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Binary numbers
We are very familiar with counting in the decimal system. This means that we have numbers
from one to nine and then go on to a number of tens.
Binary numbers are used for these sampled values.
Binary is a way of expressing numbers in ones (high voltage value) or zeros (low voltage
value) – there is nothing in between. You can only have either a 1 or a 0.
In mathematical language you are expressing numbers to the base 2 instead of our normal
decimal system where we use the base 10.
Decimal
numbers
0
1
2
3
4
5
6
7
Binary
equivalent
0000
0001
0010
0011
0100
0101
0110
0111
Decimal
numbers
8
9
10
11
12
13
14
15
Binary
equivalent
1000
1001
1010
1011
1100
1101
1110
1111
The number of digits in the group gives is the BIT NUMBER. For example all the above
numbers are FOUR BIT NUMBERS – there are only four ones or zeros. You can see from
the table that four bit binary numbers can only deal with numbers up to decimal 15. If we
want to express larger numbers we have to have 8 bit, 16 bit or 32 bit binary numbers. Many
of your computers are 32 BIT machines – they deal with numbers like:
00110011010011100011000110101011
The table below shows some examples of converting some decimal numbers into binary:
Decimal Thirty two Sixteen Eight Four Two One Binary
27
0
1
1
0
1
1
011011
53
1
1
0
1
0
1
110101
62
1
1
1
1
1
0
111110
In a computer the ones and zeros are sent in a string with one following the other so a 32 bit
number is a longish string – longer than a four bit number.
Computers and other such machines can understand binary numbers because there are only
two options – ON (1) or OFF (0).
schoolphysics 14-16/General/Binary numbers
1
The next table shows some EIGHT BIT binary numbers and their decimal equivalent.
Binary number Decimal equivalent Binary number Decimal equivalent
00011110
30
01010010
82
00100011
35
01010011
83
00100100
36
01011100
92
00101110
46
01100010
98
00111000
56
01101000
104
000111100
60
01101110
110
00111110
62
01110011
115
01000001
65
01110100
116
01000111
71
01110000
120
01001000
72
10000010
130
01001011
75
10110111
187
01001100
76
11000111
203
01001110
78
11001111
211
01010000
80
11110100
248
Why use binary numbers?
But having explained how binary numbers relate to decimal numbers we must look at why
binary numbers are so useful — especially in digital devices such as computers and in
transmitting digital information in CDs, TVs, cameras etc.
The point is that binary numbers are made up of ones and noughts as you can see from
looking at the table. If we ‘translate’ this into electricity we could have a circuit that is either
ON or OFF or a voltage that is either HIGH or LOW (the low being zero).
The real advantage of a binary system is that the voltages need not be exactly nought or one
to give a meaningful output.
1
2
3
4
5
6
7
8
Look at the people with the flags. Let’s imagine that a flag held upright means a one and a
flag held horizontal means a zero. So the number represented would be 11010110 in binary.
But if you look carefully you will see that person three does not have their flag quite
horizontal and person seven does not have theirs quite vertical. However they are still close
enough for us to take them as a 0 and a 1.
In a digital signal using binary code the output will be interpreted as a perfect version of the
input even if some of the voltages are not quite exact just like the flags.
See also 14-16/Wave properties/Text/Analogue and digital signals
schoolphysics 14-16/General/Binary numbers
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