In the Decimal system, we can multiply by 10 by adding a 0 to the end of a number. For example 4 with a zero added becomes 40, similarly 346 becomes 3460. We can expand this by adding 2 or 3 zeros to multiply by 100 or 1000.
In the Binary system used in computers, we can multiply by 2 by adding a zero at the end of the number. Thus 110 (2+4=6 decimal) becomes 1100 (4+8=12 decimal). Similarly we can add more zeros and multiply by 4,8,16 etc. (decimal). This is one form of multiplication, the process is called shifting as each bit, 1 or 0 is shifted to the next bit position, and a zero is added in at the first bit position.
Several different techniques have been used to multiply using logic elements, as before these are usually described in a logic diagram as a 'black box' labeled multiplier. In an even more sophisticated logic diagram, this would be combined with other 'black boxes' such as adders, dividers, square roots, etc. to make one big 'black box' the ALU (arithmetic logical unit). The actual 'works' inside this unit are irrelevant to the overall design of the computer. All the designer needs to know is that if he puts two numbers into the ALU, and tells it to multiply them, he will get an output of the result.
Initially these boxes would have been made up physically of vacuum tubes, in a box the size of your bedroom, these have been gradually improved, replaced, miniaturised, until nowadays that will all fit on a chip. However the basic principles are the same.
If we analyse the concept of multiplication, we see that it is one of repetition (and we know computers excel at this). Take for example 2X4. This means take 4 lots of 2, and add them together, or 2+2+2+2=8. So to make a multiplier for a computer we can use an adder, which we have, and some method of counting, which we also have, as discussed in the earlier articles we mentioned.
For the example we just looked at, 2X4, our multiplier would have one input from the 2 (10 binary) going to a 4-bit adder. The output, or result, from the adder would be looped around to form the second input to the adder.
The second of the numbers to be multiplied, 4 (100 binary) sets a flip-flop counter to count down from 4 to 1, with one count pulse every time we add. Thus the counter is 'more than 1, which is the condition for the adder output to be routed to its input. The initial add would be 10 + 10 binary (2+2 decimal), giving 100 binary.
This result is returned to the input, gated by the counter 'more than 1' to be added to 10 again, giving 110 binary. We perform another add of 110 + 10 getting a result of 1000.
This time the counter has counted down to one, and blocks the adder input. At the same time it allows the adder result output to become the multiplier result.
You can see how this simple example could be used in a scaled-up version capable of multiplying multi-bit numbers. All we need is a lot more adders, and a few logic gates to control them, maybe throw in a bit of timing, so that it doesn't all get mixed up! As we've said before, when you're talking in nanoseconds, you can get through a lot of calculations very quickly.
Still to come we will look at how negative numbers are represented in computers, and how they handle very large numbers with something called Floating Point Arithmetic.
History Of The Computer
If anything could be said in favor of war, it may be that it speeds up the development of technology. Of course much of that technology is employed in killing people and destroying infrastructure, but there are also things which could be considered beneficial.
Electronics was around in the early part of the 20th Century, wireless, or radio, was in its infancy at the time of the first world war. Radio broadcasting came into prominence in the '20s and 30's, Television started in the '30's.
The second world war, from 1939 in Europe, and a couple of years later involving the USA, ended in 1945. Radar (Radio Aid To Detection And Ranging) was developed from early experiments, just prior to the war, in Britain and Germany. There was rapid development in the field, and, by the end of the war, Radar was being used in several fields.
Aircraft Navigation - Using ground transmitters in sets of 3, widely spaced, to give an aircraft using a receiver a method of pinpointing its position. This is a similar system to that used in gps today, substituting satellites for the ground stations.
Targeting - A beam was transmitted from a Radar station in England so that it intercepted a target in Germany. An aircraft could fly along the beam, guided by signals, dots or dashes, if it strayed off the beam, left or right. Known as flying on the beam.
Interception - a series of ground stations around the South-East coast of England, feeding into a central control room, where their tracks could be displayed, significantly assisted in the Battle of Britain (1940).
Airborne Interception (AI) - Developed towards the end of the war, used a Tranceiver (transmitter/receiver) in a night fighter to find a target in the dark, or bad weather, and track it to within firing range.
Beacon - A tranceiver was located at the end of a runway so that ground staff could guide a returning aircraft to land in bad weather, this became more and more sophisticated, developing into GCA or Ground Controlled Approach.
Shipping - radar equipped vessels could track other vessels in darkness or fog, whether peacefully or aggressively.
Many other sytems were developed or initiated in that 6 year period. Knowledge of electronics, and what it could be used for vastly increased. In parallel with the development of radar, other fields of electronics were also advancing, under pressure from the requirement to improve the technology.
Long range guns on ships or in the field needed to be aimed accurately. The calculations required in ballistics to aim a gun so that you can hit the target, or aim a V2 rocket so that it hits London from continental Europe, are phenomenal.
This, then was the scene at the end of the war. We knew how to tackle large calculations with speed, and we had developed a new concept in electronics, Pulse Technology. This is so called because radar uses short pulses of high energy, for two main reasons.
1. The pulses can be coded. For example, in the Navigation example we looked at, 3 ground stations transmit a signal whose source needs to be identified. One transmitter could transmit a series of single pulses spaced say 10 milliseconds. A second could transmit a pair of pulses at 10 millisecond spacing, and the third 3 pulses. A chart would tell the navigator where the pulse sets were transmitted from, and the distances obtained from the radar set used to locate the position on the chart.
2. The power, or strength, of the signal. A continuous radio signal, like a radio broadcast, takes a given amount of power. However, a 1 millisecond pulse every 10 milliseconds, uses only one tenth of the power, on average. So a radar transmitter can have a much greater range for the same power. This is is especially important in a primary (transmit and receive) radar system, where we must detect the reflection of the signal we transmit. Likewise a secondary (receive) radar system, for example the navigation system above, will have a bigger range.
Next we will look at how early computers were now possible due to these developments.