Food For Thought

Are you excited about the approach of the year 2000? What would you say if I told you that you missed it? You also missed the year 3000 - they both took place hundreds of years ago! Crazy, you say? Not if you're a 'toon!

Because humans have ten fingers, we have a decimal (base-10) number system. 'Toons only have eight fingers, so their number system is octal (base-8). Instead of the ten decimal digits (0-9), they have eight octal digits, 0-7. That means their numbers aren't quite the same as our numbers.

If your car was new when you bought it, the odometer read 00000.0. When you drove your car off the lot, the odometer clicked off the distance: 00000.1, 00000.2, 00000.3, and so on. What followed 00000.9? 00001.0! The tenths counter re-started at zero, and the ones counter clicked up one.

A 'toon odometer never reads 00000.8 or 00000.9 - for 'toons digits 8 and 9 don't exist! Therefore, 00001.0 follows 00000.7. If you drive a 'toon car from home to the store, you'll travel the same distance, but that distance will be represented as an octal number.

We can convert 'toon numbers to human numbers once we understand places. Each odometer number has a place value which is a power of the base. Base-10 place values look like this:

        place number      place value     place name
        ------------------------------------------------
            -1          10^-1 =     0.1   tenths
             0          10^0  =     1     ones
             1          10^1  =    10     tens
             2          10^2  =   100     hundreds
             3          10^3  =  1000     thousands
             4          10^4  = 10000     ten-thousands

For example, let's say you've had your car a couple years, and you've driven it 32,185.9 (decimal) miles. That's the same as saying

          (10000 * 3) + (1000 * 2) + (100 * 1) + (10 * 8) + (1 * 5) + (0.1 * 9)
             30000    +    2000    +    100    +    80    +    5    +     0.9

This seems like a roundabout way of stating the obvious, but that's only because we've done it so much that we no longer think about it.

Here's how octal (base-8) place values look:

        place number    place value     place name
        -------------------------------------------------------
            -1          8^-1 =    0.125  one-eighths
             0          8^0  =    1      ones
             1          8^1  =    8      eights
             2          8^2  =   64      sixty-fours
             3          8^3  =  512      five-hundred-twelves
             4          8^4  = 4096      four-thousand-ninety-sixs

This looks stupid because the place values are shown in decimal; in octal the table looks the same as the decimal table (remember that 10 (octal) equals 8 (decimal)):

        place number    place value (octal)
        -----------------------------------
            -1            10^-1 =     0.1
             0            10^0  =     1
             1            10^1  =    10
             2            10^2  =   100
             3            10^3  =  1000
             4            10^4  = 10000

Let's look at what 'toon year 2000 is in decimal. To convert octal numbers to decimal, multiply each digit by its octal place value, just as we did for the decimal number:

        octal   decimal
         2000 = 512*2 + 64*0 + 8*0 + 1*0
              = 1024  +    0 +   0 +   0
              = 1024

For 'toons, the second millennium ended 974 (decimal) years ago. Their third millennium ended 512*3 + 0 + 0 + 0 = 1536, 462 (decimal) years ago. Start planning now for 4000, only 48 decimal-years after our second millennium ends: 4000 (octal) = 2048 (decimal).