The ark lands after The Flood. Noah lets all the animals out. Says, “Go and multiply.” Several months pass. Noah decides to check up on the animals. All are doing fine except a pair of snakes. “What’s the problem?” says Noah. “Cut down some trees and let us live there”, say the snakes. Noah follows their advice. Several more weeks pass. Noah checks on the snakes again. Lots of little snakes, everybody is happy. Noah asks, “Want to tell me how the trees helped?” “Certainly”, say the snakes. “We’re adders, and we need logs to multiply.”
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Theorem : All positive integers are equal.
Sufficient to show that for any two positive integers, A and B, A = B. Further, it is sufficient to show that for all N > 0, if A and B (positive integers) satisfy (MAX(A, B) = N) then A = B.
Proceed by induction.
If N = 1, then A and B, being positive integers, must both be 1. So A = B.
Assume that the theorem is true for some value k. Take A and B with MAX(A, B) = k+1. Then MAX((A-1), (B-1)) = k. And hence (A-1) = (B-1). Consequently, A = B.
Q: What’s purple and commutes?
A: An abelian grape.
Q: What’s yellow, and equivalent to the Axiom of Choice?
A: Zorn’s Lemon.
What is “pi”?
Mathematician: Pi is thenumber expressing the relationship between the circumference of a circle and its diameter.
Physicist: Pi is 3.1415927plus or minus 0.00000005
Engineer: Pi is about 3.
Theorem: All positive integers are interesting.
Proof: Assume the contrary. Then there is a lowest non-interesting positive integer. But, hey, that’s pretty interesting! A contradiction.