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Sign/magnitude notation

• Binary digits can be used to represent not only whole numbers but also other forms of data, including signed integers, decimal numbers and characters.
• To represent signed integers, we can use the leftmost bit to represent the sign, 0 meaning + and 1 meaning -

Examples:

• The number –49 would be represented as:
  1  110001
  -    49
• What about the binary number 1000000 and the binary number 0000000?

Ones Complement

• Possible solution to the problem
• The names comes from the fact that it is obtained by subtracting each digit of the input number from 1
• However, two’s complement is the better solution – this is when 1 is added to the ones-complement

Adding Negative Numbers

• Let’s calculate: 4 + (-6) using twos complement:
  

  - 8	4	2	1
  0	1	0	0
  1	0	1	0

Try another

• Calculate 5 + (-2) in binary

PAUSE

• Pause the tape to do the calculation. When done, come back to see how it is done.

Compare

• Is 10 the same as 110?

Floating Point

• Also known as scientific notation
• The number 1,023,487₁₀ is 1.023487 * 10⁶
• The number 0.1023487₁₀ is 1.023487 * 10^-1

Binary

• The number 10100100111₂ is 1.0100100111 * 2¹⁰
• The number .0011₂ is .11 * 2^-2

Conversion table (floating pt.)

• 1/2	1/4	1/8	1/16	1/32
  .5	.25	.125	.0625	.03125

Examples:

• Convert .625 into binary
  .625 * 2 = 1.250 (extract the 1)
  .250 * 2 = 0.500 (extract the 0)
  .500 * 2 = 1.000 (extract the 1)
• The digits extracted are taken in the order extracted. In this case, the result is .101 (1/2 + 1/8 = 5/8 = .625)

Errors

• One source of errors is converting back and forth between decimal and binary
• Example:
  calculate .6 + .6
  first convert to binary .6
  .1001100110011……

Do the math

• Find the sum
      .10011001
    +.10011001
    1.00110010

Convert back

• So, 1.00110010 converts to 1 + 1/8 + 1/16 + 1/128 = 1.195 (actual sum = 1.2)
• Error = 1.2 – 1.195 = .005 due to round off error made during conversions

Normalized form

• In normalized form the leading 1 appears next to the decimal point.
• Example:
  .11001001

Scientific notation

• Avagadro’s number: N_a = 6.022 * 10 ²³

Floating point

• A BBBB C DD form (8 bits)
• A = sign of the mantissa | BBBB = mantissa | C = sign of the exponent | DD = exponent
• Example:
  +.1011 * 2⁺³
  0 | 1011 | 0 | 11

Example

• sign of mantissa					sign of exponent
  A	B	B	B	B	C	D	D
  	digits of mantissa					digits of exponent