A Pure And An Applied Mathematician Are Asked To Calculate 2 * 2.

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A pure and an applied mathematician are asked to calculate 2 * 2.

The applied mathematician's solution: We have
2 * 2 = 2 *1/(1-1/2).

The second factor on the right hand side has a geometric series expansion
1/(1-1/2) = 1 + 1/2 +1/4 + 1/8 + ....

Cutting off the series after the second term yields the approximate solution
2 * 2 = 2 *(1 +1/2) = 3.

The pure mathematician's solution: We have
2 * 2 = (-2) *1/(1-3/2).

The second factor on the right hand side has a geometric series expansion
1/(1-3/2) = 1 + 3/2 +9/4 + 27/8 + ...,

which diverges. Hence, the solution to 2 * 2 does not exist.

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