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"The
wise person makes learning a joy. . . Proverbs 15:2"
Binary Numbers - An
intro along with conversion formulas
Welcome to our binary numbers guide, in this section we will explain what
binary is and show you how to convert between binary and decimal (denary)
numbers.
Binary Numbers Overview
Binary is a number system used by digital devices like computers, cd
players, etc.
Binary is Base 2 unlike our counting system decimal which is Base 10
(denary).
In other words, Binary has only 2 different numerals
(0 and 1), unlike Decimal which has 10 numerals
(0,1,2,3,4,5,6,7,8 and 9).
Here is an example of a binary number: 10011100
As you can see it is simply a bunch of zeroes and ones, there are 8
numerals in all which make this an 8 bit binary number, bit is short
for Binary Digit, and each numeral is classed as a
bit.
The bit on the far right (in this case a zero) is known as the
Least significant bit (LSB), and the bit on the far left (in
this case a 1) is known as the Most significant bit (MSB)
notations used in digital systems:
4 bits = Nibble
8 bits = Byte
16 bits = Word
32 bits = Double word
64 bits = Quad Word (or paragraph)
When writing binary numbers you will need
to signify that the number is binary (base 2),
for example take the value 101, as it is written it would be
hard to work out whether it is a binary or decimal (denary) value,
to get around this problem it is common to denote the base to which
the number belongs by writing the base value with the number, for
example:
1012 is a binary number and 10110
is a decimal (denary) value.
Once we know the base then it is easy to work out the value, for
example:
1012 = 1*22 + 0*21 + 1*20
= 5 (five)
10110 = 1*102 + 0*101 + 1*100
= 101 (one hundred and one)
One other thing about binary numbers is that it is common to signify
a negative binary value by placing a 1 (one) at the left hand side
(most significant bit) of the value, this is called a sign bit,
we will discuss this in more detail in the next part of the
tutorial.
Electronically binary numbers are
stored/processed using off or on electrical pulses, a digital system
will interpret these off and on states as 0 and 1, in other words if
the voltage is low then it would represent 0 (off state), and if the
voltage is high then it would represent a 1 (on state).
Converting binary to
decimal
To convert binary into decimal is very simple and can be
done as shown below:
Say we want to convert the 8 bit value 10011101 into
a decimal value, we can use a formula like that below:
|
128 |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
|
1 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
As you can see we have placed the numbers 1, 2, 4, 8, 16,
32, 64, 128 (powers of two) in reverse numerical order and
then written the binary value below, to convert you simply
take a value from the top row wherever there is a 1 below
and add the values together, for instance in our example we
would have 128 + 16 + 8 + 4 + 1 = 157. For a 16 bit
value you would use the decimal values 1, 2, 4, 8, 16, 32,
64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768
(powers of two) for the conversion.
Because we know binary is base 2 then the above could be
written as:
1*27 + 0*26 + 0*25 + 1*24
+ 1*23 + 1*22 + 0*21 + 1*20
= 157.
Converting decimal
to binary
To convert decimal to binary is also very simple,
you simply divide the decimal value by 2 and then
write down the remainder, repeat this process until
you cannot divide by 2 anymore, for example let's
take the decimal value 157:
157 ÷ 2 = 78
78 ÷ 2 = 39
39 ÷ 2 = 19
19 ÷ 2 = 9
9 ÷ 2 = 4
4 ÷ 2 = 2
2 ÷ 2 = 1
1 ÷ 2 = 0 |
with a remainder of 1
with a remainder of 0
with a remainder of 1
with a remainder of 1
with a remainder of 1
with a remainder of 0
with a remainder of 0
with a remainder of 1 |
<--- to convert write this remainder first.
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Next write down the value of the remainders from
bottom to top (in other words write down the bottom
remainder first and work your way up the list) which
gives:
10011101 = 157
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