1 1 number theory

Unit 1 Functions and Relations
 1-1 Number Theory
 Number Systems
 Rational and Irrational Numbers
 1-2 Functions and Linear Graphs
 Functions and Function Notation
 1-1 and Onto
 Graphing
 1-3 Equations and Inequalities
 Solving Linear and Quadratic Equations and Inequalities
 Solving for a Variable

1-1 Number Theory

Unit 1 Functions and Relations

Number Systems
 What we currently know as the set of real numbers was
only formulated around 1879. We usually present this
as sets of numbers.

Number Systems
 The set of natural numbers () and the set of integers
() have been around since ancient times, probably
prompted by the need to maintain trade accounts. They
also used ratios to compare quantities.
 One of the greatest mathematical advances was the
introduction of the number 0.
 The Greeks, specifically Pythagoras of Samos, originally
believed that the lengths of all segments in geometric
objects could be expressed as ratios of positive integers.

Rational Numbers
 A number is a rational number () if and only if it can be
expressed as the ratio (or quotient) of two integers.
 Rational numbers include decimals as well as fractions.
The definition does not require that a rational number
must be written as a quotient of two integers, only that it
can be.

Examples
 Example: Prove that the following numbers are
rational numbers by expressing them as ratios of
integers.
(1) 2-4
(2) 64-½
(3) 20.3
(4) –5.4322986
(5) 0.9
6.3
(6)
4


Irrational Numbers
 Unfortunately, the Pythagoreans themselves later
discovered that the side of a square and its diagonal
could not be expressed as a ratio of integers.
 Prove 2 is irrational.
Proof (by contradiction): Assume 2 is rational. This
means that there exists relatively prime integers a and b
such that
a a2
2  2 2
b b
2b2  a2 , therefore, a is even

Irrational Numbers
This means there is an integer j such that 2j=a.
2b   2 j 
2 2

2b2  4 j 2
b2  2 j 2  b is even
If a and b are both even, then they are not relatively
prime which is a contradiction. Therefore, 2 is
irrational.
 Theorem: Let n be a positive integer. Then n is either
an integer or it is irrational.

Real Numbers
 The number line is a geometric model of the system of
real numbers. Rational numbers are thus fairly easy to
represent:

 What about irrational numbers? Consider the following:
(1,1)


2

Real Numbers
 In this way, if an irrational number can be identified
with a length, we can find a point on the number line
corresponding to it.
 What this emphasizes is that the number line is
continuous—there are no gaps.

Intervals
Name of Inequality
Notation Number Line Representation
Interval Description
a b
finite, open
(a, b) a<x<b a b
a b
finite, closed
[a, b] axb
a b
a b
b
finite, half-
open
(a, b] a<xb a b
[a, b) ax<b
a b
a b
infinite, open
(a, ) a < x < 
a
(-, b) - < x < b
b
a b
infinite,
closed
[a, ) a  x < 
a
[-, b] -< x  b
b

Finite and Repeating Decimals
 If a nonnegative real number x can be expressed as a
finite sum of of the form
d1 d2 dt
x  D   2  ...  t
10 10 10
where D and each dn are nonnegative integers and
0  dn  9 for n = 1, 2, …, t, then D.d1d2…dt is the finite
decimal representing x.

 If the decimal representation of a rational number does
not terminate, then the decimal is periodic (or
repeating). The repeating string of numbers is called the
period of the decimal.
a
 It turns out that for a rational number where b > 0,
the period is at most b – 1. b


 Example: Use long division (yes, long division) to find
462
the decimal representation of and find its period.
13
462
 35.538461
13

What is the period of this decimal? 6

 The repeating portion of a decimal does not necessarily
start right after the decimal point. A decimal which
starts repeating after the decimal point is called a
simple-periodic decimal; one which starts later is called a
delayed-periodic decimal.

Type of Decimal Examples General Form
terminating 0.5, 0.25, 0.2, 0.125, 0.0625 0.d1d2 d3 ...dt (dt  0)

simple-periodic 0.3, 0.142857, 0.1, 0.09, 0.076923 0.d1d2 d3 ...dp

delayed-periodic 0.16, 0.083, 0.0714285, 0.06 0.d1d2 d3 ...dt dt 1dt 2 dt 3 ...dt  p

Decimal Representation
 If we know the fraction, it’s fairly straightforward
(although sometimes tedious) to find its decimal
representation. What about going the other direction?
How do we find the fraction from the decimal, especially
if it repeats?
 We’ve already seen how to represent a terminating
decimal as the sum of powers of ten. More generally, we
can state that the decimal 0.d1d2d3…dt can be written as
M
t , where M is the integer d1d2d3…dt.
10

 For simple-periodic decimals, the “trick” is to turn them
into fractions with the same number of 9s in the
denominator as there are repeating digits and simplify:

3 1 9 1 153846 2
0.3   0.09   0.153846  
9 3 99 11 999999 13

To put this more generally, the decimal 0.d1d2d3 ...dp
M
can be written as the fraction , where M is the
10  1
p

integer d1d2d3…dp.

 For delayed-periodic decimals, the process is a little
more complicated. It turns out you can break a delayed-
periodic decimal into a product of terminating and
simple-periodic decimals, so the general form is also a
product of the general forms: The decimal
0.d1d2d3 ...dt dt 1dt 2dt 3 ...dt p can be written as the fraction
M , where M is the integer (note the difference)

10t 10p  1 
d1d2d3 ...dt dt 1dt 2dt 3 ...dt p  d1d2d3 ...dt .

 Example: Convert the decimal 0.467988654 to a
fraction.
467988654  467 467988187
0.467988654  
3 6

10 10  1 
999999000

It’s possible this might reduce, but we can see that there
are no obvious common factors (2, 3, 4, 5, 6, 8, 9, or 10),
so it’s okay to leave it like this.

Absolute Value
 The absolute value of a real number a, denoted by |a|, is
the distance from 0 to a on the number line. This
distance is always taken to be nonnegative.

x if x  0
x 
 x if x  0

Absolute Value
 Example: Rewrite each expression without absolute
value bars.

1. 3 1

2. 2 

x
3. , if x  0
x

1 1 number theory

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