Radiometric dating equation example
Therefore, in general, the number of elements in a subgroup divides the number of elements in a whole group.
To sum up: In order to calculate a decimal expansion (in base 10) you need to raise 10 to higher and higher powers and divide by the denominator, M.
The individual terms may get bigger and smaller, but the total stays the same.
For example, the equation used to describe the energy of a swinging the pendulum is where the variables are mass, velocity, gravitational acceleration, and height of the pendulum.
A group is a set of things and an operation, with four properties: closure, inverses, identity, and associativity. Yet again: no biggy, that’s just how multiplication works.
In this case the set of numbers we’re looking at are the numbers coprime to M, mod M. This is a consequence of Bézout’s lemma (proof in the link), which says that if a and M are coprime, then there are integers x and y such that xa y M=1, with x coprime to M and y coprime to a. It turns out that the number of elements in a subgroup always divides the number of elements in the group as a whole. It comes about because cosets (which you get by multiplying every element in a subgroup by the same number) are always the same size and are always distinct.
There are more terms that can be included, like the heat of the pendulum or its chemical potential, but since those don’t much and the whole point of energy is to be constant, those other terms can be ignored (as far as the swinging motion is concerned).
The useful thing about energy, and the only reason anyone ever even bothered to name it, is that energy is conserved.
If you sum up all of the various kinds of energy one moment, then if you check back sometime later you’ll find that you’ll get the same sum.
What’s important about these numbers is that they each determine the next number in the decimal expansion, and they repeat every 6. If you want to change the numerator to, say, 4, then very little changes: So the important bit to look at is the remainder after each step.
More generally, the question of why a decimal expansion repeats can now be seen as the question of why repeats every P-1, when P is prime.That is to say: just do long division and see what happens. : If you’re not familiar with modular arithmetic, there’s an old post here that has lots of examples (and a shallower learning curve).