Octal And Hexadecimal

The octal numbering system is sometimes used in computing, but the hexadecimal system, or  base-16 numbering system, is more common.  It is based on 16 digits.  As there are only 10 decimal digits, 0, 1, …, 9, in a 16-digit system, additional characters must be used.  In hexadecimal numbering, these characters are uppercase letters, starting with “A” and ending with
“F”.  That is, the hexadecimal digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.  That is, hexadecimal digits: 0 – 9 denote decimal digits 0 – 9, and hexadecimal digits A – F denote decimal digits 10 – 15.  Since 16 = 2⁴, binary numbers can be easily converted to hexadecimal by grouping the binary number into groups of 4 bits, each 4-bit number representing a hexadecimal digit.  Hexadecimal numbers are often distinguished from other numbering systems by the 16 subscript, or, more commonly in computer science, by “h” or “H”.  The previous example, 1011 1100₂, is represented as BC in hexadecimal.  Since 1011₂ = 11₁₀ = B₁₆ = BH, and 1100₂ = 1210 = C₁₆ = CH.  Converting BCH to decimal is also straightforward.  BCH = (11 x 16¹) + (12 x 16⁰) = 176 + 12 = 188, as was shown in the decimal conversion for binary above.

As another example, the hexadecimal number FACE₁₆ =  (15 x 16³) + (10 x 16²) + (12 x 16¹) + (14 x 16⁰) = 64206.  Alternatively, the number can be converted easily to binary, and then to decimal, if necessary.  FACE₁₆ = 1111 1010 1100 1110₂.  Note that because 16 = 2⁴, a hexadecimal number with h digits has 4 x h bits.

Hexadecimal numbers are commonly used to represent RGB colours (colours using the “Red, Green, Blue” colour model).  This colour model will be described in detail in a subsequent section.  For now, it suffices to note that these hexadecimal number are prefaced with a “#” symbol, especially in program code and HTML, and that two hexadecimal digits represent a single 8-bit colour.  For example, the each of the three RGB components in the colour “blue” (Red = 0, Green = 0, Blue = 255 in an 8-bit representation) is denoted in binary.  For “blue”, this yields R = 0000 0000, G = 0000 0000, B = 1111 1111, and therefore the hexadecimal representation of “blue” is #0000FF.  For the colour “yellow”, R = 255, G = 255, B = 0, and therefore, in hexadecimal, yellow is represented as #FFFF00.  From tables of colours (e.g. Web Colours), it is seen that “Sienna” has RGB value R = 160, G = 82, G = 45.  To convert to hexadecimal, the R, G, and B components will first be converted to binary.

For the R component, 160 = 128 + 32 = 2⁷ + 2⁵ = 10100000₂ = 1010 0000₂ = A0H.

For the G component, 82 = 64 + 16 + 2 = 2⁶ + 2⁴ + 2¹ = 1010010₂ = 0101 0010₂ = 52H.

For the B component, 45 = 32 + 8 + 4 + 1 = 2⁵ + 2³ + 2²  + 2⁰ = 101101₂ = 0010 1101₂ = 2DH.

Therefore, the colour “sienna” is represented in hexadecimal as #A0522D.

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