NUMBER SYSTEMS: TUTORIAL
This tutorial explains the different number systems, converting a number which is represented in a specific
number system to its value in another number system and number systems arithmetic.
The "base" of number systems is determined by how many numerical digits the number system uses:
Binary
Octal
Hexadec
(base2)
(base8)
(base 10)
Study the table below:
Decimal
Hexadecimal
Octal
Binary
0
0
0
0000
1
1
1
0001
2
2
2
0010
3
3
3
0011
4
4
4
0100
5
5
5
0101
6
6
6
0110
7
7
7
0111
8
8
10
1000
9
9
11
1001
10
A
12
1010
11
B
13
1011
12
C
14
1100
13
D
15
1101
14
E
16
1110
15
F
17
1111
Decimal uses 0-9 (base 10), hexadecimal uses 0-F (base 16) octal uses 0-7 (base 8) and binary uses only 0
or 1 (base 2)
This tutorial covers 3 number systems: Binary, Hexadecimal and Octal
Well, 4 if you include decimal, but we all know decimal arithmetic, hopefully. Firstly let's look at binary.
The Binary Number System
Firstly, in order to understand how to convert number systems, we need to understand the relationship of the
digits in a number and their position of those digits for their base number. Let's use the decimal number
system (base 10), one that we're all familiar with:
103
102
101
100
2
4
6
1
Now we all know by looking at that number that it's Two thousand, 4 hundred and sixty one. But how do we
know? We know by the number's position, we can find this value like this:
1
2 x 103 = 2 x 1000 = 2000
4 x 102 = 4 x 100 = 400
6 x 101 = 6 x 10 = 60
1 x 100 = 1 x 1 = 1 +
---2461
Apply this to the binary system and we can easily convert a binary value to its decimal form
Binary to Decimal Conversion
Let's do the same we did above, only this time with the base 2 number system
23
22
21
20
1
1
0
1
We have the binary number 11012 (the little 2 represents the base number). Let's convert it to decimal:
1 x 23 = 1 x 8 = 8
1 x 22 = 1 x 4 = 4
0 x 21 = 0 x 2 = 0
1 x 20 = 1 x 1 = 1 +
---13
So the 4-bit binary number 1101 is equal to 13 in decimal. Let's convert an 8-bit binary number.
27
26
25
24
23
22
21
20
1
0
0
1
1
1
1
0
So now we have the binary number 100111102
1 x 27 = 1 x 128 = 128
0 x 26 = 0 x 64 = 0
0 x 25 = 0 x 32 = 0
1 x 24 = 1 x 16 = 16
1 x 23 = 1 x 8 = 8
1 x 22 = 1 x 4 = 4
1 x 21 = 1 x 2 = 2
0 x 20 = 0 x 1 = 0 +
---158
Pretty simple to grasp, make up a few binary numbers and practice on them. You can check your answers
using a scientific calculator. If you're using windows go to start > programs > accessories > calculator. At the
top go to "view" and pick "scientific".
Decimal to Binary Conversion
Converting a decimal number to its binary equivalent is a little fiddlier than vice-versa, but it isn't too bad. We
repeatedly divide the decimal number by 2 (the base number of the binary system) collecting the
remainders. If we divide an even number by 2 there is no remainder, if we divide an odd number by 2 there
is a remainder of 1.
Decimal number: 36
2
36 / 2 = 18 - remainder 0
18 / 2 = 9 - remainder 0
9 / 2 = 4 - remainder 1
4 / 2 = 2 - remainder 0
2 / 2 = 1 - remainder 0
1 / 2 = 0 - remainder 1
Now collect the remainders from the bottom up and we have the binary number:
1001002 = 3610
Isn't so bad is it? Let’s do one more to drill it in.
Decimal number: 227
227 / 2 = 113 - remainder 1
113 / 2 = 56 - remainder 1
56 / 2 = 28 - remainder 0
28 / 2 = 14 - remainder 0
14 / 2 = 7 - remainder 0
7 / 2 = 3 - remainder 1
3 / 2 = 1 - remainder 1
1 / 2 = 0 - remainder 1
Binary equivalent: 22710 = 111000112
Not too bad, try it on some numbers of your own and check your answers with a calculator or whatever.
Binary Arithmetic
Addition
Now before we start there are four rules to remember:
0+0=0
0+1=1
1+0=1
1 + 1 = 10
Now they all seem pretty normal, except for the last one. Well when we're adding in decimal we don't carry a
number onto the next column until the addition of the numbers in that column exceeds 9 (the highest singledigit value in decimal).
14
17+
-211 = 31
We carry the extra 1 from 11 to the next column and add it to 2 giving us 31, the same applies in binary, only
we carry the one when the addition value exceeds 1 (the highest single-digit binary number).
1
1+
-10 = 10
We carry the one over to the next column making 10. Now another rule to remember is:
1 + 1 + 1 = 11
With this in mind, let's add two binary numbers together.
3
1101
1001 +
----10110
Let's go through it from the right column to the left. 1 + 1 = 10, carry the one over to the next column which is
0 + 0 = 0, but we have the carried 1, so it's 1. We have 10. 1 + 0 = 1, no carry. We have 110. 1 + 1 = 10,
carry the one to the next column and we have 10110 2
Multiplication
Binary multiplication is very easy; we use the same rules as we do in decimal multiplication. Let's look at an
example.
101
10 x
---000
1010 +
---1010
Now to explain, firstly we multiply the binary number on the top by the first digit on the right, on the bottom.
101 x 0 = 000 (anything x 0 is 0), then we write the result.
The we multiply the top number by the second digit on the right, on the bottom.
101 x 1 = 101 (anything x 1 is itself) we then write the result under the first result, but we shift it one place to
the right and fill in the missing space with a zero.
We then add the values together and we have the result of the multiplication.
Let's do one more.
1011
1001 x
------1011
00000
000000
1011000 +
------1100011
Follow these steps:
Copy the numbers if the bottom multiplier is 1, or else just write a row of zeros.

