By Edward Barbeau

Blunders in mathematical reasoning can variety from outlandish errors to deep and refined oversights that avert even the main watchful eye.

This publication represents the second one choice of such blunders to be compiled through Edward Barbeau. Like Barbeau's earlier booklet, *Mathematical Fallacies, Flaws and Flimflam*, fabric is drawn from a number of assets together with the paintings of scholars, textbooks, the media, or even expert mathematicians.

The error awarded right here serve either to entertain, and to stress the necessity to topic even the obvious assertions to rigorous scrutiny, as instinct and facile reasoning can usually be deceptive. every one merchandise is thoroughly analysed and the resource of the mistake is uncovered. All scholars and academics of arithmetic, from college to college point, will locate this publication either enlightening and exciting.

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X C b2 /. 3. x C b2 =a/. 4. x C u2 =v2 /: 5. v2 x C u2 /. Here is how the method goes on an example: factor 6x 2 7x 1. Remove the 6 and multiply it into the constant term: x 2 2. x 3. x 3. 7x 18. x C 2/. x 6 4. x C 26 /. x C 13 /. 5. 2x which is the factored form of 6x 2 7x 3. 9. Continuing a sequence This factorization method for quadratics drew comments from three readers, Phil Wood, Ioana Mihaila and David Bloom. They thought that the method was very good, and Wood in particular regretted the title (“the illegal moves method”) which unnecessarily put it under a cloud.

2x 6x 2D x2 x x 3 x2 1 3 1/ D x 2 3 1Dx 2 3 2 2 1D 2 6x D x 2 2 6x x 2 D 2x 2x x2 2 x 2 3Dx 1 3D 4 D x: Contributed by Carl Libis. (g) Problem 2dk 2 . Prove that, if dk D 2k 1 for all k, then dk D 3dk 1 ✐ ✐ ✐ ✐ ✐ ✐ “master” — 2013/9/26 — 21:34 — page 24 — #40 ✐ ✐ 24 2. 2 / 3 C2 2 22 6k 4k D 1 D 3k 1k 2 4 D 2 k 1 D dk : D 1 Contributed by Carl Libis. (h) Problem n. Prove that 42n 22n is divisible by 6 for each positive integer Solution 42n 22n D 16n 4n D 8 2n 2 2n D 6 2n D 12n : Since 12 is divisible by 6, so is any power of 12, and the proof is complete.

Right on target! x C 5/: Squaring again and simplifying leads to an identity. So really, after all of this work, we have gained absolutely no information about the solution, and everything is open for checking. Why should this occur? Noting that the equation is equivalent to p p j x 1 2j C j x 1 3j D 1; we find that it is satisfied by 5 Ä x Ä 10, so that it has infinitely many solutions. The strategy of eliminating the surds by squaring leads to a polynomial equation that has at least this many roots, and so must be an identity.