Number Bases: Octal and Hexadecimal

A few New-World tribes used base-8 numbering systems; they counted by using the eight spaces between their fingers, rather than the ten fingers themselves. The blue natives in the movie "Avatar" used octal because their hands had only four fingers.

Let's dig right in:

• Convert 357₁₀ to the corresponding base-eight number.

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• Convert 545₈ to the corresponding decimal number.

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Hexadecimal

If you work with computer programming or computer engineering (or computer graphics, about which more later), you will encounter base-sixteen, or hexadecimal, math.

As mentioned before, decimal math does not have one single solitary digit that represents the value of "ten". Instead, we use two digits, a 1 and a 0: "10". But in hexadecimal math, the columns stand for multiples of sixteen! That is, the first column stands for how many units you have, the second column stands for how many sixteens, the third column stands for how many two hundred fifty-sixes (sixteen-times-sixteens), and so forth.

In base ten, we had digits 0 through 9. In base eight, we had digits 0 through 7. In base 4, we had digits 0 through 3. In any base system, you will have digits 0 through one-less-than-your-base. This means that, in hexadecimal, we need to have "digits" 0 through 15. To do this, we would need single solitary digits that stand for the values of "ten", "eleven", "twelve", "thirteen", "fourteen", and "fifteen". But we don't. So, instead, we use letters. That is, counting in hexadecimal, the sixteen "numerals" are:

In other words, A is "ten" in "regular" numbers, B is "eleven", C is "twelve", D is "thirteen", E is "fourteen", and "F" is fifteen. It is this use of letters for digits that makes hexadecimal numbers look so odd at first. But the conversions work in the usual manner.

• Convert 357₁₀ to the corresponding hexadecimal number.

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• Convert 165₁₆ to the corresponding decimal number.

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• Convert 63933₁₀ to the corresponding hexadecimal number.

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• Convert F9BD to decimal notation.

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Computer Graphics

If you work on web pages and graphics programs, you may find it helpful to convert between the RGB values (for an image in your graphics program) and the hexadecimal values (for a matching background color on the web page).

Graphics programs deal with the RGB (red-green-blue) values for colors. Each of these components of a given color have values somewhere between 0 and 255. These values may be converted to hexadecimal values between 00 and FF. If you list the RGB components of a color as a string of three numbers, you might get, say, R:204, G:51, B:255, which translates into a light-purplish #CC33FF in HTML coding. Note that 204₁₀ = CC₁₆, 51₁₀ = 33₁₆, and 255₁₀ = FF₁₆.

On the other hand, if you have some coding for #990033, this would translate into a dark-reddish R:153, G:0, B:51 in your graphics program. That is, to convert between your graphics program and your web-page coding, deal with the hexadecimal number not as one six-digit number, but as three two-digit numbers, and convert these pairs of digits into the corresponding RGB values.

For a discussion of the history of "web safe" colors, including why they involve only the hexadecimal equivalents of 0, 51, 102, 153, 204, and 255, look here. For a demonstration of the different text and background colors in HTML, look here.