We wish to help students “parachute into” or sneak up on mathematics before any “rocket science” (i.e., focus on high school and don’t get lost in the weeds).
Think of square roots. The square root of 4 is 2, of 9 is three of sixteen is 4.
But the moment someone asks: what about the square root of 17, you will find that crystalline simplicity and obviousness are long since gone.
If you “keep your nerve” and calmly get into the “complexity jumps” you may well find it enchanting that numerical understanding has to be coaxed forth and doesn’t offer itself up readily. But why would the physical universe, if it is really mathematical in its very “fabric,” be so ready to “jump away from you” in its complexity?
Go back to the square root of 17. Suppose you disallow logarithms and log tables. Suppose you say that “approximation theory” is too approximate. There’s no HP Scientific Calculator. One insight you might need is that 17 is “not far” from 16 so that the square root must be 4 plus a little. Call this “little” x.
Then (4+x) (4+x) = 17. You solve for x with the quadratic formula you had in high school. If you don’t remember the formula or cannot derive it, you’d have to look it up which might be disallowed in this “game.”
If you stand back, “meta-intelligently” (i.e., asking, “what does this tell me?”), you wonder whether the universe and its math fabric are an endless “onion” of such layers and complexity and not “boil-down-able.”
Another such example is Grandi’s series.
Grandi’s series and its trickiness:
In 1703, the mathematician Luigi Guido Grandi studied the addition: 1 – 1 + 1 – 1 + … ( 1-1, infinitely many, always +1 and –1). You find if you group the numbers in certain valid and legitimate way, you could different results. How can that be?
In mathematics, the infinite series, 1 − 1 + 1 − 1 + ⋯, can also be written:
It is sometimes called Grandi’s series, after Italian mathematician, philosopher, and priest Luigi Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it lacks a sum in the usual sense. (On the other hand, its Cesàro sum is ½.)
One obvious method to attack the series (
One obvious method to attack the series (1 − 1 + 1 − 1 + ⋯) is to treat it like a telescoping series and perform the subtractions in place:
(1 − 1) + (1 − 1) + (1 − 1) + … = 0 + 0 + 0 + … = 0
On the other hand, a similar bracketing procedure leads to the apparently contradictory result:
1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + … = 1 + 0 + 0 + 0 + … = 1
Thus, by applying parentheses to Grandi’s series in different ways, one can obtain either 0 or 1 as a “value.” (Variations of this idea, called the Eilenberg–Mazur swindle, are sometimes used in knot theory and algebra.)