1520 W12 section S Number systems conversions
Page 1 of 3
Number systems and conversions from one system to another.
The 4 number systems are:
Binary 2 .
§ Uses 2 symbols: the digits 0 and 1.
§ Some numbers in this system: 0 000 1010.
Decimal 10
§ Uses 10 symbols: the digits 0 1 2 … 9.
§ Some numbers in this system: 111 0 1010.
Octal 8
§ Uses 8 symbols: the digits 0 1 2 3 4 5 6 7.
Hexadecimal 16
§ Uses 16 symbols: 0 1 2 …. 9 A B C D E F.
Notes:
§ When written down a number may be ambiguous regarding which system
it belongs to. So we will associate a subscript to clear such ambiguities.
1010 in the binary system will be denoted as 1010
to distinguish it from
1010 of the decimal system which would be denoted as 1010
.
Similarly 1010 of the octal system would be denoted as 1010
and
1010 of the Hex system would be denoted as 1010
.
Conversions from one system to another.
It is possible to convert between any of the number systems.
2 à 10
1010
1010
Based on multiplying with powers of 2.
1010
1∗2
+ 0∗ 2
+1∗2
+0∗2
2
+2
8+2 10
i.e. 1010
10
10 à 2
10
10

slide 2:

1520 W12 section S Number systems conversions
Page 2 of 3
Based on dividing by 2. Record the quotients and the remainders. Stop diving if quotient
is 0 and then take the remainders bottom to top. This would be the binary number.
Number Operation Quotient Remainder
10 10 / 2 5 0
5 5/2 2 1
2 2/2 1 0
1 ½ 0 1
Taking the remainders bottom-to-top on the above table forms the number 1010. This
would be the binary number that is equal to 10
i.e. 10
1010
.
2 à 8
1010
Take chunks of 3 digits from right to left pad with 0’s at the left if there are
not enough binary digits and write the octal digit that corresponds to each
chunk.
1010
001 010
1
2
12
.
8 à 2
12
Take each octal digit and map it to its binary representative with 3 binary
digits.
12
→ 001
010
→ 001010
1010
2 à 16
11010
Take chunks of 4 digits from right to left pad with 0’s at the left if there are
not enough binary digits and write the hex digit that corresponds to each
chunk.
11010
0001
1010
1
1
.

slide 3:

1520 W12 section S Number systems conversions
Page 3 of 3
16 à 2
1
Take each hex digit and map it to its binary representative with 4 binary
digits.
1
0001
1010
00011010
11010
The diagram below shows all the conversion cases covered above.
This means that we can now handle any type of conversion even conversions between
the number systems that are not directly connected on the above diagram. For example to
convert 8 à 10 we can convert 8 à 2 followed by 2 à 10 . Or to
convert 8 à 16 we can convert 8 à 2 followed by 2 à 16 . Etc.
2
10
16
8

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