median
don steward
mathematics teaching 10 ~ 16

## Sunday, 29 March 2009

### powers of 2

powers of 2 are reasonably easy to calculate

to appreciate the function 'power of 2' it can be helpful for students to write out the powers of 2, without using a calculator
(maybe a good Friday afternoon task...)

1
2
4
8
16
32
...

this is the way that cells replicate

students may notice that the differences in powers of 2 are, again, powers of 2

if you jump two spaces the differences are always multiples of  ?

if you jump three spaces the differences are always multiples of  ?

graphing the function is a little awkward but it can be a helpful way of students appreciating the speed of growth:

students could think of the function in monetary terms: if your money doubled every year (i.e. 100% increase per annum) how long would it take for you to become a millionaire?

the numbers of digits in the powers of 2 seems to follow a pattern:
you get 4 of them with a length (3n - 2)
then 3 with a length (3n - 1) and
then 3 with a length (3n)

so to work out how many digits there are in a particular power of 2: you divide the whole number part of the power by 10, multiply this by 3 and
add 1 if the power ends in 0, 1, 2 or 3
add 2 if the power ends in 4, 5 or 6
add 3 if it ends in 7, 8 or 9

to check this, a pebibyte is 2 ^ 50 = 1125899906337504
50 divided by 10 is 5
5 x 3 is 15
this is the number of digits in the result

the lead digits seem to follow a similar 'block' pattern: 1 2 4 8 then 1 3 6 then 1 2 5