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To form a negative number, we simply invert the positive number, in binary, and add 1. For example
0000 0110 - (6) inverted becomes
1111 1001 - now add 1 to get
1111 1010 - (-6)
This is how a computer records -6. The Most Significant Bit - in this case bit 7 or bit value 128 decimal is used as the Sign Bit. If this bit is a zero, the number is positive, if it is a one, the number is negative. This notation is known as 'Twos Complement'.
(There is also 'Ones Complement', without the added 1, which was used in some proprietary systems, but it required more complicated logic to get around problems like having + and - zero. It is not necessary to go into the detail of this notation for an understanding of the logic.)
The Sign bit uses one of the bit positions, meaning that our range of numbers is reduced. For example 0111 1111 is the highest positive 8-bit number (127 decimal). The next number, 128 decimal would be 1000 0000, however the bit value 128 in bit 7 cannot be used as it is now the sign bit.
SUBTRACTION
Now we can have a negative number it will be easy to subtract. We can use an adder, which we already have. This is possible because 9-3 is the same as 9+(-3). For instance 9-3 decimal is 00001001-00000011 in 8 bit binary. We take the 9 and add it to the twos complement of 3, thus we can use our adder with a bit of extra logic to invert the negative number and add 1 to it. The twos complement of 3 (decimal) or in binary
of 0000 0011
is 1111 1100
add 1 + 0001
is 1111 1101
Now add to the 9 (decimal) using an 8 bit adder:-
7654 3210 - bit #
0000 1001 - (9)
1111 1101 - (-3)
Remember your binary addition?
Least significant bit (bit 0), 1+1 is 0 and a carry.
Next bit (bit 1) is 0+0+carry = 1 no carry.
Next bit (bit 2) is 1+0 = 1 no carry.
Next bit (bit 3) is 1+1 = 0 and a carry.
Next bit (bit 4) is 1+0+carry = 0 and a carry.
Next 3 bits (5,6,7) are the same as 4 - 1+0+carry = 0. The last carry falls off the end.*
So we get:-
0000 0110 (6 Decimal - 9-3=6)
Which, fortunately, if you followed it, is the right answer! Are these computers smart or what? This system works with negative results as well, for instance try 3-9. You should end up with the
twos complement of 6. This is not a compulsory exercise!
* The carry from the most significant bit, in this example bit 7, but could be bit 31 or 63 in some computers, is ignored in this case. However it is used in some functions, for instance a circular bit shift, which we will look at in a later article.
Beginning in the 1950s, magnetic tape has progressed from the original half inch wide ferrite coated tape on open reels 10inch diameter and 2400ft long to the current cassettes mounted on drives in stacks.
The first drives recorded 7 bits across the tape, being 6 data bits + parity. This was increased to 9 bits to accommodate the 8 bit byte, the backing medium was changed to mylar for strength, and allowed 4800 ft to be placed on the reel with thinner tape. The main change was in the recording systems. The first universally adopted 'standard' was NRZ (Non Return to Zero) referring to the crossing from negative to positive of the magnetisation at speeds up to 800 BPI.
The initial recording systems had the tape starting and stopping at high repetition rates, so that it was necessary to provide a buffer to prevent the tape being dragged of the reel, or spilled onto the floor! This was normally accomplished by having two large chambers for the tape to run in, one between each reel (supply and take-up) and the read write head next to the capstan (drive) assembly.
Each reel was driven by a reversible dc servo motor, which was controlled from sensors in the chambers indicating the chamber was filling or emptying, due to the action of the capstan feeding the tape, and the reel taking it up or feeding it out. The tape was kept under tension by vacuum pumps in the chamber, and the sensors could be vacuum switches or photocells, indicating the presence or absence of the tape in that part of the chamber. An alternative system to vacuum chambers used swinging mechanical arms.
Improvements in magnetic coatings allowed improved heads with smaller gaps to record more densely. PE or Phase enabled was the next standard at 1600 bits or characters per inch. GCR (Group Controlled Recording) was a high density recording system, and the last to be extensively used on open reel tapes.
A later development, removing the requirement to have rapid start-stop times, was streaming tape, which writes or reads continuously. A disadvantage of this system is an overshoot on an error condition. Time is then lost relocating the data on the tape.
Current tape drives take a stack of cassettes and record at high density, using compression techniques. This makes the tape suitable for its main purpose, backup of important, indeed all, data. However it emphasises the main disadvantage of tape, the relatively long access time.
Tape backups also take the form of an 'audit trail' used on systems where 'real time' transactions are taking place, for instance airline reservation systems. These require a continual backup of the data being updated on the system disks to be written to the tape, so that an up to the second recovery can be made in the event of a system failure.
Another development in tape handling has been the use of libraries to handle tape cartridges. These store up to 5,000 cartridges, and are accessed by robots, and controlled by a dedicated computer system.