Dear Dirichlet, Issue 01

Chalkdust

24 March 2015

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Moonlighting agony uncle Professor Dirichlet answers your personal problems. Want the Prof’s help? Send your problems to deardirichlet@chalkdustmagazine.com.

Dear Dirichlet,

This week’s problem sheet asks me to show that $\log(x) < x-1$ for all $x > 0$. To me this seems obviously true – a quick sketch backs it up as well. I think I’m supposed to use Taylor series or something but I’m not sure where to start. Can you help?

Dirichlet says: Starting is always very difficult, but if you don’t make a move now, you may later regret it. You are operating on the impression that you ought to use Taylor series, but this is not a certainty. Perhaps enquire about other approaches and see whether the problem opens up to you.

Of the other examples you encounter, do you feel there are any that you click with? If so, why not try and introduce yourself to them? The more contact you have, the more likely it is that you connect with something that can really help.

If you don’t feel like you’re getting anywhere, perhaps think up some new and exciting things to try. Ask your friends for their recommendations! The less you focus on this one aspect of the problem, and the more you focus on just enjoying your time together, the more rewarding you will find it.

Failing all that, of course, you can always just copylast year’s solutions.

Dear Dirichlet,

From the moment I met my husband, I knew my soulmate search was over. We caught each other’s eyes at a coffee shop eighteen months ago, got married six months later, and have barely been apart since. But last month I found out that my job is posting me to a cruise liner for three months, while he is still here in Britain. Naturally, I am worried that the distance and intermittent contact will tear at the bonds between us. How do I know that our relationship effects will be continuous at sea?

Dirichlet says: Your relationship $f(x)$ will be continuous at $c$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that $|x – c| < \delta \implies |f(x) – f(c)| < \varepsilon$.

Dear Dirichlet,

I’ve been with my girlfriend for three years, and I can’t imagine being with anyone else. We lived in France for two years before moving to England, and every day is made better with her in it. I’ve made the easy decision to propose to her. I’m normally quite competent but am having difficulty choosing a ring. Any advice?

Dirichlet says: Have you considered the Gaussian integers? This ring is related to Pythagorean triples (romantic), and which numbers are the sums of squares. Furthermore, the only invertible integers are $\pm 1$ and $\pm i$, symbolising the give and take that every relationship needs.

Bonus exercise: Show that there are no elements in the Gaussian integers whose norm (defined as the square of the modulus) is of the form $4n+3$. Hence show that if $p=4n+3$ is prime in $\mathbb{Z}$, then $p$ is also a Gaussian prime.