ch05MGF1106 - Chapter 5 Number Theory and the Real Number...

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Unformatted text preview: Chapter 5 Number Theory and the Real Number System Slide 5 - 1 5.2 The Integers Slide 5 - 2 Number Theory (from Chapter 5.1) The study of numbers and their properties. The numbers we use to count are called natural numbers, ¥ , or counting numbers. ¥ = {1, 2, 3, 4, 5, ...} Slide 5 - 3 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 5 - 4 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 5 - 5 Inequality A number line can be used to determine the greater (or lesser) of two integers. Numbers on a number line increase from left to right. We us inequality symbols to indicate the greater (or lesser) of two integers. < “less than” > “greater than” Slide 5 - 6 Writing an Inequality Insert either > or < in the box between the paired numbers to make the statement correct. a) −3 −1 −3 < −1 c) 0 −4 0 > −4 b) −9 −7 −9 < −7 d) 6 8 6 <8 Slide 5 - 7 Addition of integers Add the following integers. 3 + -6 = -3 + -7 = -7 + 3 = 8 + (-12) = Slide 5 - 8 Addition of integers Add the following integers. 3 + -6 = -3 -3 + -7 = -10 -7 + 3 = -4 8 + (-12) = -4 Slide 5 - 9 Subtraction of Integers a – b = a + (−b) Evaluate: a) –7 – 3 = –7 + (–3) = –10 b) –7 – (–3) = –7 + 3 = –4 Slide 5 - 10 Subtraction of integers Rule: 1. 2. 3. 4. a – b = a + (-b) & -(-a) = a -6 – 8 = 6 – (-8) = 6–8 = -6 – (-8) = Slide 5 - 11 Subtraction of integers Rule: 1. 2. 3. 4. a – b = a + (-b) & -(-a) = a -6 – 8 = -14 6 – (-8) = 14 6 – 8 = -2 -6 – (-8) = 2 Slide 5 - 12 Properties Multiplication Property of Zero a ×0 = 0 ×a = 0 Division a For any a, b, and c where b ≠ 0, = c means b that c • b = a. Slide 5 - 13 Rules for Multiplication The product of two numbers with like signs (positive × positive or negative × negative) is a positive number. The product of two numbers with unlike signs (positive × negative or negative × positive) is a negative number. Slide 5 - 14 Examples Evaluate: a) (3)(−4) b) (−7)(−5) c) 8 • 7 d) (−5)(8) Solution: a) (3)(−4) = −12 b) (−7)(−5) = 35 c) 8 • 7 = 56 d) (−5)(8) = −40 Slide 5 - 15 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 5 - 16 Example Evaluate: 72 a) 9 −72 −8 c) Solution: a) 72 = 8 9 c) −72 =9 −8 b) −72 9 d) 72 −8 b) −72 = −8 9 d) 72 = −9 −8 Slide 5 - 17 5.3 The Rational Numbers Slide 5 - 18 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: 13 7 2 15 , , − , 1 , 2, 0, 34 8 3 7 Slide 5 - 19 Fractions Fractions are numbers such as: 12 9 , , and . 39 53 The numerator is the number above the fraction line. The denominator is the number below the fraction line. Slide 5 - 20 Reducing Fractions In order to reduce a fraction to its lowest terms, we divide both the numerator and denominator by the greatest common divisor. 72 Example: Reduce to its lowest terms. 81 Solution: 72 = 72 ÷ 9 = 8 81 81 ÷ 9 9 Slide 5 - 21 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 5 - 22 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. 12 An example of an improper fraction is . 5 Slide 5 - 23 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 5 - 24 Example 7 Convert 5 to an improper fraction. 10 7 (10 • 5 + 7) 50 + 7 57 5 = = = 10 10 10 10 Slide 5 - 25 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 5 - 26 Example 236 Convert to a mixed number. 7 33 7 236 21 26 21 5 5 The mixed number is 33 . 