Rational and Real numbers

1
Chapter 9

• Rational and Real numbers

2
Introduction
Fractions Integers
Counting numbers
Whole numbers
is a subset of
is a subset of
3
Definition A rational number is a quotient of
the form where a and
b are integers with b ? 0.
Fractions Integers
Whole numbers
Rational numbers
4
Closure properties of rational numbers

• The set of rational numbers is closed under all 4
  operations , - , , .
• In other words,
• The sum of any two rational numbers is still a
  rational number.
• The difference of any two rational numbers is
  still a rational number.
• The product of any two rational numbers is still
  a rational number.
• The quotient of any rational number by any
  non-zero rational number is still a rational
  number.

5
Decimal representations
Every rational number has either a terminating or
a repeating decimal representation. However, the
representation may not be unique. For example, ¼
0.25, but it is also equal to 0.24999999
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Real numbers
A real number is a number that either has a
finite decimal representation or has an infinite
decimal representation. In particular, the set
of real numbers include those numbers with
non-terminating and non-repeating decimal
representations such as p
3.141592653589793
These number are called irrational numbers.
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Is there any useful irrational number at all?
is irrational and it is the length of any
diagonal in a unit square.
1
1
never terminates and never repeats.
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Theorem The square root of any whole number is
either a whole number or an irrational number
Example Since we know that 23 is not a perfect
square, will not be a whole number. The
above theorem tells us that is not even a
rational number, that means its decimal
representation will not be repeating or
terminating.
9
All decimals
Irrational numbers
Rational numbers
(non-repeating, non-terminating decimals)
Terminating decimals
Repeating decimals
10
Converting Fractions to Decimals
Fact The decimal expansion of any fraction a/b
is either terminating or repeating.

• Theorem
• If the fraction a/b is in its reduced form, then
  its decimal expansion is terminating if and only
  if b is one of the following forms.
• a product of 2s only,
• a product of 5s only
• a product of 2s and 5s only.

Examples
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Converting Fractions to Decimals
Now we know what kind of fractions will
have terminating decimal expansions, but can we
predict how many decimal places there will be in
the expansion?
Theorem If the fraction a/b is in its reduced
form, and
b 2m5n
then the decimal expansion of a/b is
terminating with number of decimal places exactly
equal to
maxm, n
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Converting Fractions to Decimals
One more question If we know that a
certain fraction has repeating decimal expansion,
can we predict its cycle length?
Unfortunately there is no formula to
calculate the precise cycle length. All we know
is an upper bound and a small (not too helpful)
property.
Theorem If p is a prime number other than 2 and
5, then the cycle length of 1/p is at most (p
1), and the cycle length must divide (p 1).
Example The cycle length of 1/31 is at most 30,
and it must divide 30. In fact, the cycle length
of 1/31 is 15.
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Converting Fractions to Decimals
More examples
There is no obvious pattern on the cycle length,
and a large denominator can have a small cycle
length.
prime number p cycle length of 1/p
7 6
11 2
13 6
17 16
19 18
23 22
29 28
31 15
37 3
41 5
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Converting Fractions to Decimals

• More facts (optional)
• If p is a prime other than 2 or 5, then the cycle
  length of 1/(p2) is at mostp(p 1) and the
  cycle length must divide p(p 1).

Example cycle length of 1/7 is 6, cycle length
of 1/49 is 42 ( 76).
2. If p and q are different primes other than 2
and 5, then the cycle length of 1/pq will be
at most (p 1)(q 1) and divides (p 1)(q 1).
Example Cycle length of 1/(711) is less than
610 60, and must divide 60. It turns out that
the cycle length of 1/77 is only 6.
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Converting Decimals to Fractions
From the previous theorem, we see that only
repeating or terminating decimals can be
converted to a fraction.

• Procedures
• terminating decimal, eg.
  0.35742 35742 /100000The number of 0s in the
  denominator is equal to the number of decimal
  places.
• repeating decimals of type I, eg
• 0.2222 2 /9
• 0.47474747 47 /99
• 
  0.528528528 528 /999

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Converting Decimals to Fractions

• Procedures
• repeating decimals of type I, eg.
  0.2222 2 /9 0.47474747
  47 /99 0.528528528
  528 /999
• repeating decimals of type II, eg.
• 0.0626262 62 /990
• 0.00626262 62 /9900
• 
  0.000344934493449 3449 /9999000

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Converting Decimals to Fractions

• Procedures
• repeating decimals of type II, eg.
• 0.0626262 62 /990
  
• 0.00626262 62 /9900
• 0.000344934493449
  3449 /9999000
• 4) repeating decimals of type III, eg.
  0.576666 0.57 0.006666
  

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Converting Decimals to Fractions

• Procedures
• repeating decimals of type II, eg.
• 0.0626262 62 /990
  
• 0.00626262 62 /9900
• 0.000344934493449
  3449 /9999000
• 4) repeating decimals of type III, eg.
  0.576666 0.57 0.006666
  

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Irrational numbers
Even though most irrational numbers have
unpredictable decimal expansions, some do have
certain patterns, except that the patterns are
not repeating. Examples 1. 0.123456789101112131
41516 2. 0.1010010001000010000010000
001
We can therefore deduce that there must be many
irrational nmbers. In fact, there are more
irrational numbers than rational numbers. In
addition, there is at least one irrational number
between any two given rational numbers. (see lab
10)