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من طرف wafa السبت نوفمبر 27, 2010 10:27 am. Real Numbers
The real numbers are the numbers that can be written in decimal notation, including those that require an infinite decimal expansion. The set of real numbers includes all integers, positive and negative; all fractions; and the irrational numbers, those whose decimal expansions never repeat. Examples of irrational numbers are
2
= 1.414213562373...
= 3.141592653589...
It is very useful to picture the real numbers as points on the real line , as shown here.
Note that larger numbers appear to the right: if a < b then the point corresponding to b is to the right of the one corresponding to a.
Intervals
Some subsets of the set of real numbers, called intervals , show up quite often and so we have a compact notation for them.
Interval Notation
Here is a list of types of intervals along with examples.
Interval Description Picture Example
Closed [a, b] Set of numbers x with
a x b
(includes end points) [0, 10]
Open (a, b) Set of numbers x with
a < x < b
(excludes end points) ( - 1, 5)
Half-Open (a, b] Set of numbers x with
a < x b ( - 3, 1]
[a, b) Set of numbers x with
a x < b [ - 4, - 1)
Infinite [a, + ) Set of numbers x with
a x [0, + )
(a, + ) Set of numbers x with
a < x ( - 3, + )
( - , b] Set of numbers x with
x b ( - , 0]
( - , b) Set of numbers x with
x < b ( - ,
( - , + ) Set of all real numbers ( - , + )
Here is a little quiz on intervals.
Q1 2 is
is not an element of ( - 5, 2)
Q2 - 5 is
is not an element of [ - 5, 2)
Q3 1 is
is not the largest number in ( - 5, 2)
Q4 [0, + ) is
is not the set of all positive numbers
Q5 ( - , 0) is
is not the set of all negative numbers
Operations
The five most common operations on the set of real numbers are:
addition
subtraction
multiplication
division
exponentiation
"Exponentiation" means the raising of a real number to a power; for instance, 2 3 = 2 . 2 . 2 = 8.
When we write an expression involving two or more of these operations, such as
2(3 - 5) + 4 . 5, or
2 . 3 2 - 5
4 - ( - 1)
,
we agree to use the following rules to decide on the order in which we do the operations:
Standard Order of Operations
1. Parentheses and Fraction Bars
Calculate the values of all expressions inside parentheses or brackets first, using the standard order of operations, and working from the innermost parentheses out. When dealing with a fraction bar, calculate the numerator and denominator separately and then do the division.
2. Exponents
Next, raise all numbers to the indicated powers.
3. Multiplication & Division
Next, do all the multiplications and divisions from left to right.
4. Addition and Subtraction
Last, do the remaining additions and subtractions from left to right.
Notes on Technology
Most graphing calculators and spreadsheets use an asterisk * for multiplication and a caret sign ^ for exponentiation. Thus, for instance, 3 5 would be entered as 3*5 , 3x as 3*x , and 3 2 would be entered as 3^2 .
When using nested parentheses in technology, always use ordinary parentheses, since square brackets and braces usually have special meanings. Thus, for instance, enter 2[(4 + 3)/2] as 2*((4+3)/2)
Q1 A valid first step in the calculation of (2 3 - 4) . 5 is
(6 - 4) . 5 (8 - 4) . 5 2 3 - 20
Q2 Thus, the complete calculation gives (2 3 - 4) . 5 =
20 - 12 36 42
Q3 The quantity 2/3 2 - 5 is
is not the same as
2
3 2 - 5
.
Q4 The quantity 3*2/3+1 is
is not the same as (3*(2/3))+1 .
Q5 The quantity
4(1 - 4) 2
- 9(5 - 3) 2
is equal to
Q6 The quantity
2
3
4 - 5
2
is equal to
Q7 If x = 2, then
2*(1+0.1)^2*x
is equal to
Q8 If x = 2, then
(2-6/4-2)^x
is equal to
Entering Formulas
Any good calculator or computer program will respect the standard order of operations. However, we must be careful with division and exponentiation and often must use parentheses. The following table gives some examples of simple mathematical expressions and their calculator equivalents in the functional format used in most graphing calculators and computer programs. It also includes some for you to do.
Mathematical Expression Technology Equivalent Comments
2
3 - 5
2/(3-5) Note the use of parentheses instead of the fraction bar. If we omit the parentheses, we get the expression shown next.
2
3
- 5
2 - x
3 + x
2
3
5
(2/3)*5 Putting the fraction in parentheses ensures that it is calculated first. Some calculators will interpret 2/3*5 as 2/(3*5)
2
3x - 5
2x + y
1 + xy
2
3
x
y
2 3x - 2 2^(3*x) - 2 The caret "^" is commonly used to denote exponentiation. The parentheses tell the utility where the exponent ends. Enclose the entire exponent in parentheses.
2 3 - 2 5
2 - 7
2^(3-2)*5/(2-7)
or
(2^(3-2)*5)/(2-7) Notice again the use of parentheses to hold the denominator together. We could also have enclosed the numerator in parentheses, although this is optional (why?).
3
8
2 3x - 4
e 2x - 1
2+e 2x - 1
Note on Accuracy
Here is one more fact about calculators (and calculations in general): A calculation can never give you an answer more accurate than the numbers you start with. As a general rule of thumb, if you have numbers measuring something in the real world (time, length, or gross domestic product, for example) and these numbers are accurate only to a certain number of digits, then any calculations you do with them will be accurate only to that many digits (at best). For example, if someone tells you that a rectangle has sides of length 2.2 ft and 4.3 ft, you can say that the area is (approximately) 9.5 sq ft, rounding to two significant digits. If you report that the area is 9.46 sq ft, as your calculator will tell you, the third digit is probably meaningless
The real numbers are the numbers that can be written in decimal notation, including those that require an infinite decimal expansion. The set of real numbers includes all integers, positive and negative; all fractions; and the irrational numbers, those whose decimal expansions never repeat. Examples of irrational numbers are
2
= 1.414213562373...
