Binary Representation

CSCI N35100

Andy Harris

Indiana University / Purdue University - Indianapolis

Computers and Information

• Information limited to on/off impulses
• Like a light switch
• Infinite positions
• Two outcomes

Working with Multiple Switches

• The light switch demo
• One switch - two messages
• Two switches - four messages
• Three switches - eight messages

Assigning values to messages

• Looking at the pattern
• 1, 2, 4, 8, 16, 32, 64, 128...
• Seeing how it works
• The brick counter demo

Converting from Binary to base 10

• Start with Grid
• Right to left
• Powers of Two
• Substitute 1 with appropriate value

Converting from Base 10 to Binary

• Start with grid
• Find highest value
• Subtract
• Repeat

Binary is good...

• Because it's the underlying format of computers
• But
• it's a pain to work with.
• 11001100001010
• Base 10 is more natural for humans
• Conversion is tedious

Base 8 (octal)

• Rather than raising 10 or 2 to powers,
• Use 8
• 123 base 8 =

•   3 *   8^0 (1)      3
  + 2 *   8^1 (8)     16
  + 1 *   8^2 (64)    64
                      ==
                      83 base 10 
  

Base 8 characteristics

• Max: 7
• Min: 0
• Possible values: 8

3 Digits of binary

• Max: 7
• Min:0
• Possible values: 8

PATTERN!

• Whatever can be expressed in 3 digits of binary...
• Can be expressed in one digit of octal!
• 101 binary is 5 octal
• 010 binary is 2 octal

3 digits binary = 1 digit octal

• For three digits of binary,
• converting to octal -
• is just like converting to decimal

Converting Binary to Octal

• 111011
• Break into groups of 3 (from right)
• 111 011
• Solve each group of three as an independent problem

• 421  421
  111  011
    7    3   octal
  

Octal to Binary

• 45 base 8
• Turn each octal digit to 3 digits of binary:
• 421 421
• 100 101 binary

The problem with Octal

• Computers tend to work in bytes (8 bits)
• Octal prefers lengths of 6 or 9
• Eight bits fits in three digits of octal....

But what about 643 octal?

• It's possible to have a three digit octal
• that doesn't fit in a byte.

Four digits of Binary

• Max: 15
• Min: 0
• Number of values: 16
• This would be like base 16. (Hexadecimal)

Converting Binary to Hex

• 10010111
• Break to groups of four from right

• 8421  8421
  1001  0111
     9     7 hex
  

But there's a problem...

• What about this value?
• 1100
• It comes out to 12 decimal,
• But 12 hex means 16 +2 = 18!!

We can't use base 10 place value here.

• We must 'invent' new numerals
• Letters can be used for numbers!

The Sneaky Solution

• 10: A
• 11: B
• 12: C
• 13: D
• 14: E
• 15: F