Not necessarily. If your favourite chosen unit was kilometre, then it's easy to make these marks distinct and far apart. But let me know when you find a kilometre of paper or blackboard for this excellent demonstration. The distance between 0.001 and 0.1 is 0.099. The distance between 0.1 and 10 is 9.9. The distance between 0.001 to 10 is 9.999. The 0.099 extra is nothing from the perspective of 10, but from the perspective of 0.1 it is big and for 0.001 it is massive. For the elephant, the ant and the rat are practically the same tiny size. For the rat, the elephant is a giant and the ant is tiny. For the ant, even the rat is a giant. So what's the solution?
But 0.1 and 0.001 aren't "practically the same" are they? The first is 100 times larger than the second. And 100 times larger than 0.1 is 10, which we all agree is large. So in a way, the "separation" between 0.1 and 0.001 is tiny, but it is also really large when seen in another perspective. So what's the way forward here for our original problem - to make a pencil mark at 0.001 and 0.1?
This is where the logarithm (log for short) comes in. You draw a line of length 4. You mark one end as (-3) and one end as 1. (-3) corresponds to 0.001 and 1 corresponds to 10. Because 10 raised to the minus third power is 0.001. And 10 raised to the first power is 10 itself. What you've done here is you have made "scale" more important than the absolute separation. This somehow recognises the fact 0.1 is equally separated from 0.001 and 10. It's like Gulliver peering close to the Lilliput world and finding that is all exactly identical to his world, but just at a lower scale.
Let me ask another question (rhetorically). How many numbers are there between 0 and 10? Our first instinct would be to say 9 numbers. But these are 9 whole numbers. If we draw a line of length 10, how many numbers can we mark with our pencil? These are called "Real numbers", and there are infinite real numbers between 0 and 10. Take any number from 0 to 9, put a decimal point after it and then cook up any sequence you want of how many ever digits - that's a real number between 0 and 10, from 0.78346873647823 to 9.2 to 3.000000000001. Best of luck marking them with a pencil though.
But what does this "infinite" mean? How many numbers lie between 0 and 100? Obviously, infinite again. But surely, this infinite is different in some way. A line of length 10 has infinite numbers. Now, a line of length 100 has 10 lines of length 10, each of which has infinite numbers. So this new infinite should be 10 times more than the old infinite.
There was something that I learnt in my first year maths course that amazes me - the number of numbers (real) between 0 and 100 is the same as the number of numbers between 0 and 10. Not 10 times greater, as one would expect. This is a mathematically exact fact. Of course, it is also a very basic set theory fact, but maybe I am still so stuck-up on this because I'm a physicist.
But let's go further. What about between 0 and 1000? The same. Between 0 and 1000000000000000000000000000? Exactly the same. The same as between 0 and 0.00000000000000000000001. It doesn't matter what (positive) number you stick there, the number of numbers you will find in between them are exactly the same.
Is this related to the log thingy we discussed earlier? I think so - if not directly, then at least as a way of visualising and understanding this. You saw how we first took the interval from 0.001 to 10, worked our log magic and made it in such a way 0.001 and 10 are equidistant from 0.1. So we made the intervals from 0.001 to 0.1 exactly equal to 0.1 to 10. For any such interval, we can take combinations of logs and additions and multiplications and to make them equal. This is an implicit acceptance of the fact that between any 2 numbers, there exist the same number of numbers, so we can always scale it in such a way that the distance between them can be a number of our choosing. We did exactly this by taking a log and making the distance between 0.001 and 0.1 the same as the distance between 0.1 and 10.
Ok, so this is for finite numbers. What about if we call in the big guy, the guy who changes the rules when he walks in? Infinity. How many numbers are there between 0 and infinity? The answer is again the same. The same as the number of numbers between 0 and 1. And 0 and 0.00001. And between 0 and 1000000.
What is infinity? Infinity is that number which trumps all other numbers. You give me any number and ask me which number is greater, I can say infinity and I would be right. In fact, it was basically invented to do this job. It exists only in our heads.
Now let us go back to the numbers between 0 and 1. ALL the numbers.Can you give me a number greater than all these numbers? Yes, of course. 1.1 is greater than all numbers that are between 0 and 1. So is 1.000001. So is 100000. and so is infinity. In this regard, for the all the numbers between 0 and 1, the number 1.1 or 2 play the same role that infinity plays for the entire number line. Pick any number and infinity is greater. Pick any number between 0 and 1 and 1.1 is greater. We can extend this to say 100000000+1 is like the infinity for the numbers between 0 and 100000000. And maybe, we can extend this is ALL numbers and infinity itself. Usually, just invoking infinity changes the rules, but here it doesn't. And with my slight mathematical training, I am confident that if I sat down and read about it for a few days I could get it, but this kind of abstraction has never been my strong suit, so here I will believe the mathematicians blindly. (nothing new for a physicist to do)
Each little section of the real line, no matter what the length, in a way, contains the whole real number line. Every chunk has this property, as does, quite obviously, the whole real line, which stretches to infinity. So the next time someone at a party asks you to draw a line all the way to infinity, just draw a straight line and label the 2 ends with your favourite real numbers and tell them it's practically the same thing. You'll surely be invited to the next one.
My grandfather has studied Vedanta and I love to pick his brain whenever I get the chance. I remember him quoting a mantra during a discussion about something -
पूर्णमदः पूर्णमिदम्
पूर्णात् पूर्णमुदच्यते |
पूर्णस्य पूर्णमादाय
पूर्णमेवावशिष्यते ||
This is the "Shanti Mantra" of the Ishavasya Upanishad. Poornam could be translated as the whole, or the complete, or everything. The gist of the translation my grandfather gave me is this (with help from the first couple of results when you google this) -
That is the whole. This is the whole. From the whole comes the whole. Taking the whole from the whole, what remains is still whole.
This is a view on the cosmos (or maybe Brahman, I'm not sure, I'm not even close to being an expert). And I found it really cool that the imagination of the cosmos here is very similar to the number line, which when you consider the infinity at the end of it, is also an abstraction, a figment of our collective imagination.
This little section of the real number line is the whole real number line. That also is the whole real number line. If you take a section out of it (which is whole), the remaining section is still the whole real number line. It's a beautiful idea, that there is the entire cosmos in every small bit of matter. It reminds me of a (probably fake) conversation where someone asked Michelangelo how he made such a beautiful sculpture of David out of a shapeless chunk of rock and Michelangelo said that the David has always been in the rock, his job was merely to scrape out the rock that was covering it. Every rock is a potential David, every stone, made of stardust, contains within it as many secrets as the entire cosmos itself. And by knowing this stone perfectly, you can understand the entire cosmos. By knowing your own self completely, you can know the higher self.
But there's something deeper in this. Every bit of the real number line can be identified with the whole real number line, but it isn't identical. They have some properties which make them identical in some scale. But each number, or each collection of numbers has its own identity and properties. It is why Kanaka Dasa could compose thousands of Krithis in dedicated specifically to Kaginele Adikeshava or Basavanna could write his vachanas addressing Kudala Sangama Deva. These are really local deities, perhaps the temples and the forms of the deities here not known to people outside their district, not far outside Karnataka for sure. Yet, these learned men spoke of these deities as the masters of the universe. Not because they are one and the same as the entire cosmos, but simple because they are a part of it. It does not diminish the deity in the temple two streets away, who is also local, but also the master of the universe.
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