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Beginning Problem-Solving Concepts for the Computer

Lesson 2

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3 Types of Problems

• Computational– problems involving some kind of

mathematical processing• Logical

– Problems involving relational or logical processing

• Repetitive– Problems involving repeating a set of

mathematical and/or logical instructions.McManus

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Fundamental Concepts

• The building blocks of equations and expressions– Constants– Variables– Operators– Functions (predefined—more about user-

defined functions later)

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Constants

• A value– a specific alphabetical and/or numeric value – Does not change during the processing of all

the instructions in a solution• Can be of any data type

– Numeric, alphabetical or special symbols• Examples

– 3, 5, 10, “Mulder”, “Scully”, “+”, “-”, “/”– Note: “” indicates datum rather than variable

name

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Constants

• In some programming languages, constants can be named– Provides protection to the constant

value and other areas of the program.– Cannot be changed once given initial

value– Note: no spaces allowed within names– Examples

• SALES_TAX_RATE = 6• COMMISSION_RATE = 6

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Constant name are usually in ALL CAPS

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Variables

• The name references the memory or storage location where the value is stored– May change during processing– May be of any data type– A name is assigned to each variable

used in a solution.• Examples

– Age, LastName, Address, PayRate

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Rules for Naming Variables

• Use mnemonic terms--the name is meaningful– Use PayRate not Z

• Do not use spaces• Do not use special

symbols, except the underscore– Examples

• PayRate, Pay_Rate, PAYRATE, or PAY_RATE

• Use appropriate naming convention

• Be consistent! – Some languages

are case sensitive– Either combine

terms (PayRate) or include an underscore to separate (Pay_Rate)McManus

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So, what do we store?

Data or Information?

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Data Vs. Information

• Data– The raw, organized facts

• Information– The meaningful output

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Data Types

• Most common types– Numeric– Character– Logical

• Most languages include other data types– Date– Currency– User-defined– Strings

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Rules for Data Types

– All data to be used in calculations must be declared as a numeric data type

– Each data type uses a data set or domain

– Integers - -32768 -> 0 -> 32767 but is programming language dependent

– Characters – all alphanumeric characters included in the computer’s coding scheme

– Logical – the words “true” and “false”

– Data types cannot be mixed• Called data typing

– Programmer designates the data type McManus

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Data Types & Data Sets

Data Type Data Set ExamplesInteger All whole numbers 3456

-43

Real All real numbers (whole + decimal)

3456.78-123.45

0.000123

Character (uses quotation marks)

All letters, numbers, and special symbols

“A”, “a”, “1”, “+”, “%”

String (uses quotation marks)

Combinations of more than one character

“Mulder”“123-45-6789”

Logical True or False TrueFalse

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Numeric Types

• Includes all types of numbers– Natural Numbers - The set of positive

(counting) numbers • Ex. {1,2,3,4,5,…}

– Integers - The set of real numbers consisting of the natural numbers, their additive inverses, and zero• Ex. 3, 5, -5, 0, -1 and 32,767

– Real numbers (floating point numbers)• Ex. 3.1459, 1.618039887, -5745 & Scientific

NotationMcManus

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Can I Confuse You?

• Which is larger?

The set of Integers or the set of Real Numbers?

• Well, actually…• There are several different levels of

infinity! Confused?

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Levels of Infinity

Complex Numbers

Real Numbers

Irrationals

Rationals

Integers

Naturals

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Naturals: Counting Numbers

Integers: Naturals, Additive Inverses & 0

Rationals: Numbers expressed as a ratio between 2 Integers; Ex. 1/10

Irrationals:

Reals: All numbers on number line

Complex Numbers: Real Numbers, Imaginary Numbers

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Character Types

• Alphanumeric data set– Consists of

• all single-digit numbers, • letters, and • special characters available to the computer

– contained within quotation marks

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Coding Systems

– EBCDIC • Extended Binary Coded Decimal Information

Code– ASCII

• American Standard Code for Information Interchange

• 256 characters (1 byte)– First 128 – standard– Second 128 – specific to computer

– Unicode (2 bytes)• Replacing ASCII as standard on PC’s

McManus

See Appendix

B or online

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String Data Type

• Made up of one or more characters.• Contained within quotation marks• Cannot be used within calculations

– There is some debate about using strings…• The new consensus is that if the value will be used in

a mathematical calculation, make it a number, otherwise make it a string.

• The alternative view (older view) is to look at the amount of storage that is needed for the value. A number generally takes up less memory space than a string.