Move the result one column to the left as you move onto a new multiplier.

Add the results together using binary addition to find the answer.

That about wraps up binary, there is also converting binary numbers to hexadecimal and octal, but we'll not
get into that until we have got used to hexadecimal and octal.
Hexadecimal
Hexadecimal is base 16, it uses 16 unique digits to represent values.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
4
Remember that if we compare this to decimal, A is 10, B is 11, C is 12, D is 13, E is 14 and F is 16. We
need to remember this when converting hexadecimal to decimal.
Hexadecimal to Decimal Conversion
We do this in the same way we convert binary to decimal, only we use hexadecimal's base number (16).
161
160
3
E
Easy enough, now we just multiply the values as we did with binary.
Hex Number: 3E
3 x 161 = 3 x 16 = 48
E is equal to 14 decimal, see above.
14 x 160 = 14 x 1 = 14
48 + 14 = 6210
Now with a larger number we do the same as in binary again.
163
162
161
160
5
E
B
7
5 x 163 = 5 x 4096 = 20480
14 x 162 = 14 x 256 = 3584
11 x 161 = 11 x 16 = 176
7 x 160 = 7 x 1
=7
add the results and we get: 24247
Decimal to Hexadecimal Conversion
This, again, is similar to decimal-to-binary converting, only, again, we use hexadecimal's base number.
Remember we repeatedly divided the decimal number by 2 (binary's base) and collected the remainders.
We do the same here, only we divide by 16 (hex's base).
Decimal number: 97
97 / 16 = 6 - remainder 1
6 / 16 = 0 - remainder 6
remainders from BOTTOM UP: 61
That's it, this is trickier to do the sums in your head, especially when dealing with larger numbers, let's do a
larger one.
Decimal Number: 934
934 / 16 = 58 - remainder 6
58 / 16 = 3 - remainder 10
3 / 16 = 0 - remainder 3
remember, 10 in decimal is A in hex
5
remainders in reverse order: 3A6
It's as simple as that.
Hexadecimal and Binary
A very simple conversion, we simply use this table:
Binary
Hex
0000
0
0001
1
0010
2
0011
3
0100
4
0101
5
0110
6
0111
7
1000
8
1001
9
1010
A
1011
B
1100
C
1101
D
1110
E
1111
F
There you go, the conversions are in front of you.
Binary number 100101102
now we split the binary number into groups of four.
1001 - 0110
Now we refer to the above table, look up each 4-bit binary value and find its corresponding hex value.
1001 = 9
0110 = 6
Hex: 96
This is the easiest conversion, all you need is that table. But what if the binary number can't be split equally
into groups of four?
Binary: 1110101101
11 - 1010 - 1101
We simply add zeros on the left until it is four digits.
0011 - 1010 - 1101
hex: 3AD
6
This works the same the other way, just split the hexadecimal value into digits and look up its corresponding
binary value.
hex: 6AF
6 - 0110
A - 1010
F - 1111
binary: 0110 1010 1111
we can lose the zero on the left: 11010101111
There we go, that wraps it up for hex.
Octal
We do octal to decimal the same way we do bin and hex to decimal.
81
4
80
6
4 x 81 = 4 x 8 = 32
6 x 80 = 6 x 1 = 6
32 + 6 = 3810
I'm sure you get the drift of that, it's the same as the other number systems to decimal conversions.
Decimal to Octal
This again is the same as decimal to bin and decimal to hex, we just repeatedly divide the decimal value by
octal's base number (8) and collect the remainders in reverse order.
Decimal: 76
76 / 8 = 9 - remainder 4
9 / 8 = 1 - remainder 1
1 / 8 = 0 - remainder 1
octal = 114
if you've played around converting the other number systems to decimal you should have the hang of this by
now.
Octal and Binary
Remember how we converted hex to binary? We split the binary number into groups of 4 and looked up their
corresponding value in the table. With octal, we split the binary values into groups of three.
Binary
Octal
000
0
001
1
010
2
011
3
100
4
7
101
5
110
6
111
7
There's the table, let's do a conversion.
Octal: 561
5 = 101
6 = 110
1 = 001
binary: 101110001
For binary to octal:
Binary = 10100111011
010 - 100 - 111 - 011
2 4 7 3
octal: 2473
A very easy conversion, just have the table handy and you'll have no problems.
Hexadecimal and Octal
When converting hexadecimal to octal and vice versa we convert the number to a binary number first, and
then convert that binary number.
Octal to Hex
Octal > binary > Hexadecimal
Octal: 433
4 = 100
3 = 011
3 = 011
binary: 100011011
0001 - 0001 - 1011
1
1
B
Hex: 11B
Hex to Octal
Hexadecimal > binary > Octal
hex: A6F
A = 1010
6 = 0110
F = 1111
101 - 001 - 101 - 111
5 1 5 7
Octal: 5157
8