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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.
 Spring '10
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