Numeric Processing Chapter 6, Exploring the Digital Domain

Digital Number Representations Integers infinite discrete subset of the number line are represented with a limited range Decimal numbers (real numbers) infinite and continuous are represented with limited range and limited precision

Integer Storage 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 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 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 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 Arianne 5 Rocket Failure Patriot Missile Failure during Gulf War

Software for Numerical Work Software Libraries Spreadsheets Mathematical Software symbolic manipulation data analysis data visualization

Spreadsheets: Brief History 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 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) Formatting Data 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 cell addresses (column, row) can be used like variable names in formulas formulas begin with an = or other special symbol

Using Spreadsheet Formulas An Example

Spreadsheets: Basic Features (cont’d) 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 Using absolute references and problem parameters Using built-in functions Using logical functions Displaying data in charts Spreadsheets as decision support tools

Spreadsheet Models Using Problem Parameters 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 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 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 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 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 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 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