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

Slide Copyright © 2009 Pearson Education, Inc. Slide Copyright © 2009 Pearson Education, Inc. WHAT YOU WILL LEARN 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 Arithmetic and geometric sequences The Fibonacci sequence

Slide Copyright © 2009 Pearson Education, Inc. Slide Copyright © 2009 Pearson Education, Inc. 1.1 Number Theory

Slide Copyright © 2009 Pearson Education, Inc. Number Theory The study of numbers and their properties. The numbers we use to count are called natural numbers,, or counting numbers.

Slide Copyright © 2009 Pearson Education, Inc. Factors 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 Copyright © 2009 Pearson Education, Inc. Divisors If 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.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. Rules of Divisibility 285The number ends in 0 or since 44 / 4 The number formed by the last two digits of the number is divisible by since = 18 The sum of the digits of the number is divisible by The number is even.2 ExampleTestDivisible by

Slide Copyright © 2009 Pearson Education, Inc. Divisibility Rules, continued 730The number ends in since = 18 The sum of the digits of the number is divisible by since 848 / 8 The number formed by the last three digits of the number is divisible by The number is divisible by both 2 and 3. 6 ExampleTestDivisible by

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. Finding Prime Factorizations Branching Method: – Select any two numbers whose product is the number to be factored. – If the factors are not prime numbers, continue factoring each number until all numbers are prime.

Slide Copyright © 2009 Pearson Education, Inc. Example of branching method Therefore, the prime factorization of 3190 =

Slide Copyright © 2009 Pearson Education, Inc. Division Method 1. Divide the given number by the smallest prime number by which it is divisible. 2.Place the quotient under the given number. 3.Divide the quotient by the smallest prime number by which it is divisible and again record the quotient. 4.Repeat this process until the quotient is a prime number.

Slide Copyright © 2009 Pearson Education, Inc. Example of division method Write the prime factorization of 663. The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is

Slide Copyright © 2009 Pearson Education, Inc. Greatest Common Divisor The greatest common divisor (GCD) of a set of natural numbers is the largest natural number that divides (without remainder) every number in that set.

Slide Copyright © 2009 Pearson Education, Inc. Finding the GCD of Two or More Numbers Determine the prime factorization of each number. List each prime factor with smallest exponent that appears in each of the prime factorizations. Determine the product of the factors found in step 2.

Slide Copyright © 2009 Pearson Education, Inc. Example (GCD) Find the GCD of 63 and 105.

Slide Copyright © 2009 Pearson Education, Inc. Example (GCD) Find the GCD of 63 and = = Smallest exponent of each factor: 3 and 7 So, the GCD is 3 7 = 21.

Slide Copyright © 2009 Pearson Education, Inc. Least Common Multiple The least common multiple (LCM) of a set of natural numbers is the smallest natural number that is divisible (without remainder) by each element of the set.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. Example (LCM) Find the LCM of 63 and 105.

Slide Copyright © 2009 Pearson Education, Inc. Example (LCM) Find the LCM of 63 and = = Greatest exponent of each factor: 3 2, 5 and 7 So, the LCM is = 315.

Slide Copyright © 2009 Pearson Education, Inc. Example of GCD and LCM Find the GCD and LCM of 48 and 54

Slide Copyright © 2009 Pearson Education, Inc. Example of GCD and LCM Find the GCD and LCM of 48 and 54. Prime factorizations of each: 48 = = = = GCD = 2 3 = 6 LCM = = 432

Slide Copyright © 2009 Pearson Education, Inc. Slide Copyright © 2009 Pearson Education, Inc. 1.2 The Integers

Slide Copyright © 2009 Pearson Education, Inc. Whole Numbers The set of whole numbers contains the set of natural numbers and the number 0. Whole numbers = {0,1,2,3,4,…}

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. Writing an Inequality Insert either > or < in the space between the paired numbers to make the statement correct. a) -3 ___ -1 b) -9 ___ -7 c) 0 ___ -4d) 6 ___ 8

Slide Copyright © 2009 Pearson Education, Inc. Writing an Inequality Insert either > or < in the box between the paired numbers to make the statement correct. a) -3 < -1 b) -9 < -7 c) 0 > -4d) 6 < 8

Slide Copyright © 2009 Pearson Education, Inc. Subtraction of Integers a – b = a + (-b) Evaluate: a) –7 – 3 = –7 + (–3) = –10 b) –7 – (–3) = –7 + 3 = –4

Slide Copyright © 2009 Pearson Education, Inc. Properties Multiplication Property of Zero Division For any a, b, and c where b ≠ 0, means that c b = a.

Slide Copyright © 2009 Pearson Education, Inc. Rules for Multiplication The product of two numbers with like signs (positive x positive or negative x negative) is a positive number. The product of two numbers with unlike signs (positive x negative or negative x positive) is a negative number.

