Slide 5 - 1 Copyright 2009 Pearson Education, Inc. Slide 5 - 1 Copyright 2009 Pearson Education, Inc. Chapter 1 Number Theory and the Real Number System.

Slide 5 - 1Copyright © 2009 Pearson Education, Inc. Slide 5 - 1Copyright © 2009 Pearson Education, Inc.

Chapter 1 Number Theory and the Real Number

System

• An introduction to number theory• Prime numbers• Integers, rational numbers, irrational numbers, and real

numbers• Properties of real numbers• Rules of exponents and scientific notation

WHAT YOU WILL LEARN

Slide 5 - 2Copyright © 2009 Pearson Education, Inc.

1.1 Number Theory

• The study of numbers and their properties.• The numbers we use to count are called natural

numbers, or counting numbers.

N = { 1, 2, 3, 4, }• The natural numbers that are multiplied together to

equal another natural number are called factors of the product.

• Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

Slide 5 - 3Copyright © 2009 Pearson Education, Inc.

DivisorsIf a and b are natural numbers and the quotient of b

divided by a has a remainder of 0, then we say that a is a divisor of b or a divides b.

Prime and Composite Numbers• A prime number is a natural number greater than 1

that has exactly two factors (or divisors), itself and 1.• A composite number is a natural number that is

divisible by a number other than itself and 1.• The number 1 is neither prime nor composite, it is

called a unit.

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The Fundamental Theorem of Arithmetic• Every composite number can be expressed as a

unique product of prime numbers.• This unique product is referred to as the prime

factorization of the number.

There are two ways to do this as we show in the next two slides.

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Example of branching method for the number 3190

Therefore, the prime factorization of

3190 =

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Example of division method

• Write the prime factorization of 663.

• The prime factorization of 663 is

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Finding the LCM of Two or More Numbers• Determine the prime factorization of each number.• List each prime factor with the greatest exponent that

appears in any of the prime factorizations.• Determine the product of the factors found in step 2. Example (LCM)• Find the LCM of 63 and 105.

• So, the LCM is

Slide 5 - 8Copyright © 2009 Pearson Education, Inc.

Example of LCM

• Find the LCM of 48 and 54.• Prime factorizations of each:

48 =54 =

LCM =

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1.2 Integers • The set of integers consists of 0, the natural

numbers, and the negative natural numbers. • Integers = {…–4, –3, –2, –1, 0, 1, 2, 3 4,…}• On a number line, the positive numbers extend to

the right from zero; the negative numbers extend to the left from zero.

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Addition of Integers Subtraction of Integers Multiplication of Integers Division of Integers

Will all be done using your calculator.Do not confuse the (-) sign with the subtraction sign

when doing these calculations!

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Reducing Fractions

• In order to reduce a fraction to its lowest terms, we divide both the numerator and denominator by the greatest common divisor.

• Example: Reduce to its lowest terms.

• Solution:

7281

Slide 5 - 12Copyright © 2009 Pearson Education, Inc.

Improper Fractions• Rational numbers greater than 1 or less than –

1 that are not integers may be written as mixed numbers, or as improper fractions.

• An improper fraction is a fraction whose numerator is greater than its denominator. An example of an improper fraction is .

To write this as a mixed number we write it

as 2

125

53

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We can do the reverse process:• Convert to an improper fraction.

• An Example of going the other way: • Convert to a mixed number.

5

710

7236

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Multiplication of Fractions

• Division of Fractions

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Example: Multiplying Fractions

• Evaluate the following.

a)

b) 23

7

16 1

34

2

12

Slide 5 - 16Copyright © 2009 Pearson Education, Inc.

Example: Dividing Fractions

• Evaluate the following.a)

b) 23

67

58

45

Slide 5 - 17Copyright © 2009 Pearson Education, Inc.

Example: Add or Subtract Fractions with alike denominators

Add:

Subtract:

49

39

1116

3

16

Slide 5 - 18Copyright © 2009 Pearson Education, Inc.

Example with unalike denominators:

• Evaluate:

• Solution: first find the LCD:

7

12

110

.

Slide 5 - 19Copyright © 2009 Pearson Education, Inc.

