Irrational exuberance Pink also used the phrase in 2009 in his book "" in the chapter discussing how extrinsic motivation can encourage short-term thinking at the cost of long-term health: "This is the nature of economic bubbles: What seems to be irrational exuberance is ultimately a bad case of extrinsically motivated myopia". Irrational exuberance "Irrational exuberance" is a phrase used by the then-Federal Reserve Board chairman, Alan Greenspan, in a speech given at the American Enterprise Institute during the dot-com bubble of the 1990s. The phrase was interpreted as a warning that the market might be overvalued. Greenspan's comment was made during
Proof that π is irrational Laczkovich to find a new and simpler proof of the fact that the tangent function has the continued fraction expansion that Lambert had discovered. Laczkovich's result can also be expressed in . In fact, Γ("k")"J"(2"x") = "x""f"("x"). So Laczkovich's result is equivalent to: If "x" ≠ 0 and if "x" is rational, then Proof that π is irrational In the 1760's, Johann Heinrich Lambert proved that the number (pi) is irrational: that is, it cannot be expressed as a fraction "a"/"b", where "a" is an integer and "b" is a non-zero integer. In the 19th century, Charles Hermite found a