Binary Numbers Are Used for These Sampled Values

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

Binary equivalent

Decimal numbers

Binary equivalent

0 / 0000 / 8 / 1000
1 / 0001 / 9 / 1001
2 / 0010 / 10 / 1010
3 / 0011 / 11 / 1011
4 / 0100 / 12 / 1100
5 / 0101 / 13 / 1101
6 / 0110 / 14 / 1110
7 / 0111 / 15 / 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

011011

110101

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).

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.

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