Complements Essay, Research Paper

COMPLEMENTS

INTRODUCTION:

Basic binary digits are described as UNSIGNED values. This is because if you

look at the string of bits (zeros and ones) there is no indicator whether or

not this value is positive or negative. You must use an additional symbol “+”

or “-” to indicate the sign of the number. There are some types of number

systems where you can determine the sign of number without using an addition

symbol. You can tell whether it is positive or negative by looking at the

bits. Such representations are referred as SIGNED representations because you

can determine the sign of the number from the number itself.

In signed representations you the MOST SIGNIFICANT BIT (MSB) indicates the sign

of the number. The MSB is the left most bit. If the MSB is equal to 0 then the

number is positive. If the MSB is equal to 1 then the number is negative.

e.g. 001111101 is positive

^

|

The MSB is equal to zero

e.g. 101101010 is negative

^

|

The MSB is equal to one

At this point you might ask yourself why bother with signed representations –

we perform decimal based subtractions in every day life using base ten

unsigned representations and we’re all happy and well adjusted people. The

reason why it’s important to learn about signed representations is because

when the computer tries to subtract one number Y from another number X, it

doesn’t do so in the same way that we do: i.e X – Y

Instead it uses a technique known as negating and adding:

i.e. X + (-Y) which is still equal to X – Y

CONVERSIONS (UNSIGNED BINARY TO SIGNED VALUES)

This is summary of what you must do to convert an unsigned binary value to a

complemented value.

1’s complement 2’s complement

Binary value >= 0 No change No change

Binary value < 0 Swap the bits Swap the bits and add

one to the result.

Carry out? Add it back in Ignore it

Swapping the bits means we substitue 1’s for 0’s and 0’s for 1’s.

DOING SUBTRACTIONS VIA COMPLEMENTS (ala Negate and Add)

How do we do this with the negate and add? First convert these numbers to

binary:

Base 10 Base 2 (must be signed!)

eg 4 0100

-6 -0110*

= -2

*I added one extra zero to the left hand side as an extra placeholder. When

I convert these numbers to signed values, this digit it will represent the

sign of the number. We cannot do a mathematical operation in the computer on

minus six in this form. It must be converted to a SIGNED VALUE. You can

use either a 2’s complement representation or a 1’s complement representation.

But make sure that you keep straight which one you use (don’t switch halfway

thru a computation between a 1’s and 2’s complement representation or vice

versa).

e.g.

Number to convert 1’s complement 2’s complement

(flip bits)* (flip bits + 1)*

-0110 1001 1010

Add the complemented values with to the original number above (negate and

add remember).

1’s complement way 2’s complement way

Number from above (4) 0100 0100

The complemented number 1001 1010

—- —-

Summed Result 1101 1110

Convert to this value

from a complemented

form to regular binary -0010 -0010

Convert from binary -2 -2

to decimal

Notice that there is no carry out so we con’t have to worry about addding in

the overflow or ignoring it.

*But this conversion only occurs for negative numbers. Look again at the

circles that I drew in lab, a positive number is a positive number number

matter what binary representation that you use. That means that if the

MSB is equal to a zero, when we get to the second last step, when we try

to find the equivilent unsigned binary value no conversions are necessary.

A positive unsigned digit will look the same in signed (1’s and 2’s

complement form).

Here’s another example on complements that does have an overflow bit (carry

out):

Done using 1’s complement:

Base 10 Base 2

10 1010

-3 -0011

Now here’s comes the fun stage, molding the unsigned base two numbers into

signed one’s and two’s complement representations.

Base 2 Add zero to left* Convert to 1’s complement

1010 01010 01010**

0011 00011 11100

Add the numbers 10 + (-3):

01010

+11100

——

Overflow=> 1 00110

Goes here -> +1

——

000111