CSE 378 - Autumn 2002

Machine Organization and Assembly Language Programming

Tutorial 1 - Thursday, October 3
Some notes from the binary representation portion of tutorial

Simple table of equivalences:

Decimal Binary Hexadecimal
0 0000 0
1 0001 1
2 0010 2
3 0011 3
4 0100 4
5 0101 5
6 0110 6
7 0111 7
8 1000 8
9 1001 9
10 1010 A
11 1011 B
12 1100 C
13 1101 D
14 1110 E
15 1111 F

Some conversion formulas:

"base" to decimal:
where our number = ...d7d6d5d4d3d2d1d0
  


two's complement binary to decimal:
assuming a 32-bit word as in MIPS, where our
number = (dsignd30...d7d6d5d4d3d2d1d0)2
 

 


One algorithm for converting a positive decimal to two's complement binary:

Convert decimal  X10 > 0 to two-'s complement binary: (dsignd30...d7d6d5d4d3d2d1d0)2

1)  Initially set di = 0  for all i
2)  What is the largest power of 2 <= X10?  Call is 2k.
3)  Set dk = 1
4)  Set X'10  =  X10 % 2k
5)  If X'10 = 0, stop, otherwise repeat steps 2-4 with X'10

Example: Convert 53010 to two's complement binary.

  29 = 512 <= 530, so 530 % 29 = 18, set d9 = 1
  24 = 16 <= 18, so 18 % 24 = 2, set d4 = 1
  21 = 2 <= 2, so 2 % 21 = 0, set d1 = 1
  stop

So we have: 53010 = 0000 0000 0000 0000 0000 0010 0001 0010


A similar algorithm for converting a negative decimal to two's complement binary:

Convert decimal Y10 < 0  to to two's complement binary: (dsignd30...d7d6d5d4d3d2d1d0)2

1)  Set Y'10 = | Y10 |
2)  Set Y''10 = Y'10 - 1
3)  Find the two's complement of Y''10 using the algorithm for positive decimals, call it D2
4)  Flip the bits of D2, that is switch 1's to 0's and 0's to 1's

Example: Convert -102310 o two's complement binary.

  1023 = | -1023 |
  1022 = 1023 - 1
  29 = 512 <= 1022, so 1022 % 29 = 510, set d9 = 1
  28 = 256 <= 510, so 510 % 28 = 254, set d8 = 1
  27 = 128 <= 254, so 254 % 27 = 126, set d7 = 1
  26 = 64 <= 126, so 126 % 26 = 62, set d6 = 1
  25 = 32 <= 62, so 62 % 25 = 30, set d5 = 1
  24 = 16 <= 30, so 30 % 24 = 14, set d4 = 1
  23 = 8 <= 14, so 14 % 23 = 6, set d3 = 1
  22 = 4 <= 6, so 6 % 22 = 2, set d2 = 1
  21 = 2 <= 2, so 2 % 21 = 0, set d1 = 1
  stop

So we have 102210 = 0000 0000 0000 0000 0000 0011 1111 1110

Now flip the bits to get -102310 = 1111 1111 1111 1111 1111 1100 0000 0001

note: For binary to hex conversions, just use the table.