**Why:**- This an area that we don't normally give much thought to, even though we have no trouble understanding it. First, let's explain why this concept is so important. From a tender age we are taught the subtraction process, but for some reason it isn't until much later that we are taught that adding a negative number is the same as subtraction. In fact, I have run across not a few folks who don't remember ever being taught that as a skill.
- Now, for the point of discussion, consider an important device, with which we can do basic addition but cannot do basic subtraction. We would certainly find ourseves in a quandry with this limitation. Consider that for early computers where every part of a computer was terribly expensive, and having to build a subtractor as well as an adder would be a significant expense.
- Now if we could design a system where a number could be easily converted to a negative equivelant, and then simply added, we have accomplished subtaction with an adder. This is a considerable savings because we only need one adding device, rather than an adder and a subtractor.

**How do we get a Complementing Number:**- Because in this instance we will be dealing with the "decimal" system, we need to limit ourselves to the numbers 0-10 and 0-9.
- Consider that the number "0" is at the opposite end of the "decimal" range where the other number is "10".
- We say that the "10's Complement" is obtained by simply subtracting the original number from 10.
- We also say that the "9's Complement" is obtained by simply subtracting the original number from 9.
**Original #****10's Comp****Original #****9's Comp**0 10 0 9 1 9 1 8 2 8 2 7 3 7 3 6 4 6 4 5 5 5

**How do we use Complementing Numbers**(they allow us to to perform subtraction by adding them)**:**- We know that we can subtract 2 from 6 and get 4, but consider adding the complement of 2 (which is 8) to 6.
- In adding the 8 to 6 we would get 14, but when we subtract by adding complements we throw away the carry.
- Now I know that all of this sounds either terribly silly
or useless, but it is actually terribly significant. Most important
of all is that it works, and
, then that makes it all the more important.*if all we have is devices that will only add*

- The real reason for all of this last part of the presentation, is that since we are dealing with Binary Computing Systems, we need to consider that there was a real need for simple methods of obtaining binary number compliments.
- There were a number of mainframe computers that were developed for business computations, and were developed along the line of Binary configured in Decimal - called basically "Binary Coded Decimal" (BCD).
- As a result there were a series of special binary numerical systems, otherwise known as various forms of special BCD values or variants. Each one was designed to serve a special purpose to fill a particular need.
- The BCD Excess-3, the BCD 2'421, and the BCD 5'421 were just a few of those that made serious efforts to solve some of the dilemas.
- The BCD Excess-3 and the 2'421 codes allowed direct 9's Complement by simple inversion, and also provided automatic rounding up for any number 5 or above.
- However, each number system, in the attempt to solve the problem to allow a "binary digital system" to do decimal arithmetic brought their own problem into the fray.
*Now for those of you who think that all this has been a lot of***drivel**or a bunch of**nonsense**, you are very much mistaken.*The problem still exists!*