Slide Copyright © 2009 Pearson Education, Inc. Examples Evaluate: a) (3)(-4)b) (-7)(-5) c) 8 7d) (-5)(8)

Slide Copyright © 2009 Pearson Education, Inc. Examples Evaluate: a) (3)(-4)b) (-7)(-5) c) 8 7d) (-5)(8) Solution: a) (3)(-4) = -12b) (-7)(-5) = 35 c) 8 7 = 56d) (-5)(8) = -40

Slide Copyright © 2009 Pearson Education, Inc. Rules for Division The quotient of two numbers with like signs (positive ÷ positive or negative ÷ negative) is a positive number. The quotient of two numbers with unlike signs (positive ÷ negative or negative ÷ positive) is a negative number.

Slide Copyright © 2009 Pearson Education, Inc. Example Evaluate: a) b) c) d)

Slide Copyright © 2009 Pearson Education, Inc. Example Evaluate: a) b) c) d) Solution: a) b) c) d)

Slide Copyright © 2009 Pearson Education, Inc. Slide Copyright © 2009 Pearson Education, Inc. 1.3 The Rational Numbers

Slide Copyright © 2009 Pearson Education, Inc. The Rational Numbers The set of rational numbers, denoted by Q, is the set of all numbers of the form p/q, where p and q are integers and q ≠ 0. The following are examples of rational numbers:

Slide Copyright © 2009 Pearson Education, Inc. Fractions Fractions are numbers such as: The numerator is the number above the fraction line. The denominator is the number below the fraction line.

Slide Copyright © 2009 Pearson Education, Inc. 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:

Slide Copyright © 2009 Pearson Education, Inc. Mixed Numbers A mixed number consists of an integer and a fraction. For example, 3 ½ is a mixed number. 3 ½ is read “three and one half” and means “3 + ½”.

Slide Copyright © 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.

Slide Copyright © 2009 Pearson Education, Inc. Converting a Positive Mixed Number to an Improper Fraction Multiply the denominator of the fraction in the mixed number by the integer preceding it. Add the product obtained in step 1 to the numerator of the fraction in the mixed number. This sum is the numerator of the improper fraction we are seeking. The denominator of the improper fraction we are seeking is the same as the denominator of the fraction in the mixed number.

Slide Copyright © 2009 Pearson Education, Inc. Example Convert to an improper fraction.

Slide Copyright © 2009 Pearson Education, Inc. Example Convert to an improper fraction.

Slide Copyright © 2009 Pearson Education, Inc. Converting a Positive Improper Fraction to a Mixed Number Divide the numerator by the denominator. Identify the quotient and the remainder. The quotient obtained in step 1 is the integer part of the mixed number. The remainder is the numerator of the fraction in the mixed number. The denominator in the fraction of the mixed number will be the same as the denominator in the original fraction.

Slide Copyright © 2009 Pearson Education, Inc. Example Convert to a mixed number.

Slide Copyright © 2009 Pearson Education, Inc. Example Convert to a mixed number. The mixed number is

Slide Copyright © 2009 Pearson Education, Inc. 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, Examples of repeating decimal numbers … which may be written

Slide Copyright © 2009 Pearson Education, Inc. Multiplication of Fractions Division of Fractions

Slide Copyright © 2009 Pearson Education, Inc. Example: Multiplying Fractions Evaluate the following. a) b)

Slide Copyright © 2009 Pearson Education, Inc. Example: Multiplying Fractions Evaluate the following. a) b)

Slide Copyright © 2009 Pearson Education, Inc. Example: Dividing Fractions Evaluate the following. a) b)

Slide Copyright © 2009 Pearson Education, Inc. Example: Dividing Fractions Evaluate the following. a) b)

Slide Copyright © 2009 Pearson Education, Inc. Addition and Subtraction of Fractions

Slide Copyright © 2009 Pearson Education, Inc. Example: Add or Subtract Fractions Add: Subtract:

Slide Copyright © 2009 Pearson Education, Inc. Example: Add or Subtract Fractions Add: Subtract:

Slide Copyright © 2009 Pearson Education, Inc. Fundamental Law of Rational Numbers If a, b, and c are integers, with b ≠ 0, c ≠ 0, then

Slide Copyright © 2009 Pearson Education, Inc. Example: Evaluate:

Slide Copyright © 2009 Pearson Education, Inc. Example: Evaluate: Solution:

Slide Copyright © 2009 Pearson Education, Inc. Slide Copyright © 2009 Pearson Education, Inc. 1.4 The Irrational Numbers and the Real Number System

Slide Copyright © 2009 Pearson Education, Inc. Pythagorean Theorem Pythagoras, a Greek mathematician, is credited with proving that in any right triangle, the square of the length of one side (a 2 ) added to the square of the length of the other side (b 2 ) equals the square of the length of the hypotenuse (c 2 ). a 2 + b 2 = c 2

Slide Copyright © 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:

Slide Copyright © 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.

Slide Copyright © 2009 Pearson Education, Inc. Principal Square Root The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n. For example,

Slide Copyright © 2009 Pearson Education, Inc. 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 Copyright © 2009 Pearson Education, Inc. Product Rule for Radicals Simplify: a) b)

Slide Copyright © 2009 Pearson Education, Inc. Addition and Subtraction of Irrational Numbers To add or subtract two or more square roots with the same radicand, add or subtract their coefficients. The answer is the sum or difference of the coefficients multiplied by the common radical.

