Download Information in the Digital Domain

Numeric Processing
Chapter 6,
Exploring the Digital Domain
Digital Number Representations
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Integers
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infinite discrete subset of the number line
are represented with a limited range
Decimal numbers (real numbers)
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infinite and continuous
are represented with limited range and
limited precision
Integer Storage
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All integers between two values (one negative and
one positive) are stored with exact precision
The specific values marking the range limits depend
on the particular computer system being used
If calculations with integers give rise to numbers
outside the allowable range, we say that an integer
overflow error has occurred
Real (Decimal) Number Storage

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Real numbers are stored in floating point
representation
 a sign
 an exponent
 a mantissa (normalized decimal fraction)
 no digits to the left of the decimal
 first digit to the right of the decimal is nonzero
Limited precision because most real numbers have an
infinite decimal expansion (this holds no matter what
number base is used in the representation)
Real Number Storage
Limited Range and Precision

There are three categories of numbers left out when
floating point representation is used
 numbers out of range because their absolute value
is too large (similar to integer overflow)
 numbers out of range because their absolute value
is too small (numbers too near zero to be stored
given the precision available
 numbers whose binary representations require
either an infinite number of binary digits or more
binary digits than the bits available
Real Number Storage
Limited Range and Precision Illustrated
Limited Range and Precision
Some Consequences

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Limited range will invalidate certain calculations
 If integers are involved, this can often be avoided by
switching to real numbers
 For real number calculations, this problem arises
infrequently and in those cases can sometimes be handled
by special methods. It is not a common occurrence in
non-scientific work.
Limited precision for real numbers is very pervasive
 Assume that most decimal calculations will, in fact, be in
error!
 Evaluate and use computer calculations with this in mind
Social Themes:
Risks in Numerical Computing
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Almost all computer calculations involve
roundoff error (limited precision error)
If not monitored and planned for
carefully, such errors can lead to
unexpected and catastrophic results

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Arianne 5 Rocket Failure
Patriot Missile Failure during Gulf War
Software for Numerical Work
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Software Libraries
Spreadsheets
Mathematical Software
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symbolic manipulation
data analysis
data visualization
Spreadsheets: Brief History
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First spreadsheets appeared in early 1980s
 ran on personal computers
 helped popularize personal computers
Basic Organization Hasn’t Changed
 computations organized on a two-dimensional
worksheet
 both built-in and user-supplied formulas used to
facilitate computation
Spreadsheets: Basic Features

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Worksheet Organization
 rectangular grid of cells
 cells are identified by the row (indicated by an
integer) and column (indicated by a letter) in which
they appear
Entering Data
 one cell is active at a time (called the current cell)
 a separate data entry bar is associated with the
current cell
 text, numerical data, dates, and formulas can be
entered into the current cell through the entry bar
Worksheet Organization: Illustrated
Spreadsheets: Basic Features (cont’d)
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Formatting Data
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variety of formats depending on the type of data
data format is associated with the cell and can be
changed later
Using Formulas in a Worksheet
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cell addresses (column, row) can be used like
variable names in formulas
formulas begin with an = or other special symbol
(like @)
Using Spreadsheet Formulas
An Example
Spreadsheets: Basic Features (cont’d)
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Copying (Replicating) Formulas
 formulas can be copied to perform repetitive calculations
 especially useful when similar calculations take place on a group
(block,row, column) of contiguous data
Cell Referencing
 cell references will be adjusted automatically when formulas are
copied if relative cell addressing is used in formulas
 cell references will remain the same if absolute cell addressing
is used in formulas
 relative cell addressing is the default addressing scheme
Copying Spreadsheet Formulas
An Example
The formula
in cell B12
has been
copied to
cells C12 and
D12. Note
how the cell
references are
automatically
adjusted.
Spreadsheets
Additional Features
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Using absolute references and problem
parameters
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Using built-in functions
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Using logical functions
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Displaying data in charts
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Spreadsheets as decision support tools
Spreadsheet Models
Using Problem Parameters
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Most spreadsheet models will require modifications over
time as underlying assumptions and important problem
values change
Good spreadsheet design will minimize the danger of
making errors when updating spreadsheet models
Problem parameters are important problem values that
are subject to change over time
Separating problem parameters, placing them in clearly
identified cells, then referring to them by absolute
references in formulas within the model makes changing
them relatively error-proof
Spreadsheet Models
Decision Support
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Modeling problem parameters as clearly identified
separate entities enables convenient “what if”
analysis in spreadsheet models
“What if” analysis involves observing modified
calculations and results when problem parameters
change. In other words, the model is examined
in response to questions of the form “What if …
changes to … ?”
Spreadsheet Models
Using Built-in Functions
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Built-in functions act on arguments to produce resulting values
A user of a built-in function need not know all the details of how the
function does its calculation
The user needs to understand only what input arguments are required
and what type of result is produced
An extensive library of financial and statistical functions adds powerful
problem-solving capabilities for the average user
Spreadsheet Models
Using Built-in Logical Functions
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Logical functions allow
spreadsheet models to make
decisions during calculations
Such processing is called
conditional processing and
is a fundamental property of
programming languages
For example, the IF function
acts on a logical (or Boolean)
expression to take one of two
actions, as shown here
Spreadsheet Models
Displaying Data in Charts
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Graphical display and summary of data is
often easier to interpret than the raw data
itself
Spreadsheets typically provide several types
of charts
 bar charts
 pie charts
Bar Chart
An Example
Pie Chart
An Example
Summary
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Integers are stored with limited range: real numbers
(decimals) with limited range and limited precision
Almost all decimal calculations involve roundoff errors
because of limited precision
Spreadsheets provide powerful, yet easy to use,
calculational environments
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formulas and relative referencing provide for easy calculation
extension
absolute referencing and problem parameters enable modeling
built-in functions (arithmetic and logical)
displaying data in charts
decision support tool