Hexadecimal System
The hexadecimal system, or base-16, is another way to represent numbers using 16 unique symbols:
\[\mathsf{0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}\]
After the digits \(\mathsf{0}\) through \(\mathsf{9}\), the letters \(\mathsf{A}\) to \(\mathsf{F}\) represent values from \(\mathsf{10}\) to \(\mathsf{15}\). So:
- \(\mathsf{A=10}\)
- \(\mathsf{B=11}\)
- \(\mathsf{C=12}\)
- \(\mathsf{D=13}\)
- \(\mathsf{E=14}\)
- \(\mathsf{F=15}\)
Building Multi-Digit Hexadecimal Numbers
As in other number systems, we form larger numbers by combining these digits, and each position has a place value based on powers of \(\mathsf{16}\):
\[\mathsf{16^0,16^1,16^2,16^3,\dots}\]
Just like in the decimal, binary and octal systems, we combine hexadecimal digits to build larger numbers.
- After \(\mathsf{F}\), we move to the next number: \(\mathsf{10_{16}}\) (which means sixteen in decimal).
- Similarly, after \(\mathsf{FF_{16}}\) comes \(\mathsf{100_{16}}\).
In general, the hexadecimal system follows the same positional pattern—each place increases in weight by a power of \(\mathsf{16}\), allowing efficient representation of large values.
Fractional Values in Hex
To represent fractions, we use a hexadecimal point (similar to a decimal or binary point). Digits to the right of the point have negative powers of \(\mathsf{16}\):
\[\mathsf{16^{-1},16^{-2},16^{-3},\dots}\]
Why is Hexadecimal Important?
Hexadecimal is widely used in computer science and programming — especially for:
- Representing binary data more compactly (since \(\mathsf{2^4=16}\), one hex digit = four binary bits).
- Writing memory addresses, color codes, and machine instructions.
- Example: HTML color
#FF5733— each pair of hex digits represents a color channel.
- Example: HTML color