Binary Numbers
We are used to using the decimal number system for counting and performing calculations.
Nowadays, it is very rare for any other number system to be used, but you may have come across
examples of other number systems such as Roman Numerals that are not decimals.
Decimal Number
1
5
10
50
100
1000
1984
Roman Numeral Equivalent
I
V
X
L
C
M
MCMLXXXIV
The decimal system has been used for around 5000 years because humans find it easy to
understand. Look at 1984 in the table of Roman Numerals above to see why people prefer decimal!
What does decimal mean?
The "dec" in decimal comes from the word "decem" in Latin which is the equivalent of the English
word "ten". Decimal is named after the number 10 because:
There are ten digits:
0 1 2 3 4 5 6 7 8 9
The value of a digit depends upon where it appears in a number. The place values of the digits are
multiplied by ten each time you move one digit to the left.
Look at the number 5633 below. The place values are written above the number.
x 10
x 10
1000
100
10
1
5
6
3
3
The first three in 5633 is worth ten times as much as the second three as it is one place to the left of
it.
The six in the 100s column is worth 100 times as much as a six would be in the 1s column because it
is two places (10*10=100) to the left of it.
If you understand mathematical powers, you will see that the place values in decimal are powers of
ten as 100 = 1, 101 = 10, 102 = 100, 103 = 1000 and so on.
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Binary
Computers don't like decimal they like binary
Human's like decimal, but computers do not. This is because computers are digital devices. It is
easier, cheaper and more reliable to make computers that work using a different number system
called binary.
The binary number system has just two digits. They are:
01
A binary number is just a sequence of 0s and 1s, for example 01101011000101 is a binary number.
A digit in binary is known as a binary digit or bit for short.
Binary is called binary because "bi" means "twice" or "two" in Latin.
Why do computers like binary?
Computers are electronic devices that store and process data as electrical voltages. Using
electronics, it is quite an easy job to represent two different values (such as the digits 0 and 1 in
binary) but very hard to represent 10 different values (such as the ten digits in decimal).
This is because 0 and 1 can be represented as an electrical current being off (0) or on (1). All of the
electrical components inside the computer use this principle.
If a computer were to use decimal then the electronics would need to be able to represent ten
different values (digits 0 to 9). This could be done as different voltages. For example, the digit 0
could be 0 volts, 1 could be 1 volt and so on up until 9 being 9 volts. But creating and measuring
voltages like this is much more complex than just detecting if a current is on or off, and important
components inside computers like transistors on microchips cannot do this. For these reasons, there
are no computers that use decimal.
Place values in binary
You already know that in binary there are just two digits, 0 and 1.
To represent numbers with more than one digit, place values are used so that the digits at the start
of a number are worth more than the digits at the end. In decimal, the place values start at one and
are increased by being multiplied by 10. In binary, the place values increase by being multiplied by 2.
Place values in binary are:
x2
x2
128
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64
32
16
8
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4
2
1
Binary
Conversions
As computers use binary and humans use decimal, it is necessary to be able to convert between
binary and decimal. A computer will do this automatically for you, but if you are going to work with
computers then it is also useful to be able to do this yourself.
Converting from binary to decimal
Converting numbers from binary to decimal is quite easy.
You just need to write the place values above the digits in the binary number then add up the place
values where the binary digits underneath them are 1.
Example 1
Question: Convert the binary number 1101 into decimal.
Answer:
First, write out the place values for a four-digit binary number then write the number underneath:
8
1
4
1
2
0
1
1
Place Values
Number
Now, add up the place values for the columns that have a 1 in them:
8 + 4 + 1 = 13
So, the binary number 1101 is equivalent to the decimal number 13.
Example 2
Question: Convert the binary number 11001011 into decimal.
Answer:
First, write out the place values for an eight-digit binary number then write the number underneath:
128
1
64
1
32
0
16
0
8
1
4
0
2
1
1
1
Now, add up the place values for the columns that have a 1 in them:
128 + 64 + 8 + 2 + 1 = 203
So, the binary number 11001011 is equivalent to the decimal number 203.
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Binary
Questions
Convert the following numbers from binary to decimal.
(1)
10101
To help you answer, the place values and number have been written out for you:
16
1
8
0
4
1
2
0
1
1
Answer: ...........................................................................................................................................
(2)
1011
Answer: ...........................................................................................................................................
(3)
11101
Answer: ...........................................................................................................................................
(4)
100101
Answer: ...........................................................................................................................................
(5)
10101001
Answer: ...........................................................................................................................................
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Binary
Converting from decimal to binary
There are a few different ways that you can convert a number from decimal to binary. Here is an
algorithm to explain one of the simplest ones, using the example of converting the denary number
181.
Start at 1 at the right hand side and write out the place values for binary until the place value that
you write is bigger than the number that you need to convert. For 181:
256
128
64
32
16
8
4
2
1
4
2
1
Then cross out the highest place value as you will not need this:
256
128
64
32
16
8
Next, start at the left hand end of the list of place values (128 in this case).
If the place value at the current position is less than the number you have left to convert then:


write a 0 under the place value
move right one position.
If the place value at the current position is greater than or equal to the number you have left to
convert then:



