R is a rational number
such that 2123456789
degrees Celsius is precisely equal to R degrees Fahrenheit
(I) Determine the first two digits (reading left to right) in [R].
(II) Determine the last two digits (reading left to right) in [R].
(III) Determine the first digit (reading left to right) following the decimal point in R.
: [x] denotes the greatest integer ≤ x.
Indicates that the required [R] has 37164197 digits when written in decimal, and the first few are
but written without the decimal point. That comes to
Of course, we've neglected to add 32, but that comes way at the end, so at some point there's a lack of carry, and we can safely say the first two digits are 81, and in fact, up to the terminal 6 we are safe for however many we show (the 6 itself might really be a 5 if rounding was up to make it a 6, or 7 in the unlikely event that carries percolated up this far).
Posted by Charlie
on 2010-12-18 13:35:43