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Elements, compounds, atoms and arithmetic, Part 2
Kamal Lodaya, The Institute of Mathematical Sciences, Chennai
In the last issue of Jantar Mantar, I talked about the resemblance between chemistry and arithmetic.
We talked about how prime numbers (2,3,5,7,11,...) are like the atoms of arithmetic, and composite
numbers (4,6,8,9,10,...) are like molecules or compounds. The number 1 is special among the
positive whole numbers, we do not call it prime and we do not call it composite.
Let us say something about the elements of chemistry. When we say there are 92 elements, we
really mean there are 92 kinds of atoms. Water (H2O) and hydrogen peroxide (H2O2) have the same
elements, but the number of atoms of the elements is different. There are molecules which have
lakhs of carbon and hydrogen atoms, they have the same elements as benzene (C6H6) has.
We also saw a simple argument that there is an endless list of numbers. You give me a number and I
will give you a larger one. So there cannot be any end to the list of numbers.
But perhaps, like chemists can make crores of compounds out of just 92 elements, all the numbers
could be made out of a finite list of prime numbers! After all, 2 and 4 and 64 and 232 and 267 are
made up by multiplying just one prime number over and over again. If all numbers were made of a
finite number of primes then arithmetic would be a kind of chemistry.
This question was answered by the mathematician Euclid, who worked at Alexandria under the
pharaoh Ptolemy in Egypt around 300 BCE. Egypt was then part of the Greek empire and Ptolemy
marched with Alexander all the way to the northwest of India (today's Pakistan). Euclid gave a
beautiful argument to convince us that the list of prime numbers is endless. Just like I did with the
numbers, he said that if you give me a prime number, I will give you a larger one. So there is no end
to the primes and their number is {infinite}.
How does Euclid do this? Suppose you give him 93, which you think is the largest prime number.
(Of course it is not, but it is easier for me to explain using a small number.)
Here is what Euclid does. He asks you, please multiply 2 x 3 x 4 x ... x 93, all the numbers from 2
to 93. This has all the primes on your list as factors (remember, you thought the largest one was 93).
This is a very large number and we don't really want to calculate it, so let us call it by the name E
(for Euclid's number).
Now Euclid says, please add one to it. So we get E+1. Now Euclid asks, does 2 divide it? Well, no,
because 2 divides E, so we will get a remainder of 1 if we divide E+1 by 2. Does 3 divide E+1? No,
because 3 divides E ... Relentlessly Euclid goes on, and we have to agree that none of 2,3, all the
way upto 93, can divide E+1.
Now Euclid turns to us triumphantly and says, either E+1 is a prime number or a composite
number. It is certainly not the special number 1, so he is right.
If E+1 is a prime number, it is certainly much much larger than 93. If it is a composite number, it
can be divided by a prime number, a {factor}, but this prime factor is not among 2,3,...,93, so there
must be a larger prime number than 93. Whichever choice you make, Euclid has convinced you that
there must be a prime number larger than the one you provided him. You can check that this
argument works with any number, not just 93, which we took as an example.
In the last issue of Jantar Mantar we argued that in one way, chemistry is harder than arithmetic
because it also has to “physically” fit into the world we live in. Now we see that in another way,
arithmetic is harder than chemistry because it has to deal with infinitely many “atoms”.
The chemistry of your birthday
It all sounds very depressing. Isn't there something simple left instead of everything being harder?
Actually we can turn things around. Suppose we are dealing with a chemical problem which does
not have to do with the “physical” properties of a compound, for example finding out what is its
formula. Then we turn it into the easier stuff, arithmetic, do some calculations and solve our
problem. Many school chemistry problems are of this kind.
Or suppose we are dealing with an arithmetical problem where the answer can only be one of
finitely many numbers. Then we can think of it as a chemistry problem and try to work it out.
Here is an example. Suppose your birthdate is 22 November 1997. You want to know which day of
the week that was.
The trick to this is to imagine that we are in a chemical world where there are seven kinds of
"elements", which go by the names Monday (1), Tuesday (2), Wednesday (3), ... and so on upto
Sunday (7). We will call these elements by the short forms M, Tu, W, Th, F, Sa, Su. We will suppose
that you were born on a Sunday and see where that leads us.
What kind of compound is January? It has four weeks and then three days more, so it is like Su4W
(=7x4+3=28+3=31 days). This is like saying water is H2O: two hydrogens and an oxygen. So are
March, May, July, August, October and December.
What kind of compound is April? Its formula is Su4Tu (=28+2=30), so also for June, September and
November.
The formula Su4 (=28) holds for February. In all these cases we have chosen to count from Sunday,
but that's only a choice; we could have calculated from any day.
Adding up all the months gives Su48W7Tu4 (=7x48 + 7x3 + 4x2), which we add up again to get
Su52M (7x52 + 1) or the usual 365 days. This is for a whole year.
(This is not really chemistry, but it is fun.)
Now we ask, what is the formula for 13 years, from 22 November 1997 to 22 November 2010?
Using 52x13=676, a straightforward multiplication by 13 gives Su676M13 (=7x676 + 13), which is
the same as Su677Sa (=7x677 + 6), since 13 days is 6 days more than a week.
Wait a minute! There is a mistake here! We forgot to account for the leap years in-between when
February has an extra day. These were 2000, 2004, and 2008. Adding the three extra days we have
Su678Tu.
So if 22 November 1997 were a Sunday, then 22 November 2010 would be a Tuesday.
But we look at any calendar for 2010 and find that this day is a Monday! We supposed your
birthday was Sunday and found that this year it would be on Tuesday, which is wrong by one day. If
we had started our counting from Saturday, everything would have come out right. So the real
birthday was a Saturday.
Can you now calculate and find out which day you were born on? Let me wish you a happy
birthday calculation!