Slide Copyright © 2009 Pearson Education, Inc. Example: Adding or Subtracting Irrational Numbers Simplify:

Slide Copyright © 2009 Pearson Education, Inc. Multiplication of Irrational Numbers Simplify:

Slide Copyright © 2009 Pearson Education, Inc. Quotient Rule for Radicals

Slide Copyright © 2009 Pearson Education, Inc. Example: Division Divide: Solution: Divide: Solution:

Slide Copyright © 2009 Pearson Education, Inc. Rationalizing the Denominator A denominator is rationalized when it contains no radical expressions. To rationalize the denominator, multiply BOTH the numerator and the denominator by a number that will result in the radicand in the denominator becoming a perfect square. Then simplify the result.

Slide Copyright © 2009 Pearson Education, Inc. Example: Rationalize Rationalize the denominator of Solution:

Slide Copyright © 2009 Pearson Education, Inc. Slide Copyright © 2009 Pearson Education, Inc. 1.5 Real Numbers and their Properties

Slide Copyright © 2009 Pearson Education, Inc. Real Numbers The set of real numbers is formed by the union of the rational and irrational numbers. The symbol for the set of real numbers is

Slide Copyright © 2009 Pearson Education, Inc. Relationships Among Sets Irrational numbers Rational numbers Integers Whole numbers Natural numbers Real numbers

Slide Copyright © 2009 Pearson Education, Inc. Properties of the Real Number System Closure If an operation is performed on any two elements of a set and the result is an element of the set, we say that the set is closed under that given operation.

Slide Copyright © 2009 Pearson Education, Inc. Commutative Property Addition a + b = b + a for any real numbers a and b. Multiplication a b = b a for any real numbers a and b.

Slide Copyright © 2009 Pearson Education, Inc. Example = is a true statement = is a true statement. Note: The commutative property does not hold true for subtraction or division.

Slide Copyright © 2009 Pearson Education, Inc. 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.

Slide Copyright © 2009 Pearson Education, Inc. 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 Copyright © 2009 Pearson Education, Inc. Distributive Property Distributive property of multiplication over addition a (b + c) = a b + a c for any real numbers a, b, and c. Example: 6 (r + 12) = 6 r = 6r + 72

Slide Copyright © 2009 Pearson Education, Inc. Slide Copyright © 2009 Pearson Education, Inc. 1.6 Rules of Exponents and Scientific Notation

Slide Copyright © 2009 Pearson Education, Inc. Exponents When a number is written with an exponent, there are two parts to the expression: base exponent The exponent tells how many times the base should be multiplied together.

Slide Copyright © 2009 Pearson Education, Inc. Product Rule Simplify: = = 3 13 Simplify: = = 6 9

Slide Copyright © 2009 Pearson Education, Inc. Quotient Rule Simplify:

Slide Copyright © 2009 Pearson Education, Inc. Zero Exponent Rule Simplify: (3y) 0 (3y) 0 = 1 Simplify: 3y 0 3y 0 = 3 (y 0 ) = 3(1) = 3

Slide Copyright © 2009 Pearson Education, Inc. Negative Exponent Rule Simplify: 6  4

Slide Copyright © 2009 Pearson Education, Inc. Power Rule Simplify: (3 2 ) 3 (3 2 ) 3 = 3 23 = 3 6 Simplify: (2 3 ) 5 (2 3 ) 5 = 2 35 = 2 15

Slide Copyright © 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 is a very small number.

Slide Copyright © 2009 Pearson Education, Inc. Scientific Notation continued Scientific notation is a shorthand method used to write these numbers. 9.3 x and 4.82 x 10 –10 are two examples of numbers in scientific notation.

Slide Copyright © 2009 Pearson Education, Inc. To Write a Number in Scientific Notation 1.Move the decimal point in the original number to the right or left until you obtain a number greater than or equal to 1 and less than Count the number of places you have moved the decimal point to obtain the number in step 1. If the decimal point was moved to the left, the count is to be considered positive. If the decimal point was moved to the right, the count is to be considered negative. 3.Multiply the number obtained in step 1 by 10 raised to the count found in step 2. (The count found in step 2 is the exponent on the base 10.)

Slide Copyright © 2009 Pearson Education, Inc. Example Write each number in scientific notation. a)1,265,000, x 10 9 b) x

Slide Copyright © 2009 Pearson Education, Inc. To Change a Number in Scientific Notation to Decimal Notation 1.Observe the exponent on the a)If the exponent is positive, move the decimal point in the number to the right the same number of places as the exponent. Adding zeros to the number might be necessary. b)If the exponent is negative, move the decimal point in the number to the left the same number of places as the exponent. Adding zeros might be necessary.

Slide Copyright © 2009 Pearson Education, Inc. Example Write each number in decimal notation. a)4.67 x ,000 b)1.45 x 10 –