I remember in high school when the teacher didn't allow us to write '1/0=...' . I protested. Isn't the answer infinity? No, there is no answer, you cannot even put the equal sign there, because it is undefined.
OK, that's an exaggerated version, but looking back in retrospect, it is now easy to see why. Calculus provides the answer. '1/0' is undefined, but the limit of 1/x as x goes to 0 is indeed, infinity. Conversely, the limit of 1/x as x goes to infinity is zero. I felt cheated, but it is brilliant cheating nonetheless.
What is this 'infinity ' anyway? Well, the younger and less wise me thought that it was intuitive that if some number is divided endlessly, in the end it must be zero, and conversely so. That magic denominator is infinity.
As things go by, it is clearer (or less so) that zero and infinity are problems.
Why zero you ask? "I understand that infinity is pushing the limit of human mind, but why zero?" Ah, if you consider that the earliest numeral system did not include zero, isn't that proof that zero is so elusive a concept? Let's see. The Greeks even rejected the idea of nothingness and adopt 'zero' thanks to the adoption of Arabic numeral system. The Arabs got it from the Indians. So the Arabs got nothing from the Indians (If you are not laughing, then you miss the joke, sad case). Well, enough of that, 'zero' is another story. For now let's consider infinity.
Let's do a warm-up: what's infinity minus infinity?
If you answered 'zero' with a great deal of suspicion, yes, you are correct, the answer is not that straightforward. Let me try to retell Dave Hilbert's Paradox of the Grand Hotel without sounding boring:
This illustrates how normal mathematical operations don't usually work when infinity is involved. Have yourself a set of infinite integers. Take away the infinite set of odd numbers and you are left with the infinite set of even numbers. Infinity minus infinity can be infinity.
Very curious, isn't it?
OK, that's an exaggerated version, but looking back in retrospect, it is now easy to see why. Calculus provides the answer. '1/0' is undefined, but the limit of 1/x as x goes to 0 is indeed, infinity. Conversely, the limit of 1/x as x goes to infinity is zero. I felt cheated, but it is brilliant cheating nonetheless.
What is this 'infinity ' anyway? Well, the younger and less wise me thought that it was intuitive that if some number is divided endlessly, in the end it must be zero, and conversely so. That magic denominator is infinity.
As things go by, it is clearer (or less so) that zero and infinity are problems.
Why zero you ask? "I understand that infinity is pushing the limit of human mind, but why zero?" Ah, if you consider that the earliest numeral system did not include zero, isn't that proof that zero is so elusive a concept? Let's see. The Greeks even rejected the idea of nothingness and adopt 'zero' thanks to the adoption of Arabic numeral system. The Arabs got it from the Indians. So the Arabs got nothing from the Indians (If you are not laughing, then you miss the joke, sad case). Well, enough of that, 'zero' is another story. For now let's consider infinity.
Let's do a warm-up: what's infinity minus infinity?
If you answered 'zero' with a great deal of suspicion, yes, you are correct, the answer is not that straightforward. Let me try to retell Dave Hilbert's Paradox of the Grand Hotel without sounding boring:
Suppose that there is this famous Grand Hotel. Why is it famous? Because it has infinite number of rooms, aha! On weekend you want to see this hotel for yourself. You go to the receptionist to check in but unfortunately the rooms are all full. The manager came out and tell you not to worry. He told you he would move the guest in Room 1 to Room 2, the guest in Room 2 to Room 3, and so forth. You got to stay in Room 1.
Satisfied with the hotel's excellent service, the following weekend you brought an infinite number of friends to the hotel. Again, the hotel was full. But the manager was unfazed. He moved the guest in Room 1 to Room 2, the guest in Room 2 to Room 4, the guest in Room 3 to Room 6 and so forth. Since there is infinite number of even-numbered rooms, all the guests are accounted for. The manager then put you and your retinue in the infinite number of odd-numbered rooms. Everyone is happy.
Room service is still excellent though. You know, they've got infinite number of employees.
This illustrates how normal mathematical operations don't usually work when infinity is involved. Have yourself a set of infinite integers. Take away the infinite set of odd numbers and you are left with the infinite set of even numbers. Infinity minus infinity can be infinity.
Very curious, isn't it?
No comments:
Post a Comment