Converting from one system to another
-----------------------------------------------------------

     to   Binary      Decimal    Hex
from
Binary      N            Y         Y

Decimal    Y            N         Y

Hex           Y            Y         N


From Binary to Decimal     From Hex to Decimal

Just expand the positional representation and do the math in 
base 10

Decimal to Binary
Decimal to Hex
Decimal to Base B

Binary to Hex    and    Hex to Binary
!Easy!

Addition and Subtraction in Hex or Binary



We have been assuming that numbers can be of any length.
We have been assuming that there is space for the Carry.
What about Negative integers/numbers?

The solution is predicated on the fact that
 computer memory is finite and organized in units

BITS
4 bits	NIBBLES
8 bit		BYTES
16 bit			WORDS

Remember most Registers are 16 bits or 1 word sized

Bit numbering

For 1 byte
Bit 0   is rightmost bit
Bit 7  is leftmost bit





For 1 word
Bit 0   is rightmost bit
Bit 15  is leftmost bit



Suppose we are working with numbers that must fit in 8 bits.

Bit 0 corresponds to  2^0       Least significant bit (LSB)
Bit 1 corresponds to  2^1
Bit 2 corresponds to  2^2
...
Bit 7 corresponds to  2^7       Most significant bit (MSB)

With 16 bits         Bit 0 is LSB
                              Bit 15 is MSB

so
Binary		Hex		decimal

01011101	5D 		93
11111111	FF		255
00000000	00		0
00110011	33		51

But suppose we add

	11001001		201
	01001111		 79
	--------------		------
1	00011000		 24  but shd be 280

So addition (and subtraction) cause problems.

What about the sign? positive and negative integers??


One idea is to use 1 bit out of the 8 to represent the sign
This is the SIGNED MAGNITUDE Representation

We use the MSB to represent SIGN, the rest of the bits 
represent the MAGNITUDE

0 means +ive
1 means -ive

00000101   =  +5

10000101   =  -5

10000111   =  -7

01111111   =  +127

11111111   =  -127

00000000   =  +0
10000000   =  -0   ???



Addition and subtraction:

Two rules:

Rule 1: When adding two numbers of like sign
Add the two MAGNITUDES and assign the "like" SIGN
to the result

Example:  -50 + - 25  = -75

add two magnitudes = 75  ;   sign is -ive  so  answer = -75

Rule 2: When adding two numbers of unlike sign
Subtract the smaller magnitude from the larger magnitude 
and assign the sign of the larger magnitude to the result

Example:   25  + -50  = -25

subtract 50 - 25 = 25 assign sign of 50; answer = -25


But there are problems with this kind of
representation and computers use a different kind
of representation call 2's complement.



We'll use the idea of an ODOMETER.

Imagine person sitting on an exercycle with a 2-digit mileage
indicator.

Initial Odometer reading = 00

For each mile
When person pedals forward, odometer shows 01, 02,....

When person pedals backward,odometer shows 99, 98,....


In this kind of representation, lets fix zero and say that
Pedalling Forward     =    Positive numbers
Pedalling Backward    =    Negative numbers







But there is a problem, you could get 60 by pedalling forward
60 miles or pedalling backward 40 miles. WE solve this
ambiguity by splitting the range from 00 - 99 into two parts.

00 - 49   positive numbers
50 - 99   negative numbers



TWO digit odometer numbers

ODOMETER            DECIMAL

  49                 49
  48                 48

.....               ....

   2                  2
   1                  1
   0                  0
  99                 -1
  98                 -2
  97                 -3
  96                 -4
  95                 -5

......              .....

  52                -48
  51                -49
  50                -50



Addition and Subtraction with TWO digit odometer numbers

Addition:

Add normally but ignore carry from highest order digit. 
Example:

        98 + 03 = 01    ignoring the carry

       (-2) + 3 = 1

Subtraction is simply adding the negative

       A - B  =  A + (-B)

Example:

Signed number: 
               -5  - 3 = -8

Odometer

               95 - 3  = 95 + 97 = 92  = -8 !!



Negative odometer numbers

The negative of N in the odometer representation, represents
a trip of N miles in the backward direction.

You get  -N by subtracting N from 100

	-N = (100 - N) = 10^2 - N for two digit numbers.

        This is called the 10's complement representation 
        for two digit numbers

Example:

	-10 = 100 - 10 = 90

	-23 = 100 - 23 = 77

	-47 = 100 - 47 = 63

	-57 = 100 - 57 = 43

Its easier to do the subtraction if we rewrite the equation

	-N = (99 - N) + 1

        99 - N ----> 9's complement of N



We use the same concept for binary numbers:

Size = 8 bits

2's complement         Decimal value

01111111                  127
01111110                  126
01111101                  125

.....                    .....

00000011                    3
00000010                    2
00000001                    1
00000000                    0
11111111                   -1
11111110                   -2
11111101                   -3

.....                    .....

10000010                 -126
10000001                 -127
10000000                 -128