Irrational Numbers• An irrational number is a real number whose

decimal representation is a nonterminating, nonrepeating decimal number.

• Examples of irrational numbers:• 5.12639573…• 6.1011011101111…• 0.523225222…• Please note that these are different from a

repeating decimal as shown in the next slide.

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Terminating or Repeating Decimal Numbers

• Every rational number when expressed as a decimal number will be either a terminating or a repeating decimal number.

• Examples of terminating decimal numbers are 0.7, 2.85, 0.000045

• Examples of repeating decimal numbers 0.44444… which may be written

• And 0.23232323… which can be written 0.4,

23.

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Perfect Square

• Any number that is the square of a natural number is said to be a perfect square.

• The numbers 1, 4, 9, 16, 25, 36, and 49 are the first few perfect squares.

Slide 5 - 22Copyright © 2009 Pearson Education, Inc.

Radicals• are all irrational numbers. The symbol

is called the radical sign. The number or expression inside the radical sign is called the radicand.

• The principal (or positive) square root of a number n, is the positive number that when multiplied by itself, gives n.

For example, since

and since

2, 17, 53

416 164*4

749 497*7

Slide 5 - 23Copyright © 2009 Pearson Education, Inc.

Product Rule for Radicals

• Simplify:a)

b)

40

125

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Example: Adding or Subtracting Irrational Numbers

• Simplify:• Simplify: 4 7 3 7 8 5 125

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Multiplication of Radicals

6 54

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Example: Division

• Divide:

• Solution:

• Divide:

• Solution:

16

4

144

2

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A denominator is rationalized when it contains no radical expressions.

• Rationalize the denominator of

• Solution:

8

12.

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Commutative Property• Addition

a + b = b + a for any real numbers a and b.

• 8 + 12 = 12 + 8 is a true statement.• 5 9 = 9 5 is a true statement.

• Note: The commutative property does not hold true for subtraction or division.

• Multiplication a • b = b • a for any real numbers a and b.

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Associative Property

• Addition (a + b) + c = a + (b + c),

for any real numbers a, b, and c.

• Multiplication (a • b) • c = a • (b • c),

for any real numbers a, b, and c.

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Example

• (3 + 5) + 6 = 3 + (5 + 6) is true.

• (4 6) 2 = 4 (6 2) is true.

• Note: The associative property does not hold true for subtraction or division.

Slide 5 - 31Copyright © 2009 Pearson Education, Inc.

Distributive Property

• Distributive property of multiplication over additiona • (b + c) = a • b + a • cfor any real numbers a, b, and c.

• Example: 6 • (r + 12) =

Slide 5 - 32Copyright © 2009 Pearson Education, Inc.

1.6 Exponents and Scientific Notation

• When a number is written with an exponent, there are two parts to the expression: baseexponent

• The exponent tells how many times the base should be multiplied together.

45 44444

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Rules of exponentsProduct rule: • Simplify: 34 • 39

34 • 39 = 34 + 9 = 313

• Simplify: 64 • 65

64 • 65 = 64 + 5 = 69

Quotient Rule:

• Simplify:

• Simplify

Power Rule:

Simplify: (32)3

Simplify: (23)5

aaa nmnm

aaa nmn

m

772

5

998

15

Zero exponent rule:

• Simplify:

•Simplify:

Negative Exponent Rule:

•Simplify :

•Simplify :

10 a

)3( 0y

y3 0

aa mm 1

5 8

5 6

1

aa mnnm)(

Slide 5 - 34Copyright © 2009 Pearson Education, Inc.

Scientific Notation• Many scientific problems deal with very large or very small numbers.• 93,000,000,000,000 is a very large number.• 0.000000000482 is a very small number.• Scientific notation is a shorthand method used to write these numbers.• 9.3 1013 and 4.82 10–10 are two examples of numbers in scientific notation.• Write each number in scientific notation.

a) 1,265,000,000.

b) 0.000000000432

Slide 5 - 35Copyright © 2009 Pearson Education, Inc.

Example going the other way:

• Write each number in decimal notation.a) 4.67 105

b) 1.45 10–7