Book Review: Beyond Infinity
- Numbers Numbers Numbers, natural, rational, irrational and real: Cheng explains that these number sets are infinite in nature but some are "more infinite" than others. For instance, the set of natural numbers are smaller than its superset of rational numbers, which in turn are smaller than the set of real numbers. So far so good? But...
- Infinity is but an abstract notion: It certainly is not a number to which the rules of arithmetics apply:
- ∞ + 1 = ∞ (addition/subtraction does not apply)
- 2 ⋅ ∞ = ∞ (neither does multiplication/division)
- 1/∞ = 0 (in the sense of limits which touches on the fundamentals of calculus)
- 1/0 ≠ ∞ (because 1/0 is undefined. Proof: If 1/0=r were a real number, then r⋅ 0=1, but this is impossible for any r, hence the contradiction)
- 1+ ∞ ≠ ∞ + 1 (even the sequence of addition matters when it comes to infinity. Cheng uses the book of raffle tickets to explain this conundrum)
- Some series grow to infinity but others don't. For instance, the harmonic series (1 + 1/2 + 1/3 + 1/4 + 1/5 + ...) is unbounded, even though its tail is very small. In contrast, the sum of squares of the harmonics (1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ...) is bounded and converges to a finite value of 1.64493.
- Infinity (Hilbert's) Hotel: This is a thought experiment that illustrates the counterintuitiveness of the concept. Suppose a hotel has an infinite number of rooms which are fully occupied by an infinite number of guests. If one more guest arrives, the hotel can still accommodate him by asking all guests to move up one room to n + 1. This frees up room number 1 for the new guest. Weird? We are just getting started. If an infinite number of guests arrive together, the hotel can still accommodate them by asking all current guests to move to room number 2n. This puts all current guests in even numbered rooms, thereby freeing all odd numbered rooms for the new arrivals. Other mind bending scenarios can yet be imagined. The point being Hilbert's Hotel, while always full, can still fit more tenants!
- Zeno's paradoxes: This is based on the idea that if we divide a distance or time interval into an infinite number of smaller parts, we will never be able to complete the division or reach the end. If the tortoise was given a head start over Achilles in a foot race, Achilles can never catch up to the tortoise. This is because he must first reach the point where the tortoise started, but by the time he reaches that point, the tortoise will have moved a bit farther ahead, and this process repeats forever. The resolution lies in the fact that in the real world, we can actually approximate these divisions with a finite number of steps. The final divisions (distances) become so small that the mere size of Achilles foot will be enough to complet such distance without spending any time.
It is pleasing to witness Cheng use simple-to-understand examples to illustrate abstract problems and thought experiments conducted by mathematicians, past and present. It is fun to see how infinity applies to both big and small notions. I especially enjoyed the part where the dimensions of the universe were touched upon and how one can escape a lower dimension through a higher dimension. For instance, when your path is blocked in a one-dimensional world (straight line), you can evade the obstacle by going into a two-dimensional world with a lateral movement before reverting onto the straight path. The same applies to three-dimensional space if you use time as the fourth dimension - the obstacle may not be there at a different time. This begs the question of how many dimensions are there in the universe? Infinite? Thought provoking indeed. In summary, this book was so enjoyable that I am already onto another book of hers, How to Bake Pi.