ABSTRACT
In the column on the le , the fi rst factor (75) is continually halved (remainders are consistently ignored); in the one on the right, the second factor (57) is continually doubled. Recalling that “even numbers are evil”, we delete all lines containing an even number on the le . Then we add up all the remaining fi gures in the right column that have not been deleted. The addition 57 + 114 + 456 + 3,648 yields 4,275 as the product for the multiplication of 75 · 57, which is indeed the fi gure we were expecting to obtain. For unbelievers, we will reverse the procedure by swapping round the factors: 57 75 28 150 14 300 7 600 3 1200 1 2400 Lines two and three have to be deleted, because the even numbers 28 and 14 are “evil”, and the addition of 75 + 600 + 1,200 + 2,400 again yields the same product as above. It goes without saying that we can point out why this technique always yields the correct result, without having to accept the notion of “good” or “evil” numbers. The fact remains that generations of reckoners based their multiplications on this notion, which does have an aura of magic about it. In view of how blindly we accept the results of computer operations, we should indeed beware of labeling as backward those who successfully manipulated numbers even if they did believe that odd ones were good and even ones evil. Even enlightened thinkers, such as Leibniz, found a formula seemingly proving that odd numbers were more pleasing in the sight of God. If the reciprocal values of odd numbers-i.e., 1/1, 1/3, 1/5, 1/7, 1/9, 1/11-are alternately subtracted and added, the “infi nite sum” approaches a quarter of the value of pi = 3.14159…. It goes without saying that it is actually impossible to calculate this infi nite sum. One must make do with fi nite subtotals, and consequently all one ever gets is approximate values. With reference to the infi nite sum discussed above, these approximate values converge to pi/4 pitifully slowly, as Leibniz’ adversary Newton dismissively remarked.