7 Slide 5 - 27 Negative Mixed Numbers to Improper Fraction When converting a negative mixed number to an improper fraction (or a negative improper fraction to a mixed number): Ignore the negative sign temporarily. Perform the conversion. Reattach the negative sign. Slide 5 - 28 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 0.4, and 0.2323232323... which may be written 0.23. Slide 5 - 29 Converting Decimal Numbers to Fractions Convert the following terminating decimals to a quotient of integers (fractions). a. 0.6 b. 0.079 c. 1.23 Slide 5 - 30 Converting Decimal Numbers to Fractions Convert the following terminating decimals to a quotient of integers (fractions). a. 0.6 = 6/10 or 3/5 b. 0.079 = 79/1000 c. 1.23 = 1 23/100 Slide 5 - 31 Converting Decimal Numbers to Fractions Convert the following repeating decimals to a quotient of integers (fractions). 0.6 = 0.66 = 0.666 etc. Let, n = 0.6 If we multiply both sides of the equation by 10 we get that. 10n = 6.6 Then subtract 10n = 6.6 -(n = 0.6) 9n = 6 n = 6/9 or 2/3 a. Slide 5 - 32 Converting Decimal Numbers to Fractions Convert the following repeating decimals to a quotient of integers (fractions). 0.64 = 0.6464 = 0.646464 etc. Let, n = 0.64 If we multiply both sides of the equation by 100 we get that. 100n = 64.64 Then subtract 100n = 64.64 -(n = 0.64) 99n = 64 n = 64/99 a. Slide 5 - 33 Multiplication of Fractions a c a • c ac •= = , b ≠ 0, d ≠ 0 b d b • d bd Division of Fractions a c a d ad ÷=•= , b ≠ 0, d ≠ 0, c ≠ 0 b d b c bc Slide 5 - 34 Example: Multiplying Fractions a) Evaluate the following. 3 1 14 ⋅ 2 2 27 ⋅ 3 16 b) 27 2⋅7 ⋅ = 3 16 3 ⋅ 16 14 7 = = 48 24 3 1 7 5 14 ⋅ 2 2 = 4 ⋅ 2 35 3 = =4 8 8 Slide 5 - 35 Example: Dividing Fractions Evaluate the following. a) 2 6 b) −5 4 ÷ ÷ 37 85 2 6 27 ÷=⋅ 3 7 36 2 ⋅ 7 14 7 = = = 3 ⋅ 6 18 9 −5 4 −5 5 ÷= ⋅ 85 84 −5 ⋅ 5 −25 = = 8⋅4 32 Slide 5 - 36 Addition and Subtraction of Fractions a b a+b += , c ≠ 0; cc c a b a−b −= , c≠0 cc c Slide 5 - 37 Example: Add or Subtract Fractions Add: 43 + 99 4 3 4+3 7 += = 99 9 9 Subtract: 11 3 − 16 16 11 3 11 − 3 8 − = = 16 16 16 16 1 = 2 Slide 5 - 38 Fundamental Law of Rational Numbers If a, b, and c are integers, with b ≠ 0, c ≠ 0, then a a c a ⋅ c ac =⋅= = . b b c b ⋅ c bc Slide 5 - 39 Example: Evaluate: 7 1 −. 12 10 Solution: 7 1 7 5 1 6 − = ⋅ − ⋅ 12 10 12 5 10 6 35 6 = − 60 60 29 = 60 Slide 5 - 40 5.4 The Irrational Numbers and the Real Number System Slide 5 - 41 Pythagorean Theorem Pythagoras, a Greek mathematician, is credited with proving that in any right triangle, the square of the length of one side (a2) added to the square of the length of the other side (b2) equals the square of the length of the hypotenuse (c2) . a2 + b2 = c2 Slide 5 - 42 Pythagorean Theorem In a right triangle with sides a, b and c b (leg b) c (hypotenuse) a2 + b2 = c2 or leg2 + leg2 = hypotenuse2 a (leg a) Note: hypotenuse is the side opposite the right angle. Slide 5 - 43 Example 1: Pythagorean Theorem Find the unknown side for the right triangle. 9 q 4 Slide 5 - 44 Example 1: Pythagorean Theorem Find the unknown side for the right triangle. a2 + b2 = c2 or 42 + q2 = 92 16 + q2 = 81 9 q 16 + q2 – 16 = 81 – 16 4 q2 = 65 There is no rational number that when squared will equal 65. This prompt for a new set of numbers, the irrational numbers. Slide 5 - 45 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... 