbers
• Corollary 3.2.3
– The double of a rational number is a rational
number. 2r is rational.
– corollary is a statement whose truth is deduced
from a theorem.