* email@example.com > > Obviously, one cannot give any examples of non-constructible numbers.
Here is one. Define pi_hat as follows:
1. If, in the decimal expansion of pi, there are ten thousand zeroes in a row (i.e. if pi=3.14......00000..000...): (a) if the decimal following the ten thousandth zero is 5,6,7,8 or 9 pi_hat=3.14....0000..001, that is raise the the thousandth zero to 1 and take no more decimals (making pi_hat a terminating decimal). (b) if the decimal following the ten thousandth zero is 0,1,2,3 or 4 pi_hat=3.14....0000..00, that is make it equal to the decimal expansion of pi stopping by the ten tousandth 0 (making pi_hat a terminating decimal).
2. If, in the decimal expansion of pi, there are no ten thousand zeroes in a row, pi_hat=pi.
Pi_hat is constructible in the sense that you can calculate its decimal expansion. It is also meaningful.
However, the number rho = pi-pi_hat is not constructible. What is its first decimal? And is it greater than, equal to, or less than zero?
I find it, however, meaningful.
(Example found in the excellent book "The Mathematical Experience" by Davis and Hersh.)