0.525225222... Slide 5 - 46 Definitions The number s is called a square root of the number b if s2 = b. 5 = 25 5 (s) is the square root of 25 (b) since 52 = 25 Finding the square root of a number is the opposite of squaring a number. Slide 5 - 47 Radicals 2, 17 , 53 are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand. Slide 5 - 48 Principal Square Root The principal (or positive) square root of a number n, written n is the positive number that when multiplied by itself, gives n. For example, 16 = 4 since 4 ⋅ 4 = 16 49 = 7 since 7 ⋅ 7 = 49 Slide 5 - 49 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 - 50 Product Rule for Radicals a ⋅ b = a ⋅ b, a ≥ 0, b ≥ 0 Simplify: a) 40 40 = 4 ⋅ 10 = 4 ⋅ 10 = 2 ⋅ 10 = 2 10 b) 125 125 = 25 ⋅ 5 = 25 ⋅ 5 = 5 ⋅ 5 = 5 5 Slide 5 - 51 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 5 - 52 Example: Adding or Subtracting Irrational Numbers Simplify: 4 7 + 3 7 Simplify: 8 5 − 125 4 7 +3 7 8 5 − 125 = ( 4 + 3) 7 = 8 5 − 25 ⋅ 5 =7 7 =8 5 −5 5 = (8 − 5) 5 =3 5 Slide 5 - 53 Example: Addition and Subtraction of Radicals with different radicands. Simplify the following radical expressions. a. 4 5 + 45 = 5 5 + 9x5 = 4 5 + 9g 5 = 4 5 +3 5 = (4 + 3) 5 =7 5 Slide 5 - 54 Example: Addition and Subtraction of Radicals with different radicands. Simplify the following radical expressions. b. 50 +2 32 - 7 8 25×2 +2 16×2 - 7 4×2 25× 2 +2 16× 2 - 7 4× 2 5 2 +2 ( 4 ) 2 - 7 ( 2) 2 5 2 +8 2 - 14 2 ( 5+8- 14 ) 2 -1 2 -2 Slide 5 - 55 Multiplication of Irrational Numbers Simplify: 6 ⋅ 54 6 ⋅ 54 = 6 ⋅ 54 = 324 = 18 Slide 5 - 56 Quotient Rule for Radicals a b = a , b a ≥ 0, b > 0 Slide 5 - 57 Example: Division Divide: 16 4 Solution: 16 16 = = 4=2 4 4 Divide: 144 2 Solution: 144 144 = = 72 2 2 = 36 ⋅ 2 = 36 ⋅ 2 =6 2 Slide 5 - 58 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. Simplify the result. Slide 5 - 59 Example: Rationalize 8 Rationalize the denominator of 12 . Solution: 8 8 = = 12 12 = 2 3 = 2 3 2 3 ⋅ 3 3 6 = 3 Slide 5 - 60 5.5 Real Numbers and their Properties Slide 5 - 61 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 5 - 62 Relationships Among Sets Real numbers Rational numbers Integers Whole numbers Natural numbers Irrational numbers Slide 5 - 63 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 5 - 64 Closure Determine whether the integers are closed under the operations of a. Addition b. Subtraction c. Multiplication d. Division Slide 5 - 65 Closure Addition a. Is the sum of two integers an integer? Since the answer is yes, we say that integers are closed under addition. Subtraction b. Is the difference of two natural numbers a natural number? Ex. 4 - 6 = -2 (an integer) Since the answer is no, we say that Natural numbers are not closed under subtraction. Slide 5 - 66 Closure Multiplication c. Is the product of two integers an integer? Since the answer is yes, we say that integers are closed under multiplication. Division d. Is the quotient of two integers an integer? ÷ 7 = 4 (not an integer) Ex. 4 7 Since the answer is no, we say that integers are not closed under the operation of division. Slide 5 - 67 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. Think: “Order does not matter.” Slide 5 - 68 Example, Commutative Property 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. Slide 5 - 69 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. Think: “Grouping does not matter.” Slide 5 - 70 Example, Associative Property (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 - 71 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 + 6 • 12 = 6r + 72 Slide 5 - 72 Example, Distributive Property Simplify using distributive property a. b. c. ( 4 2+ 7 ( ) 3 4+ 5 ) 3( r + 6) Slide 5 - 73 Example, Distributive Property Simplify using distributive property a. b. c. ( 4 2+ 7 ( ) = 4 •2 + 4 • 7 = 8 + 4 7 ) 3 4+ 5 = 3 •4 + 3 • 5 = 4 3 + 15 3 ( r + 6 ) = 3 • r + 3 • 6 = 3r + 18 Slide 5 - 74 ...
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This note was uploaded on 12/12/2011 for the course MGF 1106 taught by Professor Holbrook during the Spring '10 term at Santa Fe College.

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