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

Concepts and Objectives

Number Theory (Obj. #1) Identify subsets of real numbers Simplify expressions using order of operations Identify real number axioms

Rational Numbers (Obj. #2) 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 1 = a

Inverse Property a + (–a) = 0

Distributive Property a(b + c) = ab + ac

For all real numbers a, b, and c:

1

=1 aa


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.


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

operation of subtraction. Why?

–20 5 = –4. Does this show that the set of integers is closed under division?


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 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

2 21 3 1 3

Order of Operations

Work the following examples without using your calculator.

1.

2.

3.

2 5 12 3

34 9 8 7 2

8 4 6 12

4 3

Order of Operations

Work the following examples without using your calculator.

1.

2.

3.

2 5 12 3

34 9 8 7 2

8 4 6 12

4 3

1. –6

2. –60

6

3. 7

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.

if 0

if 0

x xx

x x

Absolute Value Properties

For all real numbers a and b:

1.

2.

3.

4.

5.

0a

a a

a b ab

( 0)a a

bb b

a b a b

Absolute Value

Example: Rewrite each expression without absolute value bars.

1.

2.

3.

3 1

2

, if 0x

xx

Absolute Value

Example: Rewrite each expression without absolute value bars.

1.

2.

3.

3 1

2

, if 0x

xx

1. 3 1

2. – 2

3. –1

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. –5.4322986

4

20.3

0.96.3

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. –5.4322986

4

20.3

0.96.3

116

18

4

17

1 61

203 3

5432298610000000

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 is irrational.

Proof (by contradiction): Assume is rational. This means that there exist relatively prime integers a and b such that

2

2

2

22 2a ab b

2 22 , therefore, is evenb a a

Irrational Numbers

This means there is an integer j such that 2j=a.

If a and b are both even, then they are not relatively prime. This is a contradiction. Therefore, is irrational.

Theorem: Let n be a positive integer. Then is either an integer or it is irrational.

222 2b j2 22 4b j

2 22 is evenb j b

2

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:

-2 -1 1 2

-1

1

x

y

(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 Interval

NotationInequality

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 a nonnegative real number x can be expressed as a finite sum of of the form

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.

1 22 ...

10 10 10

t

t

d d dx D


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.

ab


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

the decimal representation of and find its period.

What is the period of this decimal?

46213


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

the decimal representation of and find its period.

What is the period of this decimal?

46213

46235.538461

13

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.

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, 0.06 1 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’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

, where M is the integer d1d2d3…dt.10t

M


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 this more generally, the decimal

can be written as the fraction , where M is the

integer d1d2d3…dp.

3 10.3

9 3

9 10.09

99 11

153846 20.153846

999999 13

1 2 30. ... pd d d d

10 1p

M


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.

112

0.083

112

1 1

4 3


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


Example: Convert the decimal to a fraction.

0.467988654


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 4679881870.467988654

99999900010 10 1

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1 1 number theory