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Strings - Caution

• “123” is not the same thing as 123– The first one is a string– The second one is a number

• Becomes important when deciding when a piece of data is a string or a number

• Ex. Social Security Numbers– The first one contains 11 characters and is left

aligned whereas the second contains 9 numbers and is right aligned

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Comparing Characters & Strings

– The computer’s particular coding scheme (ASCII, Unicode, or EBCDIC) converts the character to its associated ordinal number• Note: Uppercase letters have a smaller

ordinal values than lowercase letters– “A” = 193 whereas “a” = 129 (Unicode 16-bit)– “A” = 65 whereas “a” = 97 (ASCII 8-bit)

– Once converted, each character, one at a time, can be compared

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Concatenation

• Joins character or string data together– Operators: + and &

• Dependent on programming language• & can mix data types• + cannot mix data types (also known as Join

in databbases)• Examples

– FName & “ ” & MI & “ ” & LName– “James” + “ ” + “T.” + “ ” + “Kirk” = “James T.

Kirk”– “James” & “ ” & “T.” & “ ” & “Kirk” = “James T.

Kirk”» Note the space at the end of the T.

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Logical (Boolean) Data Type

• Consists just two pieces of data– True and False– Different programming languages also

allow• Yes and No• 1 (True) and 0 (False) • On and Off

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Other Pre-defined Data Types

• Dependent on programming language, numerous other pre-defined data types are available.– Examples

• Dates 01/01/1900 January 1, 1900

• Times03:35:05 PM 15:35:05• Currency 3000.00

– Note: This is how the data will be stored, not how it is formatted.

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User-Defined Data Types

• User-defined Data types– Defined by the user as a new data type

• Example– Record

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Employee_RecordFnameLnameSSNBirthDateHireDateTerminationDate

PayRateEnd

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Functions

• Small sets of instructions that perform specific tasks and return values.

• Two types:– Pre-defined

• Built into a computer language or application

– User-defined • More about these later

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Functions

• Benefits– Reduces the amount of code that needs

to be written, thus reducing errors of repetition

– Shortens the development time– Improves readability

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Function Parameters

• Data usually placed within parentheses that the function uses without altering the data– In Sqrt(n), the n represents the

parameter, in this case a number– In Value(s), the s represents a string

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Function Types

• Mathematical • String• Conversion• Statistical • Utility• Specific to each programming

language…a partial list follows

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Mathematical Functions

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Function Form Example Result

Square Root Sqrt(n) Sqrt(4) 2

Absolute Abs(n) Abs(-3), Abs(3) 3

Rounding Round(n,n1) Round (3.7259,2) 3.73

Integer Integer(n) Integer(5.7269) 5

Random Random Random 0.239768

Sign of a number Sign(n)

Sign(7.39)Sign(0)Sign(-7.39)

10

-1

Random is a random number selected between 0 and 1

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String Functions

McManus

Function Form Example Result

Middle String Mid(s, n1, n2)Mid(S,3,2) where S = "Thomas" "om"

Left String Left(s, n)Left(S,3) where S = "Thomas" "Tho"

Right String Right(s, n)Right(S, 3) where S = "Thomas" "mas"

Length String Length(s)Length(S) where S = "Thomas" 6

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Conversion Functions

McManus

Function Form Example Result

Valued Conversion Value(s) Value("-12.34") -12.34

String Conversion String(n) String(-12.34) "-12.34"

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Statistical Functions

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Function Form Example Result

Average Average(list) Average(5, 3, 8, 6) 5.5

Maximum Max(list) Max(5, 3, 8, 6) 8

Minimum Min(list) Min(5, 3, 8, 6) 3

Summation Sum(list) Sum (5, 3, 8, 6) 22

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Utility Functions

McManus

Function Form Example Result

Date Date Date 9/14/02

Time Time Time 9:22:38

Error is a special function that appears in most languages and upon execution return control to the program when (not if) system errors occur. Such as Disk Errors…

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Operands, Resultants & Operators

• Operands– The data that the

operator connects and processes

• Resultant– The answer that

results when the operation is executed

• Operators– The data connectors

within expressions and equations.

– Two types of operations• Unary

– Negation• Binary

– Addition, Subtraction, Multiplication, Floating Point Division and Integer Division

McManus

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Operator Types

• Mathematical– +, -, *, /– \, Mod {Integer Division & Modulo}– ^, {Exponentiation} and – functions

• Relational– <, >, <=, >=, =, <>

• Logical– And, Or, Not, XOR, EQV

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Two Special Math Operators

• Integer Division (\)– Regular Floating

Point Division, except only the whole number part of the answer is returned.

