Hexadecimal (Base₁₆ )

Why learn the Hexadecimal (Base₁₆ ) number system?

Binary numbers can be cumbersome to work with as they get large. A shorthand notation is useful when working with a large string of 0's and 1's as seen in binary. Although people are most familiar with decimal because it's used on a daily basis, it does not map well to binary. Hexadecimal is extremely useful as a shorthand for binary because one single hexadecimal digit can represent four binary digits or bits. Four binary digits or bits equal a half byte or nibble. A byte can be represented with two hexadecimal digits instead of eight binary digits. For this reason, hexadecimal digits are often used to represent byte values in computer systems and networks.

What is the Hexadecimal (Base₁₆ ) number system?

The hexadecimal base₁₆ number system, often called "hex", consists of two elements:

1. Sixteen digits are used in the number system (i.e., 0 through 9 and A through F).
2. Each column represents a value based on a factor of 16.

How does the Hexadecimal (Base₁₆ ) number system work?

Counting in Hexadecimal (Base₁₆)

The standard unit of storage in computing is a byte. A byte consists of 8 bits or 2 nibbles (4 bits each). A byte can contain the decimal values 0-255 for a total of 256 different numbers or bit combinations. Below is a table that counts through all the possible bit combinations in a byte (decimal values 0-255, hexadecimal values 00-FF).

Column Values in Hexadecimal (Base₁₆)
For the base₁₆ hexadecimal number system, the 1st four columns look like the following (note that these four columns also represent sixteen bits or two bytes):

For the value of the 1st column, the base is raised to the power of zero (i.e., 16⁰). To get the next column value, one is added to the power of the current column (i.e., 16¹ for the 2nd column, 16² for the 3rd column, 16³ for the 4th column, etc.). Powers or exponents simply determine how many times the number should be multiplied by itself (note that the powers of zero and one are an exception).

16⁰ = 1, any number raised to the power of zero is 1
16¹ = 16, any number raised to the power of one is the number itself
16² = 16 x 16 = 256
16³ = 16 x 16 x 16 = 4096

The decimal equivalent of the base₁₆ hexadecimal number 13AC₁₆ is:
(1x4096)+(3x256)+(10x16)+(12x1) = 5036

In any base system, the next column value to the left can be found by multiplying the current column value by the base number. This works because one is being added to the exponent for the next column. Adding one to the exponent means that the base should be multiplied by itself one more time. In the base₁₆ hexadecimal number system, to get the column value of the next column, just multiply the current column value by 16.

It's easy to extend the column values of a hexadecimal number by either adding one to the power of the current column or multiplying the current column by sixteen. Here's what the column values are for a 32-bit or 4-byte hexadecimal number.

Because the column values begin to get quite large, it makes sense to begin using a shorthand to indicate these larger values. The number 1024 can be represented by Kilo (K). The number 1024K can be represented by Meg (M). Using this shorthand makes the multiples of 1024 much easier to remember. Below is the same table using K to represent 1024 and M to represent 1024K.

Here's what 4K and above mean:
4K = 4 x 1024 = 4,096
64K = 64 x 1024 = 65,536
1M = 1 x 1024K = 1,048,576
16M = 16 x 1024K = 16,777,216
256M = 256 x 1024K = 268,435,456

Often in computing, K is rounded to 1,000. Although this isn't quite accurate, it does make it easier to compare numbers when rough values are all that's needed. Note that many numeric shorthand letters in computing are based on the same concept as K (i.e., 1024K = 1Meg or 1M or roughly 1 million, 1024M = 1Gig or 1G or roughly 1 billion).

The decimal equivalent of the 32-bit or 4-byte base₁₆ hexadecimal number 1A5E7C2D₁₆ is:
(1x256M)+(10x16M)+(5x1M)+(14x64K)+(7x4K)+(12x256)+(2x16)+(13x1) = 442,399,789

Fortunately, hexadecimal is most often used as a shorthand for binary or for addressing. In both cases, it is usually not necessary to convert from hexadecimal-to-decimal or vice versa. However, it is often necessary to convert between hexadecimal and binary.

Converting from Hexadecimal (Base₁₆) to Binary (Base₂)

Determining the binary value of a hexadecimal number simply requires converting each hexadecimal digit to four binary digits or bits. For example, the binary equivalent of the base₁₆ hexadecimal number 1AB5₁₆ is 0001101010110101₂.

Method: Determine the decimal equivalent of each hexadecimal digit, then convert each decimal equivalent to a 4-bit binary number and combine all the binary numbers.

For example, determining the binary equivalent of the hexadecimal number 2FA0₁₆ requires that the four decimal numbers 2, 15, 10 and 0 be converted to four 4-bit binary numbers.

The leftmost digit of the hexadecimal number 2FA0₁₆ is 2 in hex and also 2 in decimal. Converting 2 in decimal to a 4-bit binary number produces 0010₂.

The next digit of the hexadecimal number 2FA0₁₆ is F in hex which is 15 in decimal. Converting 15 in decimal to a 4-bit binary number produces 1111₂.

The next digit of the hexadecimal number 2FA0₁₆ is A in hex which is 10 in decimal. Converting 10 in decimal to a 4-bit binary number produces 1010₂.

The last digit of the hexadecimal number 2FA0₁₆ is 0 in hex and also 0 in decimal. Converting 0 in decimal to a 4-bit binary number produces 0000₂.

The four 4-bit binary numbers together are 0010 1111 1010 0000.
Hexadecimal number 2FA0₁₆ = 0010111110100000₂ in binary.

Converting from Binary (Base₂) to Hexadecimal (Base₁₆)

Determining the hexadecimal value of a binary number simply requires converting every four binary digits or bits to a single hexadecimal digit. For example, the hexadecimal equivalent of the base₂ binary number 1001000100001110₂ is 910E₁₆.

Method: Determine the decimal equivalent of every four binary digits or bits, then convert each decimal equivalent to a hexadecimal digit and combine all the hex digits.

For example, determining the hexadecimal equivalent of the binary number 100100010110110₂ requires that every four bits 0100, 1000, 1011 and 0110 be converted to a hexadecimal digit. Notice that the bits must be counted from the right, then if needed, the leftmost set of bits are padded with leading zeros to create four bits for that set.

The leftmost four bits of the binary number 100 1000 1011 0110₂ are 0100 (note that the leading zero was added to 100 to pad out to four bits). Binary 0100₂ converts to decimal 4. Decimal 4 converts to hexadecimal 4₁₆.

The next four bits of the binary number 100 1000 1011 0110₂ are 1000. Binary 1000₂ converts to decimal 8. Decimal 8 converts to hexadecimal 8₁₆.

The next four bits of the binary number 100 1000 1011 0110₂ are 1011. Binary 1011₂ converts to decimal 11. Decimal 11 converts to hexadecimal B₁₆.

The last four bits of the binary number 100 1000 1011 0110₂ are 0110. Binary 0110₂ converts to decimal 6. Decimal 6 converts to hexadecimal 6₁₆.

Combining the four hex digits gives the result 48B6.
Binary 100100010110110₂ = Hexadecimal 48B6₁₆

Back to Understanding and Converting Base Number Systems