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In the previous post we discussed the number of primes less than a given number and derived some very poor estimates for this quantity. In this post, using no extra technical machinery whatsoever, we derive a slightly better estimate.
Previously, we used Euclid’s proof of the infinitude of the primes as inspiration for a way of arriving at an upper bound for the number of primes less than a given number. Here, we will come up with a slightly cleverer, more quantitative proof of Euclid’s Theorem and, as before, our estimate will grow out of the proof of Euclid’s Theorem, the improvement in the bound being testament to the fact that the proof is more illuminating.
If we are prepared to accept that every number greater than one has a prime divisor, then Euclid’s famous proof of the infinitude of prime numbers is one sentence long: Given any prime number p, if I multiply together all the prime numbers less than or equal to p and then add one, I get a number which leaves a remainder of one when divided by any of the prime numbers less than or equal to p and whence has a prime divisor greater than p. I try to explain why every number greater than one has a prime divisor in this post.
Today I would like to think a little bit more about prime numbers. Specifically, we will spend some time thinking about the number of prime numbers less than a given a number. We will start by seeing if we can get any quantitative information out of Euclid’s proof itself, before moving on to cleverer ways of achieving this.
Affinity in Generality
Let V be a vector space. Consider the set of all affine transformations of V: An affine transformation of V is a map from V to itself which can be expressed as
for some invertible linear map (automorphism) L and some vector b in V. The set of all affine transformations of V forms a group under composition and is called the affine group of V. Note that the set of automorphisms of V is a subgroup of the affine group and also that the set of translations of V is also subgroup of the affine group. Note that every element of the affine group can be expressed as a composition of an invertible linear map followed by a translation. Note also that the only affine transformation which is both an automorphism and a translation is the identity map. It is also quite easy to see that the translation group is a normal subgroup of the affine group. However, the automorphism group of V, which, since V is a vector space is known as the general linear group of V, is not.
Is the affine group the internal direct product of the translation group and the automorphism group? No, but it comes close. In the previous post, we saw that if G is the internal direct product of H and K, then not only can every element of G can be expressed (uniquely) in the form g = hk, but also the elements of H commute with the elements of K. This is the sense in which the two groups H and K do not interact with each other inside G. In the case of the affine group, there is interaction: The translation group of a vector space does not commute with its general linear group. This breakdown is evidenced by the fact that one of the two groups fails to be normal.
Recently, I have been reading some algebra. This has been immensely enjoyable; I had forgotten how much I used to like algebra. The material here is by no means advanced, but relies on some basic definitions from group theory.
Suppose I have two groups H and K. Can we combine them together? I learnt some years ago now that there is a straightforward way of forming a ‘sum’ of these two objects: One takes the cartesian product HK, and bestows it with a group operation in the most basic and obvious way possible: (h,k)(h’,k’) = (hh’,kk’). The group that results is known as the direct product of H and K.
The same can be done with vector spaces: Given two vector spaces V and W, I can form the direct sum of them, which is thought of as the vector space consisting of ordered pairs of the form (v,w) with v in V and w in W. It is easy to guess what the rules for addition and scalar multiplication must be. However, soon after learning these two definitions, I began to realise that at least a small amount of wool was obscuring my eyes: Suppose I have a vector space X and two subspaces Y and Z such that every vector x in X can be written uniquely in the form x = y + z, for y in Y and z in Z. I would then be invited to observe how this means that X was the direct sum of Y and Z. This didn’t sit well with me. It clearly wasn’t quite the same thing: There was no cartesian product; there was no way that x = y + z was the same as x = (y,z).
I want to extend the treatment to we just went through to higher dimensions. In my view, this provides a very nice way of thinking about why the generalisation of the derivative to higher dimensions is what it is. It has certainly taken me a while to appreciate the definition of the derivative of a a function of many variables and I am attempting to share some of my thoughts.
In the last post we thought about graphs of functions and tangent-lines. Now, the graph of a function is intially something 0ne thinks of a picture of that function. In my opinion, there is a sense in which one ought to continue to do so. It’s just that now, we’ll be thinking about -valued functions f defined on an open subset U of . Recall that the graph of f is the set of points in such that f(x) = y.
What might we mean by the derivative of f ?
In these posts I will not be explaining all of the basic mathematical concepts which are required to appreciate the main discussion, so they are not aimed at the novice, as it were. I would guess maybe that a second-year university student in mathematics will be able to fully appreciate these posts.
Recently, I have been thinking about differentiation and in this post I would like to discuss some ways of approaching the concept, starting right from the basics. In this first post, we’ll discuss some approaches to the idea of differentiation. I want to settle on a certain geometric perspective and then generalise it to higher dimensions. The aim is to shed light on why the definition of the derivative of a function generalises as it does.