write a 1 under the place value
subtract the place value from the number you have left to convert
move right one position.
Repeat the above until you have written values in under every place value.
For the number 181:
128
1
64
0
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32
1
16
1
8
0
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4
1
2
0
1
1
Binary
Here are some more examples to help you understand the method.
Example 1
Question: Convert the binary number 37 from decimal into binary.
Answer:
The place values to write out are 1, 2, 4, 8, 16, 32 and 64. 64 is then crossed out as it is bigger than
37. So, the place values to use are:
32
16
8
4
2
1
32
1
16
1
8
0
4
1
2
0
1
1
37>32
so write
1
leaves
37-32=5
5<16
so write
0
5<8
so write
0
5>4
so write
1
leaves
5-4 = 1
1<2
so write
0
1=1
so write
1
Leaves
1-1 = 0
And the conversion is:
So the binary equivalent of 37 is 110101.
Example 2
Question: Convert the binary number 204 from decimal into binary.
Answer:
The place values to write out are 1, 2, 4, 8, 16, 32, 64, 128 and 256. 256 is then crossed out as it is
bigger than 204. So, the place values to use are:
128
64
32
16
8
4
2
1
And the conversion is:
128
1
64
1
32
0
16
0
8
1
4
1
2
0
1
0
204>128
so write
1
leaves
204-128
= 76
76>64
so write
1
leaves
76 - 64
= 12
12<32
so write
0
12<16
so write
0
12>8
so write
1
leaves
12-8
=4
4=4
so write
1
leaves
4-4
=0
0
left to
convert
so write
0
0
left to
convert
so write
0
So the binary equivalent of 204 is 11001100.
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Binary
Questions
Convert the following numbers from decimal to binary.
(1)
21
To help you answer the first question, the place values have been written out for you:
16
8
4
2
1
Answer: ...........................................................................................................................................
(2)
43
Answer: ...........................................................................................................................................
(3)
58
Answer: ...........................................................................................................................................
(4)
214
Answer: ...........................................................................................................................................
(5)
706
Answer: ...........................................................................................................................................
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Binary
Bits and Bytes
Computers usually store numbers in fixed sized memory locations.
Remember that a bit is a single binary digit i.e. a 0 or a 1.
The most common size of memory location that a computer uses is a byte. A byte is 8 bits long. If a
number does not need 8 bits, then if it is stored in a byte extra 0s are added to the start of it.
For example, the decimal number 25 is 11001 in binary. This would be stored as 00011001 in a byte.
Three extra 0s are added to the start to make it 8 bits long.
Kilobytes, Megabytes and More
How much storage a computer has is measured in bytes. Extra storage in the RAM will let the
computer run more programs as the same time. Extra storage on the Hard Disk Drive will let you
install more programs and save more files.
The Manchester baby, the world's first stored-program electronic digital computer which
successfully ran its first program on 21st June 1948 had 128 bytes of memory.
128 bytes of
memory here!
As the amount of memory a computer has increased, bytes became too small a unit to describe
memory size, so other units such as Kilobytes, Megabytes and Gigabytes were invented.




1 Kilobyte (Kb) is equal to 1024 bytes.
1 Megabyte (Mb) is equal to 1024 Kilobytes.
1 Gigabyte (Gb) is equal to 1024 Megabytes.
1 Terabyte (Tb) is equal to 1024 Gigabytes.
The computer that flew
the rocket which landed
the Apollo 11 lunar capsule
on the moon in 1969 had
64 Kilobytes of memory.
The Apple Macintosh Plus
computer, launched in 1986,
came with 1 Megabyte of
memory which could be
expanded up to 4 Megabytes.
The Nintendo SNES
console, launched in 1990,
had 256 Kilobytes of
memory.
A modern hard disk drive can
store anything from about 256
Gigabytes up to 4 Terabytes of
data.
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Binary
Converting Memory Sizes
Sometimes it is useful to be able to convert between bytes, Megabytes, Gigabytes etc. so that you
can compare the amount of memory in two different computers. Here are two example of
converting memory sizes into bytes:
The Apollo 11 computer had 64 Kilobytes of memory.



We know that 1 Kilobyte = 1024 bytes
So 64 Kilobytes = 64 * 1024 = 65,536 bytes
Therefore the Apollo 11 computer had 65,536 bytes of memory.
An upgraded Apple Macintosh Plus computer could have 4 Megabytes of memory.





We know that 1 Megabyte = 1024 Kilobytes
So 4 Megabytes = 4 * 1024 = 4,096 Kilobytes
We know that 1 Kilobyte = 1024 bytes
So 4,096 Kilobytes = 4096 * 1024 = 4,194,304 bytes
Therefore the upgraded Apple Macintosh Plus had 4,194,304 bytes of memory.
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Binary
Questions
(1)
How many bits are there in a byte?
Answer: ...........................................................................................................................................
(2)
Show how the binary number 1011 would be stored in a byte.
Answer: ...........................................................................................................................................
(3)
What is the biggest binary number that can be stored in a byte, and what is the decimal
equivalent of this number?
Answer: ...........................................................................................................................................
(4)
An image file is saved on a computer. It takes up 100 Kilobytes. How many bytes does the file
take up?
Answer: ...........................................................................................................................................
(5)
An new computer has 2 Gigabytes of memory. How many Megabytes is this equivalent to?
Answer: ...........................................................................................................................................
(6)
Three different devices which can store data are a CD-R, a DVD-R and a USB Flash Drive.



The CD-R can store 650 Megabytes of data.
The DVD-R can store 4.7 Gigabytes of data.
The USB Flash Drive can store 8 Gigabytes of data.
Write out the names of the devices, in order, with the device that can store most data first
and the device that can store least last.
Answer: ...........................................................................................................................................
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Binary