Significant Maths

We’re interested in numbers. Calculating things. That sort of thing. Here’s a brief bit of something, trying to get back in the habit of writing.

Gelfand’s Question essentially asks interesting thing about the first (leading) digit of numbers of the form dⁿ for d from 2 to 9. Suppose we wanted to calculate the leading digit of 2^50000000. One way would be to, of course, calculate out the full number, all 15051500 digits of it. But that is inelegant, hardly practical, and not very interesting. Alternately, you could look at the sequence produced by looking at the first digit of powers of 2. It goes something like: 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1… There are some immediate patterns. For example, each digit is twice the previous … except when it’s not. And there seem to be groupings of three: (2,4,8), (1,2,5), (1,3,6), … except when they’re not: (1,2,4,8). Interestingly, the occurrences of groups of 3 or 4 is connected to the continued fraction representation of log(2). But … nothing immediately useful presents itself. We could fiddle with number-theoretic ideas for a while. But it is late and my eyes are tired.

More interestingly, consider the log (in base 10), of the number.

Writing that as 15051499 + 0.7832…, we can then do the following.

Now, not exact, certainly, but the calculation is good enough to give us the first few digits, specifically that the first digit is a 6. In general, the fractional part of the log is going to be a number at least 0 and strictly less than 1. Taking 10 raised to that power will always give a number at least 1 and less than 10, yielding the first digit of our desired number, rendering the thing into scientific notation.

But, notice we’ve significantly decreased the amount of work, in comparison to actually calculating out the full number. We merely have to multiply the exponent by log(2), calculated out to sufficiently many digits. And if there’s one thing that we can do efficiently, it’s calculate log(2). Mathematica gives 10000 digits in a fraction of a second, more than enough to calculate what we needed for this calculation!

I like this – it feels like we’re short circuiting the computation, jumping right past the mass of middle digits, and straight to the leading term, carrying and long multiplication be damned!

Next, we look at the last digits of some very, very large numbers. Very large. Bigger than you can imagine. And yet, mathematics lets us discuss and handle such numbers with near ease. I don’t know about you, but I find that fascinating.

Erdős Number – Of Sőrts

Finals are winding down. Graph theory was Monday, Thermal was today – the end of my woes will be PDE on Friday.

But, however badly graph theory might have gone, I did come out of it with something few others have.

Paul Erdős is a peculiar character in the world of Mathematics. Possibly one of the most noted mathematicians of modern times. He did some brilliant and incredible things, that are worthy of several posts each. But he was also one of the most notably prolific mathematicians, writing around 1475 different mathematical articles, with 511 different people. To commemorate such prodigious work, friends created the Erdős number – your distance away from Erdős in terms of publications. People who co-authored a paper -with- Erdős have an Erdős number of 1. People who co-authored papers with them have a number 2. Etc, etc. Most active mathematicians, I read somewhere, have an Erdős number of no more than 8. Anyone who hasn’t published a paper with someone with an Erdős number is then said to have an Erdős number of infinity.

And, while I have yet to publish anything at all, and thus find an Erdős number out of my reach, I do have something almost as good.

Consider.

This is Paul Erdős, in 1952, with the children (epsilons) of some colleagues. In his lap is Barbie Benzer, now Barbie Freidin. We therefore give her an Erdős-Lap number of 1.

Therefore, Oleg Pikhurko has an Erdős-Lap number of 2, as he is shown here sitting in Barbie’s lap.

Therefore, while I wish it were a more flattering picture, I have an Erdős-Lap number of 3.

It is surprisingly difficult to ask your professor if you can sit in his lap.