= 3.141592653589...
It is very useful to picture the real numbers as points on the real line , as shown here.
Note that larger numbers appear to the right: if a < b then the point corresponding to b is to the right of the one corresponding to a.
Intervals
Some subsets of the set of real numbers, called intervals , show up quite often and so we have a compact notation for them.
Interval Notation
Here is a list of types of intervals along with examples.
Interval Description Picture Example
Closed [a, b] Set of numbers x with
a x b
(includes end points) [0, 10]
Open (a, b) Set of numbers x with
a < x < b
(excludes end points) ( - 1, 5)
Half-Open (a, b] Set of numbers x with
a < x b ( - 3, 1]
[a, b) Set of numbers x with
a x < b [ - 4, - 1)
Infinite [a, + ) Set of numbers x with
a x [0, + )
(a, + ) Set of numbers x with
a < x ( - 3, + )
( - , b] Set of numbers x with
x b ( - , 0]
( - , b) Set of numbers x with
x < b ( - ,
( - , + ) Set of all real numbers ( - , + )
Here is a little quiz on intervals.
Q1 2 is
is not an element of ( - 5, 2)
Q2 - 5 is
is not an element of [ - 5, 2)
Q3 1 is
is not the largest number in ( - 5, 2)
Q4 [0, + ) is
is not the set of all positive numbers
Q5 ( - , 0) is
is not the set of all negative numbers
Operations
The five most common operations on the set of real numbers are:
addition
subtraction
multiplication
division
exponentiation
"Exponentiation" means the raising of a real number to a power; for instance, 2 3 = 2 . 2 . 2 = 8.
When we write an expression involving two or more of these operations, such as
2(3 - 5) + 4 . 5, or
2 . 3 2 - 5
4 - ( - 1)
,
we agree to use the following rules to decide on the order in which we do the operations:
Standard Order of Operations
1. Parentheses and Fraction Bars
Calculate the values of all expressions inside parentheses or brackets first, using the standard order of operations, and working from the innermost parentheses out. When dealing with a fraction bar, calculate the numerator and denominator separately and then do the division.
2. Exponents
Next, raise all numbers to the indicated powers.
3. Multiplication & Division
Next, do all the multiplications and divisions from left to right.
4. Addition and Subtraction
Last, do the remaining additions and subtractions from left to right.
Notes on Technology
Most graphing calculators and spreadsheets use an asterisk * for multiplication and a caret sign ^ for exponentiation. Thus, for instance, 3 5 would be entered as 3*5 , 3x as 3*x , and 3 2 would be entered as 3^2 .
When using nested parentheses in technology, always use ordinary parentheses, since square brackets and braces usually have special meanings. Thus, for instance, enter 2[(4 + 3)/2] as 2*((4+3)/2)
Q1 A valid first step in the calculation of (2 3 - 4) . 5 is
(6 - 4) . 5 (8 - 4) . 5 2 3 - 20
Q2 Thus, the complete calculation gives (2 3 - 4) . 5 =
20 - 12 36 42
Q3 The quantity 2/3 2 - 5 is
is not the same as
2
3 2 - 5
.
Q4 The quantity 3*2/3+1 is
is not the same as (3*(2/3))+1 .
Q5 The quantity
4(1 - 4) 2
- 9(5 - 3) 2
is equal to
Q6 The quantity
2
3
4 - 5
2
is equal to
Q7 If x = 2, then
2*(1+0.1)^2*x
is equal to
Q8 If x = 2, then
(2-6/4-2)^x
is equal to
Entering Formulas
Any good calculator or computer program will respect the standard order of operations. However, we must be careful with division and exponentiation and often must use parentheses. The following table gives some examples of simple mathematical expressions and their calculator equivalents in the functional format used in most graphing calculators and computer programs. It also includes some for you to do.
Mathematical Expression Technology Equivalent Comments
2
3 - 5
2/(3-5) Note the use of parentheses instead of the fraction bar. If we omit the parentheses, we get the expression shown next.
2
3
- 5
2 - x
3 + x
2
3
5
(2/3)*5 Putting the fraction in parentheses ensures that it is calculated first. Some calculators will interpret 2/3*5 as 2/(3*5)
2
3x - 5
2x + y
1 + xy
2
3
x
y
2 3x - 2 2^(3*x) - 2 The caret "^" is commonly used to denote exponentiation. The parentheses tell the utility where the exponent ends. Enclose the entire exponent in parentheses.
2 3 - 2 5
2 - 7
2^(3-2)*5/(2-7)
or
(2^(3-2)*5)/(2-7) Notice again the use of parentheses to hold the denominator together. We could also have enclosed the numerator in parentheses, although this is optional (why?).
3
8
2 3x - 4
e 2x - 1
2+e 2x - 1
Note on Accuracy
Here is one more fact about calculators (and calculations in general): A calculation can never give you an answer more accurate than the numbers you start with. As a general rule of thumb, if you have numbers measuring something in the real world (time, length, or gross domestic product, for example) and these numbers are accurate only to a certain number of digits, then any calculations you do with them will be accurate only to that many digits (at best). For example, if someone tells you that a rectangle has sides of length 2.2 ft and 4.3 ft, you can say that the area is (approximately) 9.5 sq ft, rounding to two significant digits. If you report that the area is 9.46 sq ft, as your calculator will tell you, the third digit is probably meaningless
