Introduction to Number Systems

Introduction

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By now, we all know how to count. But can you count in Binary, or Hexadecimal? How about add, subtract, multiply and divide in a different number system?

We take it for granted that we work with a Base 10 (decimal) number system, and until we encounter a different number system (such as binary or hexadecimal) we probably don’t even realise that anything other than base 10 even exists or is possible. The reality however is that the use of base 10 in our colloquial number systems is completely arbitrary and probably originated simply because we have 10 fingers (for counting with), and other number systems are just as valid. If we had 12 fingers then we would probably all be using a base 12 number system.

Base 10 (or Radix 10) means that there are 10 possible values for each digit:

0,1,2,3,4,5,6,7,8,9

Once we run out of possible values for a single digit, then we add another digit:

Decimal
1	0

And so on..

Again, we take all of this for granted since we have learnt it from the earliest age. Once we understand the basic mechanism of how this works however, then it becomes clear that other number systems are just as logical and valid.

Binary

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Computers, internally, work on switches (transistors), which can have only one of 2 states: on or off. This means that there are only 2 possible values for each digit in the computer-world (0, 1), and computers thus need to work with a Base 2 (binary) number system. When counting in binary, the same rule applies – once we run out of possible values for a single digit, then we add another digit:

Binary
1	0

Here's a quick run down comparison to help clarify:

Base 2 (Binary)	Base 10 (Decimal)
0	0
1	1
10	2
11	3
100	4
101	5
110	6
111	7
1000	8
1001	9
1010	10
1011	11
1100	12

Hexadecimal

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Hexadecimal uses Base 16, meaning that there are 16 possible values for each digit:

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

Here's a side by side comparison of all three to help clarify:

Base 2 (Binary)	Base 10 (Decimal)	Base 16 (Hexadecimal)
0	0	0
1	1	1
10	2	2
11	3	3
100	4	4
101	5	5
110	6	6
111	7	7
1000	8	8
1001	9	9
1010	10	A
1011	11	B
1100	12	C
1101	13	D
1110	14	E
1111	15	F
1 0000	16	10
1 0001	17	11
1 0010	18	12
1 0011	19	13
1 010	20	14

Hexadecimal is popular in the computer world because it translates conveniently with the binary which computers work with. Each hexadecimal digit translates to 4 binary digits:

Base 2 (Binary)	Base 16 (Heaxadecimal)
0	0
1	1
1111	F
1 0000	10
1 0001	11
1111 1111	FF
1 0000 0000	100
1 0001 0001	111
10 0010 0010	222

Prefixes

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Computers internally also generally work on multiples of 8-bits (8-bits = 1 byte), so each byte can be conveniently represented with 2 hexadecimal digits. This is easier for engineers to work with than long sequences of binary digits, however the computer still works on the base 2 binary number system internally. In programming languages, types of number systems are generally identified with a prefix – for example in the C programming language:

Base 2 (Binary)	Base 10 (Decimal)	Base 16 (Hexadecimal)
0b00000000	0	0x00
0b00000001	1	0x01
0b00001010	10	0x0A
0b11111111	255	0xFF

Note that the leading zeroes are not mandatory (0b00000001 is the same as just 0b1, and 0x01 is the same as just 0x1) however they are often used to ease readability in context.

In summary, different number systems are easy to understand once the number-system-principles are understood. You might argue that Decimal is better than Binary becase less digits are needed to represent any particular number, but by that reasoning Hexadecimal would be better than Decimal. The reality is that we like Decimal simply because that is what we are used to, and what everyone else is used to.

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