Base Number Systems

Why learn about base number systems?

The number system used by people everyday is the base₁₀ decimal number system. However, because of how they work, computers actually use the base₂ binary number system to represent data internally. Binary numbers can be quite large and difficult to read, often the base₁₆ hexadecimal number system is used as a shorthand for binary numbers. The base₈ octal number system is sometimes used as a shorthand for binary numbers as well. In order to work with computers, it's important to understand the binary (base₂), hexadecimal (base₁₆), and to a lesser extent, octal (base₈) number systems and how to convert between them and decimal (base₁₀).

What is a base number system?

The base number determines the number that the system is based upon (e.g., the base₁₀ decimal number system is based upon the number 10). The base number determines two things:

1. How many digits are used in the number system (e.g., base₁₀ uses 10 digits, 0-9)
2. What the value of each column represents (e.g., base₁₀ uses factors of 10 for each column)

Note that the base of a number is sometimes indicated by a subscripted number following the actual number:
Decimal 10 = 10₁₀ Binary 10 = 10₂ Hexadecimal 10 = 10₁₆ Octal 10 = 10₈

Generally, decimal numbers have no indication of base because that is what people are most familiar with. Computer and network systems use a variety of additional indicators to represent the base of a number (e.g., hexadecimal numbers are often preceded by "0x"). Sometimes the base of the number is not indicated, but assumed as standard (e.g., a MAC address is represented in hexadecimal).

How does a base number system work?

Digits available

The base of a number system defines how many digits can be used. The first digit used in a number system is always zero. When the highest digit is reached in a column, one is added to the next column and the columns to the right start at zero again.

Counting in Decimal (Base₁₀)
Counting in base₁₀ (10 digits available, 0-9) would be as follows:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, when 9 is reached in the 1st column, 1 is added to the next column and the right column starts at zero again
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
when 9 is reached in the 1st column, 1 is added to the next column and the right column starts at zero again
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, etc.
This continues:
90, 91, 92, 93, 94, 95, 96, 97, 98, 99,
when 99 is reached in the 1st two columns, 1 is added to the third column and the right columns start at zero again
100, 101, 102, 103, 104, 105, 106, 107, 108, 109,
when 9 is reached in the 1st column, 1 is added to the 2nd column and the right column starts at zero again
110, 111, 112, 113, 114, 115, 116, 117, 118, 119, etc.

We are familiar with how this works in the base₁₀ decimal number system because we use it all the time. This is also how it works in other base number systems, but the number of digits available is different along with the decimal value of the numbers.

Counting in Binary (Base₂)
Counting in base₂ (2 digits available, 0-1) would be as follows:
0, 1, when 1 is reached in the 1st column, 1 is added to the next column and the right column starts at zero again
10, 11, when 11 is reached in the 1st two columns, 1 is added to the next column and the right columns start at zero again
100, 101, 110, 111, when 111 is reached in the 1st three columns, 1 is added to the next column and the right columns start at zero again
1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, etc.
Note that only the two digits 0 and 1 are used in the base₂ binary number system.

Counting in Hexadecimal (Base₁₆)
Counting in base₁₆ (16 digits available, 0-9 and the letters A-F) would be as follows:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F,
when F is reached in the 1st column, 1 is added to the next column and the right column starts at zero again
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F,
when F is reached in the 1st column, 1 is added to the next column and the right column starts at zero again
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B, 2C, 2D, 2E, 2F, etc.
This continues:
F0, F1, F2, F3, F4, F5, F6, F7, F8, F9, FA, FB, FC, FD, FE, FF,
when FF is reached in the 1st two columns, 1 is added to the third column and the right columns start at zero again
100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 10A, 10B, 10C, 10D, 10E, 10F,
when F is reached in the 1st column, 1 is added to the 2nd column and the right column starts at zero again
110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 11A, 11B, 11C, 11D, 11E, 11F, etc.
Note that the letters A-F are used to represent the decimal values 10-15 because the base₁₆ number system requires six additional digits beyond the ten digits 0-9.

Counting in Octal (Base₈)
Counting in base₈ (8 digits available, 0-7) would be as follows:
0, 1, 2, 3, 4, 5, 6, 7,
when 7 is reached in the 1st column, 1 is added to the next column and the right column starts at zero again
10, 11, 12, 13, 14, 15, 16, 17,
when 7 is reached in the 1st column, 1 is added to the next column and the right column starts at zero again
20, 21, 22, 23, 24, 25, 26, 27, etc.
This continues:
70, 71, 72, 73, 74, 75, 76, 77,
when 77 is reached in the 1st two columns, 1 is added to the third column and the right columns start at zero again
100, 101, 102, 103, 104, 105, 106, 107,
when 7 is reached in the 1st column, 1 is added to the 2nd column and the right column starts at zero again
110, 111, 112, 113, 114, 115, 116, 117, etc.
Note that the digits 8 and 9 cannot be used in a base₈ number system because the base number determines how many digits the system can use. In the case of a base₈ number system, the eight digits used are 0-7 and NOT 0-9 as in the base₁₀ decimal number system we're used to.

Values represented by column

To determine the value of each column in a base number system, raise the base to the 0 power in the rightmost column and add 1 to the power for each column as you move to the left.

Column Values in Decimal (Base₁₀) For the base₁₀ decimal number system, the 1st eight columns would look like the following:

10⁰, 10¹, 10², 10³, 10⁴, 10⁵, 10⁶, and 10⁷ are powers or exponents of ten. In general:
Any number raised to the power of 0 is 1 (e.g., 10⁰ = 1)
Any number raised to the power of 1 is the number itself (e.g., 10¹ = 10)
Any number raised to the power beyond 1 is the number multiplied by itself  that many times
(e.g., 10² = 10 x 10 = 100, 10³ = 10 x 10 x 10 = 1000, 10⁴ = 10 x 10 x 10 x 10 = 10000, etc.)
The values for the 1st eight columns in the base₁₀ decimal number system would be:

The base₁₀ decimal number 12,345,678 would be equivalent to:
(1x10,000,000)+(2x1,000,000)+(3x100,000)+(4x10,000)+(5x1,000)+(6x100)+(7x10)+(8x1)

Column Values in Binary (Base₂) For the base₂ binary number system, the 1st eight columns would look like the following:

The decimal equivalent of the base₂ binary number 10101010₂ would be:
(1x128)+(0x64)+(1x32)+(0x16)+(1x8)+(0x4)+(1x2)+(0x1) = 170

Column Values in Hexadecimal (Base₁₆) For the base₁₆ hexadecimal number system, the 1st four columns would look like the following:

The decimal equivalent of the base₁₆ hexadecimal number 1A2F₁₆ would be:
(1x4096)+(10x256)+(2x16)+(15x1) = 6703

Column Values in Octal (Base₈) For the base₈ octal number system, the 1st four columns would look like the following:

The decimal equivalent of the base₈ octal number 1234₈ would be:
(1x512)+(2x64)+(3x8)+(4x1) = 668

Differences in column values between number systems
To give you an idea of the differences between  base₁₀ decimal and other base systems, below are listed the decimal, binary, hexadecimal, and octal representations of the decimal value 172.

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