CS 149

Professor: Alvin Chao

What is 20/3 + 5*8 - 4?

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"Answer to the Ultimate Question of Life, the Universe, and Everything"

Hitchhikers Guide to the Galaxy

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Circle math

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• Counting on a Line:

• x+a moves you a units to the right of x
• x−b moves you b units to the left of x
• Counting on a Circle
• (x+a) moves you a units clockwise of x
• (x−b) moves you b units counterclockwise of x

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Clock Arithmetic

• Background:
• A 24-hour clock (00 - 23)
• It is now 17. What time will it be in 12 hours?
• The Naive Approach:
• 17+12 is 29. So, we have advanced a day. That means the time is actually 29-24 or 5.
• A Shortcoming of this Approach:
• We might advance more than one day! (For example, advancing 93 hours from now.)

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Clock Arithmetic(contd.)

• A Better Way
• Use arithmetic on a circle (that goes from 0 to 23)
• Using int variables and %
• future = (current + change) % 24;

Other Time/Date examples

• Minutes
• Go from 0 to 59
• Be mindful if you also need to count hours(which can be calculated using integer division).
• Days of the Week ( 0 to 6)
• Months of the Year ( 0 to 11 not 1 to 12)
• Weights: Ounces (0 – 15) then use pounds
• Pounds ( 0 – 1999) then use tons
• Distances:
• Inches (0 – 11) then use feet
• Feet( 0 – 2 ) then use yards
• Yards ( 0 – 5279) then use miles

Another time example

“Twenty-nine days” means the same thing as “Four weeks and one day”. If days is a Python integer variable containing some number of days, develop expressions for:

• The number of weeks in days (4 in the example above). [floor division]
• The number of days that are left over. (1 in the example above). [remainder/modulus]

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Even/Odd Numbers

• Definition
• A number is even if it can be divided by 2 with no remainder
• Observe
• If we think of all numbers as being either even or odd we can conceptualize this as a circle with two items in the cycle.
• Using %:
• Does x % 2 equal 0?

Cycling through a Set

• Examples
• Turn-taking by different # of players
• Cycling through a set of colors
• Repeating a set of instructions
• Observation
• An element in a set can be identified by its number
• If we start at 0 and let n be the cardinality in the set then we can use index =(index + 1) % n

Divisibility: Census Example

• Background: The U.S. Census Bureau conducts a census every 10 years(in years ending with a zero)
• Problem: Find the previous census year for a given year
• Using // :
• censusYear = (year // 10) * 10;

Digit Manipulation

• Note an int value is ‘atomic’ meaning it has no sub-parts.
• Many times we want to find the ones digit or tens digit of a number.
• Get the left-most digits
• Use integer or floor division (i.e. // )
• Use a right-side operand of 10^N-n
• Get the right-most digits
• Use remainder after division (i.e. %)
• Use a rightside operand of 10ⁿ

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In practice

• Retrieving the Left-Most n Digits:
• The left-most digit of 7198 is 7198 // 1000
• The left-most two digits of 531768 are 531768 // 10000
• Retrieving the Right-Most n Digits:
• The right-most digit of 7198 is 7198 % 10
• The right-most two digits of 531768 are 531768 % 100

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Operations

• An operator is a symbol indicating that an operation is to be performed on one or more operands
• An operand can be a variable, literal, or expression
• Number of Operands:
• A unary operator has one operand
• A binary operator has two operands
• A ternary operator has three operands

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Binary Operators

• Numeric Operations and Operators:
• Addition (+)
• Subtraction (-)
• Multiplication (*)
• Division (/)
• Floor or Integer Division (//)
• Remainder or Modulo (%)
• Exponent(**)�

Unary Operators

• Operations and Operators:
• "Positive" (+)
• Negative (-)

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• Operand:
• A numeric type

Math module

Must do import math at top of program to use these:

• math.cos(a) Cosine of angle a(a is in radians)
• math.radians(a) converts angle a to radians from degrees
• math.degrees(a) converts angle a from radians to degrees
• math.sqrt(x) - Square root of x
• math.fabs(v) - Absolute value of v
• math.exp(x) - returns e to the power of x.
• math.pow(x,y) - returns x raised to the power of y
• math.pi - constant for pi.
• math.e - constant for e

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• Acknowledgements

Parts of this activity are based on materials developed by David Bernstein

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