Download Numbers, Data and Problem Solving

Operations
Basic Arithmetic
Operations
Notation

Operations

A
Addition ( +) , Multiplication ( • )

Subtraction ( – ) … addition of negatives

Division ( / ) … multiplication of reciprocals
Axioms
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Operations
2
Order of Operations
Inverses


Additive inverses
 For every real number x there is a real
number -x such that x + (-x) = 0
 Examples
 1. 7 + (-7) = 0
 2. -3 + (-(-3)) = 0 = -3 + 3
Axioms
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Operations
3
Order of Operations

Subtraction

The binary operation “–” is defined by
a – b ≡ a + (-b)
for any real numbers a and b

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The operation “–” is called subtraction
… which is just addition of negatives
 Examples:
 1.
7 – 4 = 7 + (-4) = 3
Axioms
 2. 10 – 12 = 10 + (-12) = -2
Operations
4
Order of Operations
Inverses



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Multiplicative inverses
 For every real number x there is a real
number x–1 such that x • x–1 = 0
 Examples
Axioms
–1 = 1
 1. 7 • 7
–1 • (3–1)–1 = 1
 2. 3
1
–1
Additive inverse x is also written
x
Operations
5
Order of Operations
Division


The division operation“ •• ” or “ / ” or “ ”
is defined by
a •• b = a = a / b ≡ a • b–1
b
for any real numbers a and b

The operation “–” is called subtraction
… which is just addition of negatives

7/9/2013
Examples:

1. 7 – 4 = 7 + (-4) = 3

2. 10 – 12 = 10 + (-12) = -2
Operations
Axioms
6
Order of Operations
Addition and Multiplication


Multiply first then add
 Examples
 1.
3•5+4•7
= 15 + 28
= 43
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2. 3 • x + 4 • 5 – 1 • 7
= 3x + 20 – 7
= 3x + 13
Operations
7
Order of Operations
Addition and Multiplication


Apply distributive property to clear groups

Examples


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1. 3 • (x + 4) = 3x + 12
2. 3 • ((x + 4) + (x – 1) • 7) + 1
= 3 • ((x + 4) + 7x – 7) + 1
= 3 • (x + 4 + 7x – 7) + 1
= 3 • (8x – 3) + 1
= 24x – 9 + 1
= 24x – 8
Operations
8
Notation
Intervals




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Open interval: (a, b) = { x | a < x < b }
 The set of all numbers between a and b
Closed interval: [a, b] = { x | a ≤ x ≤ b }
 The set of all numbers between a and b
including a and b
Half-open/half-closed: [a, b) , (a, b]
 [a, b) includes a, excludes b
 (a, b] excludes a, includes b
Operations
9
Notation

Binary Relations

Grouping Symbols



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Symbols: ≡ = ≠ < > ≤ ≥
Symbols: { } , ( ) , [ ]
Special Symbols
 ∞       ± ∆   
Operations
10
Calculations with Data

Finding Averages

Add n values, divide by n

Call Avg the average of n values of x
Avg =
x1 + x2 + x3 + ··· + xn
n
n
1
= n  xk
k=1
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Operations
11
Calculations with Data
Example
Average of test scores 73, 85, 14, 92
4

1
xk = 1 (73 + 85 + 14 + 92)
4 k =1
4
1
= 4 (264)
= 66
Works fine for small number of values
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Operations
12
Calculations with Data
Example

Travel d miles in t hours

For d = 200 miles and t = 3 hours
Average speed = d
t
200 miles
=
3 hours
= 66.67 mph

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Works for large/infinite number of values
Operations
13
Calculations with Data

Finding percentages

A per-100 proportion
a is to b as P is to 100 … that is
a
P
=
b
100
 a as a percentage of number b is
a (100)
=P
b

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Operations
14
Calculations with Data

Example

In a chain saw 12 ounces of oil are
added per gallon of gasoline
What percentage of the mixture is oil ?
Note that 1 gal = 128 oz
12
% oil =
(100) % ≈ 8.571 %
128 + 12
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For every 100 ounces of mixture 8.571
ounces are oil
Operations
15
Calculations with Data

Finding percent change

A variable p changes by amount ∆p

What percentage of p is ∆p ?

Percent change in p is
∆p
(100) %
p
where p is the initial value
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Operations
16
Calculations with Data

Example

The price of gasoline increased from
$2.56 per gallon to $3.89 per gallon

Change is: ∆p = 3.89 – 2.56 = 1.33

Percent change is
∆p
1.33
(100) =
(100) ≈ 51.95 %
p
2.56
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Operations
17
Calculations with Data

Absolute value


The unsigned “size” of a number
Definition:
For any real number a the absolute
value of a, written a, is
a , for a ≥ 0
a =
– a , for a < 0
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Operations
18
Calculations with Data

Examples

1.  7 = 7

2. – 3 = 3 Here a = – 3 so – a = – (– 3) = 3

3.  0 = 0

4. x – 5 =
x – 5 , for x ≥ 5
– x + 5 , for x < 5
Question: If a is any real number …
… is –a positive or negative ?
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Operations
19
Think about it !
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Operations
20