Figure 2.8 The fans enter the stadium
Electrons don’t lose their tickets but random effects like temperature, voltage and interactions between adjacent electrons have a very similar effect.
A single current of, say, 1 A can be split into two currents of 0.5 A when measured over the long-term, but when examined carefully, each will contain random fluctuations. This type of electrical noise is called partition noise or partition effect. The overall effect is similar to the thermal noise and, between them, would cause too much noise and hence would rule out the use of a 10-digit system.
The 10-finger system that we use is called a ‘denary’ or ‘decimal’ system. We have seen that a 5 V supply would accommodate a 10-digit counting system if each digit was separated by 0.5 V or, using the more modern choice of 3.3 V, the digits would be separated by only 0.33 V.
Question: Using a 5 V supply and a denary system, what is the highest noise voltage that can be tolerated?
Answer: Each digit is separated by only 5 V/10 = 0.5 V. The number 6 for example would have a value of 3 V and the number 7 would be represented by 3.5 V. If the noise voltage were to increase the 3 V to over 3.25 V, the number is likely to be misread as 7. The highest acceptable noise level would therefore be 0.25 V. This is not very high and errors would be common. If we used a supply voltage of 3.3 V, the situation would get even worse.
So why don’t we just increase the operating voltage to say, 10 V, or 100 V? The higher the supply voltage the less likely it is that electrical noise would be a problem. This is true but the effect of increasing the supply would be to require thicker insulation and would increase the physical size of the microprocessor and reduce its speed. More about this in Chapter 11.
If we reduce the number of digits then a wider voltage range can be used for each value and the errors due to noise are likely to occur less often.
We have chosen to use only two digits, 0 and 1, to provide the maximum degree of reliability. A further improvement is to provide a safety zone between each voltage. Instead of taking our supply voltage of 3.3 V and simply using the lower half to represent the digit 0 and the top half for 1, we allocate only the lower third to 0 and the upper third to 1 as shown in Figure 2.9. This means that the noise level will have to be at least 1.1 V (one-third of 3.3 V) to push a level 0 digit up to the minimum value for a level 1.
Figure 2.9 A better choice of voltages
Normally, we count in the system we call ‘denary’. We start with 0
then go to 1 then to a new symbol that we write as 2 and call ‘two’. This continues until we run out of symbols. So far, it looks like this:
0
1
2
3
4
5
6
7
8
9
At this point we have used all the symbols once and, to show this, we put a ‘1’ to the left of the numbers as we re-use them. This gives us:
10
11
12
13
14
… and so on up to 19 when we put a 2 on the left-hand side and start again 20, 21, 22 etc.
When we reach 99, we again add a ‘1’ on the left-hand side and put the other digits back to zero to give 100. After we reach 999, we go to 1000 and so on.
Counting is not easy. We often take it for granted but if we think back to our early days at school, it took the teacher over a year before we were happy and reasonably competent. So counting is more difficult than microprocessors – you’ve mastered the difficult part already!
The base of a number system is the number of different symbols used in it. In the case of the denary system, we use 10 different symbols, 0…9, other numbers, like 28 657, are simply combinations of the 10 basic symbols.
Since the denary system uses 10 digits, the system is said to have a base of 10. The base is therefore just the technical word for the number of digits used in any counting system.
We can count using any base that we like. In the denary or decimal system, we used a base of 10 but we have seen that microprocessors use a base of 2 – just the two digits 0 and 1. This is called the binary system.
We usually abbreviate the words BInary digiT to bit. Counting follows the same pattern as we have seen in the denary system: we use up the digits then start again. Let’s give it a try. Start by listing all the digits:
0
1
and that’s it!
We now put a ‘1’ in the next column and start again:
10
11
It is convenient at this stage to keep the number of binary columns the same and so we add a 0 at the start of the first two digits. These extra zeros do not alter the value at all. For example, the denary number 25 is not affected by writing it as 025 or 0025 or even 000 000 000 000 025.
The binary and decimal equivalents are:
| Binary | Denary |
|---|---|
| 00 | 0 |
| 01 | 1 |
| 10 | 2 |
| 11 | 3 |
We do the same again – put a ‘1’ in the next column and repeat the pattern to give:
| Binary | Denary |
|---|---|
| 100 | 4 |
| 101 | 5 |
| 110 | 6 |
| 111 | 7 |
and once more:
| Binary | Denary |
|---|---|
| 1000 | 8 |
| 1001 | 9 |
| 1010 | 10 |
| 1011 | 11 |
Here is a number: 1000. But what number is it? Is it a thousand in denary or is it eight written in binary?
I don’t know. I could take a guess but the difference between flying an aircraft at eight feet and a thousand feet is a serious matter. The only way to be certain is to say so at the time. This is done by showing the base of the number system being used to make the meaning quite clear. The base of the number system is shown as a subscript after the number.
If the 1000 were a binary number, it is written as 10002 and if it were a denary number it would be shown as 100010.
It would be easy to advise that the base of the number system in use is always shown against every number but this would be totally unrealistic. No one is going to write a base after their telephone number or a price in a shop. Use a base when it would be useful to avoid confusion, such as by writing statements like 1000 = 8 (a thousand = eight???). Write it as 10002 = 810 and make life a little easier.
Of course, if someone were to ask us for the binary equivalent of nine we could just start from zero and count up until we reach nine. This is a boring way to do it and with larger numbers like 1 000 00010 it would be very tedious indeed. Here is a better way. The method will be explained using the conversion of 5210 to binary as an example.