Astraphobia immediately identified as a phobia because children naturally go through many fears as they mature. Their fear of thunder and lightning cannot be considered a fully developed phobia unless it persists for more than six months. In this case, the child's phobia should be addressed, for it may become a serious problem in adulthood. To lessen a child's fear during thunderstorms, the child can be distracted by games and activities. A bolder approach is to treat the storm as an entertainment; a fearless adult is an excellent role model for children. The most widely used and possibly the most effective
Proof that π is irrational In 1761, Lambert proved that is irrational by first showing that this continued fraction expansion holds: Then Lambert proved that if "x" is non-zero and rational then this expression must be irrational. Since tan(/4) = 1, it follows that /4 is irrational and therefore that is irrational. A simplification of Lambert's proof is given below. This proof uses the characterization of as the smallest positive number whose half is a zero of the cosine function and it actually proves that is irrational. As in many proofs of irrationality, the argument proceeds by reductio ad absurdum. Consider the sequences ("A") and