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Binary numbers
Binary
numbersThe difference between digital and analog systemsAsk most people what the difference is between a digital and an analog
system, they will tell you that digital is more precise and accurate
than analog - whether it be digital TV which is clearer than analog TV,
digital recording systems, digital scales, you name it.
That's
because digital systems work in a world of black and white: your
computer contains millions of tiny electronic switches and circuits -
each can be either on or off. Off means around 0 volts flowing through in a particular wire, on
usually means something around 5 volts - there's no middle ground. This
is why digital systems are so accurate: electric circuits are never
perfect, there's always going to be fluctuations and glitches in
voltage here and there; but with a digital system these fluctuations
don't matter, the circuit concerned will always be correctly
interpreted as either on or off as long as the voltage stays on the
correct side of a certain critical threshold voltage.
The difference with an analog system, is that with analog, a circuit can have any
voltage in a continuous range of voltages, and often the exact voltage
is then measured or used in some way in order to produce the desired
result. The problem here is that the slightest bit of interference in a
sensitive circuit will often change the result.
Digital systems are more complicated than analog systemsIf
you think about it, having a circuit limited to being on or off
presents certain challenges. Imagine you are building an electronic
machine to count up from zero to ten - with an analog system you could
have a special 'counting circuit', and keep track of the number you
were up to changing the number of volts in that circuit.
How can you do that with a digital system if a circuit can only be on or off? The answer is that you actually need four circuits to provide the required number of possible states
To
illustrate the different possibilities, lets use the number 1 to
represent 'on', and the number 0 to represent 'off'. Computer
scientists actually do this all the time:
In general:
1 stands for 'on'
0 stands for 'off' |
If we have 4 circuits which can be on or off, these are the possible states:
| | | | Represents: | | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 1 | 1 | | 0 | 0 | 1 | 0 | 2 | | 0 | 0 | 1 | 1 | 3 | | 0 | 1 | 0 | 0 | 4 | | 0 | 1 | 0 | 1 | 5 | | 0 | 1 | 1 | 0 | 6 | | 0 | 1 | 1 | 1 | 7 | | 1 | 0 | 0 | 0 | 8 | | 1 | 0 | 0 | 1 | 9 | | 1 | 0 | 1 | 0 | 10 |
We
could keep going. By the way, four circuits actually gives us 16
different
possibilities, which means we could actually count from 0 up to 15.
That's because every time you add another on/off circuit to the system,
you practically multiply the number of possibilities by 2; but we have
4 circuits here, and 2×2×2×2 (or 24) is equal to 16.
Binary numbersWhen we humans count normally in everyday life, we use a number system called decimal,
or 'base ten'. This means that for each digit in a number, there are 10
different possible values for that digit. Let me list them: in a normal
decimal number (like, say, 355), each digit in the number be either 0,
1, 2, 3, 4, 5, 6, 7, 8, or 9. You can get up to 9 for a particular
digit, and then you have to carry over the next digit.
But how
would you count if you could only choose between 0 and 1 for each
digit? Let's say you start at 0, then 1, then what? You can't go past
1, so you have to carry over the next digit; so the next number you get
is 10, then 11, then the nearest number after 11 is 100, and so on.
Isn't this how you'd count if you could only choose between 1 and 0 for
each digit?
This number system is called 'base two', or binary,
and it's exactly how numbers are represented inside computers. Each
digit in this system of counting is known as a 'binary digit', or bit for short. Take another look at the above table. See? We were counting in binary the whole time!
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