When I ask "How many numbers are there?", I get the usual "An infinite number!"
- If there are the same number of positive numbers and negative numbers, is the number of positive numbers (∞) - the number of negative numbers (also ∞) = 0?
- What infinite group of numbers is one member greater than the set of positive integers?
- Are there more rational numbers or irrational numbers?
- The distance between 0 and 1 is equal to 1, right? If I went halfway from 0 to 1 (i.e., I moved to the 0.5 mark) and then moved halfway from there to the end, and then I moved halfway from where I was to the end, can I ever reach the 1.0 mark?
The last question is interesting because a lot of people say you CAN reach 1 and use the example to illustrate the sum 1/2+1/4+1/8+...=1, even though the answer is clearly no, you cannot reach 1. The journey can be broken into a countable number of steps. At step 1, you are not at 1. At step 2, you are not at 1. By induction, you are never at 1.
ReplyDeleteIt's like people, in their excitement over cardinality and their understandable desire to give "real world" examples, completely forget the basic mathematical fact that the partial sums of the infinite sum in question are never 1, and this has no bearing on the fact that the infinite sum IS 1. The very definition of the sum of an infinite series in terms of limits was designed to avoid such sloppy reasoning as "reaching 1 on the infinitieth step" (or something), and I'm not convinced that such "real world" examples don't do more harm than good.