binary

signed magnitude pros

• simple and intuitive representation

• easy to convert unsigned values to signed positive numbers (just add 0 as sign bit)

signed magnitude cons

• sign bit makes it difficult to do arithmetic

• 2 representations of 0, makes no sense

ones complement pros

easy to perform addition so simple circuits

ones complement cons

2 representations of 0

twos complement pros

• no special bits so simple circuits

• single representation of 0

2 reasons why computers use binary

1. computers use transistors which only have 2 states

2. efficient for boolean computational logic

decimal to ones complement method

1. decimal to unsigned binary

2. extend bits if needed

3. flip bits

ones complement to decimal method

1. if MSB if 1, flip bits

2. unsigned binary to (negative) decimal

decimal to twos complement method

1. decimal to unsigned binary

2. if decimal was negative, flip bits + add 1

twos complement to decimal method

1. if MSB = 1, flip bits + add 1

2. unsigned binary to decimal

maximum/minimum for unsigned binary

minimum = 00000000 = 0

maximum = 11111111 = 255

maximum/minimum for signed binary

minimum = 11111111 = -127

maximum = 01111111 = 127

maximum/minimum for ones complement

minimum = 1 0000000 = -127

maximum = 0 1111111 = 127

maximum/minimum for twos complement

minimum = 10000000 = -128

maximum = 01111111 = 127

signed magnitude addition method

• if operands have same sign, add magnitudes and keep bit

• else large operand - small operand and choose sign bit of larger magnitude operand

signed magnitude subtraction method

1. rearrange -b to +b by changing sign bit

2. if operands have same sign post change, add

3. else large operand - small operand with final answer sign same as large operand

signed multiplication/division method

perform as unsigned, if operands have same signs then keep, else pick from larger operand

ones complement addition overflow

end carry around

twos complement addition overflow

discard excess carry bits

• if final carry bit is not same as last answer bit, overflow is detected

ones/twos complement multiplication/division method

1. convert both operands to be positive

2. perform as unsigned

3. convert to negative is operands had different signs

IEEE-754 single precision format

32 bits

• sign bit + 8 bit exponent + 23 bit significand

• 127 bias

IEEE-754 double precision format

64 bits

• sign bit + 11 bit exponent + 52 bit significand

• 1023 bias

IEEE-754 single precision to decimal method

1. calculate exponent by exponent - bias

2. add leading 1 to significand (1.significand)

3. move floating point as per exponent, calculate and add sign

decimal to IEEE-754 single precision method

1. convert to unsigned binary

2. normalise and find exponent

3. set sign bit

0, infinity and not a number special values for IEEE-754

0: 0/1 00000000 000…

infinity: 0/1 11111111 000…

not a number: 0 11111111 any significand

IEEE-754 addition method

1. adjust floating point so significands have same exponent (2^0)

2. if different signs, add significands else subtract

3. renormalise if needed and pick sign

IEEE-754 multiplication method

1. add decimal exponents

2. multiply significands

3. renormalise and set exponent

4. select sign

IEEE-754 division method

1. subtract decimal exponents

2. divide significands

3. renormalise and set exponent

4. select sign

overflow/underflow in IEEE-754

overflow: exponent > 128

underflow: magnitude too close to 0 to be distinguished from it