I keep a list of interesting mathematical questions with me at all times. On my flight Friday from Dulles to Hartsfield, I pulled this list out and considered the following question:
For the number , what is the digit in the units place?*
Just so that we’re clear on the question, in the number 12.34, the digit 2 is in the units place. Also just for clarification, is a really big number. It is approximately equal to , or 1 followed by 19900 zeroes!
For a point of comparison, the largest number that can be represented in a computer using the 64-bit floating point standard is about equal to 1 followed by a mere 308 zeroes. We cannot therefore hope to find the answer simply by handing the question to a machine. Instead, we will have to be clever.
Our trick will be to devise an easy-to-compute approximation of and use this approximation to answer the question. Normally, of course, this strategy will only give an approximation to the answer. The clever bit is that we will show that our approximation is so good that our answer is exact.
We start be rewriting :
So far this is exact. We can rewrite still further by taking advantage of some calculus. In particular, we can expand the fraction using a Taylor series:
Setting , we get that
which means that
So far, this is still exact as long as you keep all infinity terms in the sum. Instead, we will form an approximation of by keeping only a finite number of terms. Actually, we are only going to keep one term. The question, of course, is which one.
To figure out which term is the right one to keep, remember that we don’t really care about itself; we only want to know about the units digit. Notice also that our Taylor series in parentheses gets multiplied by the really big number . For a digit to end up in the units place after having been multiplied by , it has to have started out very small. In fact, it ought to be about equal to . Do we have any such term in our Taylor series?
Yes we do. The term when is equal to . When multiplied by the out front, we get just . This seems to be the most important term. Later, we’ll see that this is the only term that matters.
In order to determine the units digit of , we will compute the units digit of . This is actually a much simpler problem. Remember that is approximately equal to 1 followed by 19,900 zeros. For comparison, is about equal to 1 followed by a mere 100 zeroes or so. In fact, this is small enough that we could reasonably just hand the question to a computer now. And yet, somehow, that would seem to be such a sad thing to do. We have been clever enough to turn a really big problem into a merely-big problem so far; let us see if we can be still more so.
While is far too large a number to compute by hand, remember that we still don’t actually care about the whole number. We only want the units digit. Let’s see if we can find a pattern in the units digits of various powers of 3. Consider the following table:
Notice the pattern in the units digit. It is always the same four numbers repeated in sequence: 1, 3, 9, 7, 1, 3, 9, 7, 1 … Notice also that the unit digit is equal to 1 when is a multiple of 4. We want to know about . Since 196 is divisible by 4, we know that the units digit of is 1. Then the units digit of is 3, that of is 9, and that of is 7.
Finally, then, we have an answer! For our number , the units digit is 7.
… Or is it? Remember that our answer has come from an approximation. In fact, we made what may have been a terrible approximation: instead of using all of the infinite terms in our Taylor series for , we used just the one term when ! How could our answer possibly be correct?
To begin with, the terms when all end with long strings of zeroes. When you add them to our term, then, the units digit doesn’t change. To see that they all end in long strings of zeroes, consider the term:
Since is an integer, this means that this number ends with a string of 100 zeroes! The term ends with 200 zeroes, the term ends with 300 zeroes, et cetera, down to the term which ends with 19900 zeroes! These terms, then, don’t change the units digit of at all.
How about the terms when ? These terms turn out to all be between 0 and 1. That is to say, they all have the form 0-point-something. None of these terms change the units digit of either! To see this, consider the term:
Since both and are big numbers, it might not be clear whether this number is bigger than 1 or less than 1. If this number is less than 1, then it has a zero in the units place and it doesn’t change our answer above. If this number is greater than 1, then it might not have a zero in the units place and we will have to worry about it. To quickly check whether it is less than 1 or not, we can rewrite the exponentials in terms of the natural exponential base :
Similarly . Thus
Since , the exponent is negative and therefore this number is less than 1. The same also holds true for all the terms with . This is to say that none of the terms with affect the ones digit of either.
So there we have it: to determine the units digit of , we only need to know the units digit of the term! We can therefore be perfectly confident in our answer:
The units digit of the number equals 7.
As suggested in the title of this post, there is an interesting lesson to learn here. Sometimes, just sometimes, in very special circumstances, approximate answers are actually exact answers. For our question, it is easy to see why this worked. While the number is very, very large, we only wanted to know about a very small portion of it—just a single digit, in fact. We were therefore justified in ignoring most of the information we had about and focusing on just the one term that matters. By doing so, we turned a question about a number so big that even a computer couldn’t handle it into question that can be solved using pen-and-paper.
* This question is a very-slightly modified version of question A2 of the Forty-Seventh William Lowell Putnam Mathematical Competition given on December 6, 1986.