– If given a Floating Point number, the number is rounded—first—before division occurs.

• Modulo Division (Mod) or Modulus– The remainder part

of the answer is returned.

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The Boolean Data Type

• Returns either the keywords True or False or

• Returns a non-zero or zero value.– -9, -1, 1, 2 are equivalent to True– 0 is equivalent to False

• Is stored in two bytes

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Relational Operators

• Perform comparisonsvariable - relational operator - variable

– Ex. If Y > Z then ...

= equal<> not equal< less than> greater than <= less than or equal to>= greater than or equal to

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Logical (Boolean) Operators

• And, Or, Not, XOR, EQV• Boolean Expressions are used

– To create more complicated Boolean Expressions.

– If a = b AND c < d Then …• They are defined by using Truth

Tables…– Know the significance of each!

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Hint!These are

on the Test!

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The AND Operator

• Significance: The only time the result is True is if both A and B are True.

McManus

A = B = then A AND BTrue True TrueTrue False FalseFalse True FalseFalse False False

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The OR Operator

• Significance: The only time the result is False is if both A and B are False.

McManus

A = B = then A OR BTrue True TrueTrue False TrueFalse True TrueFalse False False

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The NOT Operator

• Also called the “logical complement” or “logical negation”

• Significance: Whatever the value of A is, NOT A will be the opposite.

McManus

A = NOT A

True FalseFalse True

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The XOR (Exclusive Or) Operator

• Significance: The only time the result is True is if A and B are different and False if both A and B are the same.

McManus

A = B = then A XOR BTrue True FalseTrue False TrueFalse True TrueFalse False False

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The EQV (Equivalent) Operator

• Significance: The only time the result is True is if A and B are the same and False if both A and B are the different.

McManus

A = B = then A EQV BTrue True TrueTrue False FalseFalse True FalseFalse False True

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Hierarchy of Operations

• Data and Operators are combined to form expressions and equations

• To evaluate these in the proper order, a hierarchy of operations, or Order of Precedence, is used.– Note: Different programming languages

use different Order of Precedence…– Note: Whenever you want to clarify an

equation, use the parentheses!

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The Order of Precedence

McManus

OPERATOR TYPE NAME ( ) Parentheses Parentheses ^ Arithmetic Exponent (Powers) * / Arithmetic Multiplcation, floating-point division\ Arithmetic Integer division

Mod Arithmetic Modulus + - Arithmetic Addition, subtraction

= <> <= >=

> <

Comparison (all have same level of precedence

Equal to, not equal to, less than or equal to, greater than or equal to, greater than, less than

NOT Logical Logical Negation AND Logical Logical And OR Logical Logical Or

XOR Logical Logical Exclusive Or EQV Logical Logical Equivalent

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Example

• Evaluate the following:• A = True, B = False, C = True, D =

False

• To do this, we create what’s called an Evaluation Tree…

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A or B or C and D and A and B and (not C) or (not D)

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The Evaluation Tree

McManus

A or B or C and D and A and B and (not C) or (not D)

False True

False False False

False True

True True

1

2

4

3

6

5

7

8

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Expressions & Equations

• Expressions– Process the data and the operands,

through the use of the operators– Does not store the resultant (answer)– Examples

• Price * SalesTaxRate• (Hours – 40) * Wage * 1.5• Height * Width

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Equations

• Same as an expression, except the equation stores the resultant (answer) in a memory location (must be on the left side of Assignment Operator)

• “takes on the value of” not “equals”– Examples

– SalesTax is the memory location where the answer will be stored, the resultant

• Often called Assignment Statements

McManus

SalesTax = Price * SalesTaxRateOvertimePay = (Hours – 40) * Wage * 1.5Area = Height * Width

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More about Equations

• Destructive Read-In– The value stored in

the variable on the left-hand side of the equals sign is replaced by the result of the expression on the right-hand side.

• Non-Destructive Read-In

• Values used on the right-hand side are protected from being changed.

McManus

SalesTax = = Price ** SalesTaxRate D R-I ND R-I ND R-I

Ahh..but what about this one? Count = Count + 1

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Straight-Line Form

• Algebraic Expressions cannot be represented in the computer.

• They must be converted to a form the computer will recognize.– Thus, Straight Line Form

McManus

2

2

)()2(

)(2

YXYX

YXY

= ( 2 *(X – Y ) ) / ( ( ( 2 * X + Y )^2 ) / ( ( X – Y )^2 ) )

straight line form

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Next?

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