## Friday, January 2, 2009

### Counting in eights

Imagine a parallel universe where everything is the same as our world except the people occupying this alternate existence have four digits on each hand and foot, and (unsurprisingly) count in groups of eights .... how does that work ?
Lets start counting 1, 2, 3, 4 (first hand), 5, 6, 7, 10 (second hand), 11, 12, 13, 14 (first foot), 15, 16, 17, 20 (every digit has now been used).
From this little example, we can see that "16" in our base 10 world is "20" in this strange place.
How would we make sense of this world ? Lets imagine we are driving down a road in this parallel universe and we see a sign saying "Maximum Speed 65 km/hr". What does this translate to in our universe ?
65 in octal (counting in eights) = (6 x 8 ) + (5 x 1 ) = 53 in base 10
Note: 655 in octal = (6 x 64) + (5 x 8) + (5 x 1) = 429 in base 10. Converting from a another base number to base ten always follows this simple procedure. Going the other way (e.g. from a base 10 number to another base) is a little trickier but basically involves dividing the base 10 number by the base of the new number and breaking down the number into multiples of the new base.
e.g. 1 53/8 = (6 x 8) + 5 remainder = 65 in octal
e.g. 2 429/8 = (53 x 8 ) + 5 remainder = (((53/8) x 8) x 8)+ 5 remainder
= ((6) 8 + 5 remainder) x 8) + 5 remainder = (6 x 64) + (5 x 8) + (5 x 1)
= 655 in octal
Easy so far (don't ask about fractions !). What about the unit "km" ? In our system this refers to a 1000 m. Lets assume that a metre in our eight toed counter universe is the same distance as our comfortable sane base 10 world. Therefore, 1000 m in octal = (1 x 512) + (0 x 64) + (0 x 8) + (0 x 1) = 512 m in base 10.
What about the unit of hours ? Lets us assume (a common expression for mathematicians !) that an hour is the same duration of time in both universe. This means that 65 km/hr is the equivalent as 53 x 0.512 km/hr = 27 km/hr in our neck of the woods. Our parrallel universe is not a fast place; I suggest not getting out of first gear and leaving the handbrake on !
Please notice that in an octal system, numbers will tend to be "larger" i.e. need more columns of numbers. This explains why computers sometimes store very large numbers using a hexadecimal system (base 16). On the other hand, the octal system needs less symbols to express numbers ("8" and "9" are redundant in our octal universe), which taken to an extreme in the binary system ("0" and "1") reduces counting to the manipulation of on/off signals. This reduction of symbols, first envisaged by Leibniz in the 17th cenury, greatly facilitates the mechanisation of counting and, indeed, binary is used in all modern computers for the manipulation and storage of numbers. The fact that Leibniz saw this advantage some 280 years before the first computer started crunching numbers in binary, tells you something about his powers of imagination and insight. Touche Mr Leibniz (who also invented calculus in his spare time).