10 Representations

Announcements

• Problem set 1 grading is posted: check to confirm
  • Majority did not test their codes on all scenarios as described in the instruction
  • Follow the submission guidelines and do not create a separate file
  • Make sure your names are fully written at the top of the starter file
• You have Problem Set 2 due tonight at 10 pm
• Remember your section meetings and attend them.
  • Get help from your section leader
  • Also go to the QUAD Center for help: opened six days a week for in-person (drop-in) tutoring. They also offer Zoom appointments
• We are jumpting to chapter 7 today! Read the chapter along for this week
• Link to Polling https://www.polleverywhere.com/agbofred203

Review Question

def largest_factor(n):
    factor = n - 1
    while n % factor != 0:
        factor -= 1
    return factor

if __name__ == '__main__':
    fac = largest_factor(20)
    print(fac)

Representations

• Focusing today on how a computer can internally store more complex and abstract information
• Initially will look at numbers
• Then we’ll transition to strings

Bit Power

• The fundamental unit of memory inside a computer is called a bit
  • Coined from a contraction of the words binary and digit
• An individual bit exists in one of two states, usually denoted as 0 or 1.
• More complex data can be represented by combining larger numbers of bits:
  • Two bits can represent 4 (\(2\times 2\)) values
  • Three bits can represent 8 (\(2\times2\times2\)) values
  • Four bits can represent 16 (\(2\times2\times2\times2\) or \(2^4\)) values, etc
• Many laptops have 16GB of system memory, and can therefore keep track of approximately \(2^{16,000,000,000}\) states!

An old code, but it checks out

• Binary notation is an old idea
  • Described by German mathematician Leibniz back in 1703
• Leibniz describes his use of binary notation in an easy-to-follow style
• Leibniz’s paper notes that the Chinese had discovered binary arithmetic 2000 years earlier, as illustrated by the patterns of lines in the I Ching!

Back to Grade school

Now in Binary

Representing Integers

• The number of symbols available to count with determines the base of a number system
  • Decimal is base 10, as we have 10 symbols (0-9) to count with
    • Each new number counts for 10 times as much as the previous
  • Binary is base 2, as we only have 2 symbols (0 and 1) to count with
    • Each new number counts for twice as much as the previous
• Can always determine what number a representation corresponds to by adding up the individual contributions

Specifying Bases

• So the binary number 00101010 is equivalent to the decimal number 42
• We distinguish by subsetting the bases: \[ 00101010_2 = 42_{10}\]
• The number itself still is the same! All that changes is how we represent that number.
  • Numbers do not have bases – representations of numbers have bases

Binary Clocks

Other Bases

• Binary is not a particularly compact representation to write out, so computer scientists will often use more compact representations as well
  • Octal (base 8) uses the digits 0 to 7
  • Hexadecimal (base 16) uses the digits 0 to 9 and then the letters A through F
    

Base(ic) Practice

• The Java compiler has a fun quirk where every binary file it produces begins with
  
  

• What is this in decimal? octal? hexadecimal?

Representation

• Sequences of bits have no intrinsic meaning!
  • Just the representations we assign to them by convention or by building certain operations into hardware
  • A 32-bit sequence represents an integer only because we have designed hardware to manipulate those sequences arithmetically: applying operations like addition, subtraction, etc
• By choosing an appropriate representation, you can use bits to represent any value you could imagine!
  • Characters represented by numeric character codes
  • Floating-point representations to support real numbers
  • Two-dimensional arrays of bits representing images
• To be useful though, everyone needs to agree on a representation!

Representation Pitfalls

• How we choose to represent values has consequences!
• Python represents floating point (fractional) numbers using two integers
  • One to represent the significant digits
  • One to represent the exponent (where the decimal place is)
• \(1\frac{1}{4}\) Example
  • In decimal: \(\quad\displaystyle 1\frac{1}{4} = \frac{1}{1} + \frac{2}{10} + \frac{5}{100} = 1.25 = (125, -2)\)
  • In binary: \(\quad\displaystyle 1\frac{1}{4} = \frac{1}{1} + \frac{0}{2} + \frac{1}{4} = 1.01 = (101, -10)\)

Consequences

• The best we can do within the range of normal integers \[\frac{3602879701896397}{2^{55}} = 0.10000000000000000555111512312578270\]
• When doing operations on these numbers, extra decimals will sometimes get rounded off, suddenly making the number look precise, but you might always have a tiny bit of this rounding error showing up in floating point values.
• So be careful using == for numeric comparisons! Rounding might result in unexpected falsehoods
  • 0.1 + 0.1 + 0.1 != 0.3
  • Far better to check if two numbers are within a small margin of one another, or greater or less than the other

Representing Characters

• Use numeric encodings to represent character data inside the machine, where each character is assigned an integer value.
• Character codes are not very useful unless standardized though!
  • Competing encodings in the early years made it difficult to share data across machines
• First widely adopted character encoding was ASCII (American Standard Code for Information Interchange)
• With only 256 possible characters, ASCII proved inadequate in the international world, and has therefore been superseded by Unicode.

ASCII