Obj. 1 Number Theory

Unit 1 Functions and Relations

Concepts and Objectives

� Number Theory (Obj. #1)

� Identify subsets of real numbers

� Simplify expressions using order of operations

� Identify real number axioms

� Convert between fractions and decimals

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.

Ancient civilizations, such as the Babylonians, also used

ratios to compare quantities.

� One of the greatest mathematical advances was the

introduction of the number 0.

Properties of Real Numbers

� Closure Property

� a + b ∈�

� ab ∈ �

� Commutative Property

� a + b = b + a

� ab = ba

� Associative Property

� (a + b) + c = a + (b + c)

� (ab)c = a(bc)

� Identity Property

� a + 0 = a

� a i 1 = a

� Inverse Property

� a + (–a) = 0

�

� Distributive Property

� a(b + c) = ab + ac

For all real numbers a, b, and c:

i

1 =1 a

a

Properties of Real Numbers

� The properties are also called axioms.

� 0 is called the additive identity and 1 is called the

multiplicative identity.

� Notice the relationships between the identities and the

inverses (called the additive inverse and the

multiplicative inverse).

� Saying that a set is “closed” under an operation (such as

multiplication) means that performing that operation on

numbers in the set will always produce an answer that is

also in the set – there are no answers outside the set.

Properties of Real Numbers

� Examples

� The set of natural numbers (�) is not closed under the

operation of subtraction. Why?

� 5 – 7 = –2, which is not in �.

� –20 ÷ 5 = –4. Does this show that the set of integers is

closed under division?

� No. Any division that has a remainder is not in �.

Order of Operations

� Parentheses (or other grouping symbols, such as square

brackets or fraction bars) – start with the innermost set,

following the sequence below, and work outward.

� Exponents

� Multiplication

� Division

� Addition

� Subtraction

working from left to right

working from left to right

Order of Operations

� Use order of operations to explain why

� We can think of –3 as being –1 i 3. Therefore we have

It should be easier now to see that on the left side we

multiply first and then apply the exponent, and on the

right side, we apply the exponent and then multiply.

( )− ≠ −2 23 3

( )− ≠ −i i

2 21 3 1 3

Order of Operations

Work the following examples without using your calculator.

1.

2.

3.

− + ÷i2 5 12 3

( ) ( )( )− − + −3

4 9 8 7 2

( )( )

( )

− + − − ÷

− −

8 4 6 12

4 3

1. –6

2. –60

−6

3. 7

Rational Numbers

� 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.

� 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 4.

2. 64-½ 5.

3. 6. –5.4322986π

π

4

20.3

0.9

6.3

1

16

1

8

4

1

7

=1 61

203 3

−54322986

10000000

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.

� Theorem: Let n be a positive integer. Then is either

an integer or it is irrational.

n

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

ba

ba

ba ba

ba

ba

Name of

IntervalNotation

Inequality

DescriptionNumber Line Representation

finite, open (a, b) a < x < b

finite, closed [a, b] a ≤ x ≤ b

finite, half-

open(a, b]

[a, b)

a < x ≤ b

a ≤ x < b

infinite, open (a, ∞)

(-∞, b)

a < x < ∞

-∞ < x < b

infinite,

closed[a, ∞)

(-∞, b]

a ≤ x < ∞

-∞< x ≤ b

ba

ba

ba

ba

b

a

b

a

Finite and Repeating Decimals

� If the decimal representation of a rational number has a

finite number of digits after the decimal, then it is said to

terminate.

� 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.

� It turns out that for a rational number where b > 0,

the period is at most b – 1.

a

b

Finite and Repeating Decimals

� Example: Use long division (yes, long division) to find

the decimal representation of and find its period.

What is the period of this decimal?

462

13

46235.538461

13=

6

Finite and Repeating Decimals

� 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.

1 2 30. ... td d d d ≠( 0)td

0.3, 0.142857, 0.1, 0.09, 0.0769231 2 30. ... pd d d d

0.16, 0.083, 0.0714285, 0.061 2 3 1 2 30. ... ...t t t t t pd d d d d d d d

+ + + +

Type of Decimal Examples General Form

terminating 0.5, 0.25, 0.2, 0.125, 0.0625

simple-periodic

delayed-periodic

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 can state that the decimal 0.d1d2d3…dt can be written

as .

Example:

1 2 3 ...

10t

t

d d d d

= =845 169

0.8451000 200

Decimal Representation

� 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:

To put it more generally, to convert , we can

write it as .

3 10.3

9 3= =

9 10.09

99 11= =

153846 20.153846

999999 13= =

1 2 30. ...

pd d d d

1 2 3 ...

999...9

pd d d d

Decimal Representation

� For delayed-periodic decimals, the process is a little

more complicated. Consider the following:

What is the decimal representation of ?

is the product of what two fractions?

Notice that the decimal representation has

characteristics of each factor (2 terminating digits and 1

repeating digit).

1

120.083

1

12i

1 1

4 3

Decimal Representation

� 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 can be written

as the fraction , where N is the integer

d1d2d3…dtdt+1dt+2dt+3…dt+p – d1d2d3…dt .

1 2 3 1 2 30. ... ...

t t t t t pd d d d d d d d

+ + + +

( )−10 10 1t p

N

Decimal Representation

� Example: Convert the decimal to a

fraction.

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.

0.467988654

( )−

= =−

3 6

467988654 467 4679881870. 988654

99999900010 10 1467

Homework

� Assign. 1 WS

� I will post solutions to some of the homework problems

on my blog: http://mathblog.wordpress.com.

� Remember, homework is due at the beginning of class, and

I will not give credit for late work.