Data Representation

binary number system

• uses 2 as the radix

• employs two binary digits: 0 and 1

• can rep any number using the positional notation

• the MOST SIGNIFICANT BIT (MSB) is the leftmost bit of a binary number

• the LEAST SIGNIFICANT BIT (LSB) is the rightmost bit of a binary number

radix

the “base” of the number system (ex: for binary, the radix = 2)

positional notation

see “Summary of Chapter 3” notes, page 1

converting from decimal to binary (fractional part and the whole number part)

see “Summary of Chapter 3” notes, page 1, 2

convert from decimal to hexadecimal

same process as converting decimal to binary except you keep dividing by 16 instead of 2

• see “Summary of Chapter 3” notes, page 1

convert from decimal to octadecimal 

same process as converting decimal to binary except you keep dividing by 8 instead of 2

• see “Summary of Chapter 3” notes, page 1

converting from hexadecimal to binary (& vice versa)

see “Summary of Chapter 3” notes, page 2

converting from octadecimal to binary (& vice versa)

see “Summary of Chapter 3” notes, page 3

two ways to represent binary numbers

1. unsigned numbers → always non-negative (>= 0)

2. signed numbers → can be either negative or non-negative and it’s indicated by the SIGN BIT (the MSB) where 0 = non-negative num and 1 = negative num

domain of UNSIGNED binary numbers

[0, (2^n) - 1] with n = max number of bits

EX. if max num of bits (n) = 4, then the domain is 0 to (2^4)-1 (0 to 15) and overflow occurs when the result of an operation (such as addition) is outside of that domain.

complements

• in digital computers, complements are mainly used to rep neg nums

• there are 2 types of complements for each base r system:

  • (r - 1)’s comp

  • r’s comp

  • for instance, for base 2 (the binary sys), there are 2’s comp and 1’s comp representation

to find the 2’s complement and 1’s complement of a binary num

see “Summary of Chapter 3” notes, page 4

how to detect overflow when adding signed bits

• when performing addition of two non-negative numbers, the result has a sign bit of 1 (1 = neg num, which wouldn’t make sense)

• when performing addition of two negative nums, the result has a sign bit of 0 (0 = non-neg num, which wouldn’t make sense since neg + neg cannot equal a pos num)

• if the operands have the same sign bit, then the expression [MSB of the result] xor [next-MSB of the result] determines whether you had overflow

• if the operands have diff sign bits, then use the expression [carry-in of the MSB of the result] xor [carry-out of the MSB of the result]

BCD

binary bits are in groups of 4

ex. 0010 = 2

floating point representation

see “ Summary of Chapter 3” notes, page 5

converting a decimal num to a binary num in floating point rep

see '“Summary of Chapter 3” page 6

converting a binary num in floating point rep to a decimal num

see “Summary of Chapter 3” page 7

IEEE 754 standard for representing floating point numbers

see “Summary of Chapter 3” page 5

the rule for the exponent part of a floating point num (32 bits)

e - 127 = exp

where:

e = the number expressed in binary in the register reserved for the exponent

exp = the actual exponent