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Perfect Numbers

Olly / / Updates

Today I read about perfect numbers and some related concepts in an online textbook called Elementary Number Theory that is available from LibreTexts. It was good to have a refresher on some of the finer points.

I will not be studying or blogging Tuesday or Wednesday because I will be busy wrapping up my house sitting and traveling home. I will return either Thursday or Friday, depending on how long it takes me to get settled again.

ETA: I have decided to return Monday, as some new commitments came up after I made this post.

The Great Wide Open

Olly / / Math Posts

My mental health was poor today, and I spent my study time watching math videos. This included one on open problems that are relatively simple to state but have proven difficult to solve. It was quite interesting, but had several errors (some of which I noticed and some of which I found pointed out in the comments). Therefore I won’t link it here.

My favorite open problem is the first one that caught my imagination: the conjecture that there are infinitely many perfect numbers. This turns out to be closely related to the conjecture that there are infinitely many Mersenne primes, a type of prime number that is one less than a power of two. So a conjecture about summing factors is related to the existence of a type of prime numbers. Cool huh?

Boom Day

Olly / / Math Posts

Happy 4th of July, readers.

Today I finished the exercises and reading from yesterday. The latter explained a way I could use parametric equations to instruct a graphing calculator to draw the inverse of Tuesday’s wild function. Desmos does not require this indirect method, however. It can graph $x=3+y^2+\tan(\frac{\pi y}{2})$ as written. Whatever the benefits for graphing, I don’t think converting the wild function into parametric equations makes it any easier to test whether it has an inverse. That may require calculus, as a reader suggested earlier in the week.

Day of the Naps

Olly / / Updates

Today was another of my periodic sleepy days, and my study time was compressed. I did not quite finish the section of exercises I worked on yesterday, but I still spent some time reading the following section, which covers parametric curves. Those were always an afterthought in my mathematical education, and I found them a bit mysterious. They are making more sense this time, I think.

Wild Functions

Olly / / Math Posts

Today went well. I’m nearly finished with the exercises for the section I’m working on.

One of the exercises I did today reminded me what a walled garden I am playing in when working from a textbook. The functions encountered there are always tame ones susceptible to the techniques being taught. Today one of the exercises gave me a glimpse of a wild function, though.

$f(x)=3+x^2+\tan(\frac{\pi x}{2})$ for $-1<x<1$

I was asked to find $f^{-1}(3)$. It’s not too difficult to find that particular value, but I could find no way to do it by the first method I tried: by deriving a formula for the inverse, $f^{-1}(x)$. I don’t know how to solve $x=3+y^2+\tan(\frac{\pi y}{2})$ for $y$. That’s what makes this a wild function. It cannot be bidden the way most of the functions used in textbook exercises can.

(I believe the function does have an inverse, though I would like to come up with a better test for that than I have.1 Right now, I have the word of the textbook and an examination of the graph, which you can see here and which does not show any two $x$ values with the same $y$ value.)

  1. To do. ↩︎

Can’t Live With ‘Em…

Olly / / Math Posts

My struggle to make progress on my review over the past couple of weeks was caused mostly by issues with my medication. Problems with my regimen as prescribed were the main thing, with a little user error thrown in, as well. I’m finally recovered from that, but I complicated today with a little more user error by forgetting to take my medication last night. As a result, I slept a good part of the day and only had time and energy for an hour of study. Yet I did manage both to work on calculus exercises and to explore my question about composition of functions.

I didn’t make any new progress in the explorations. So far, I have only the fairly low-hanging observation that, if $f(x)$ and $g(x)$ each apply a single algebraic operation to $x$ and both operations are of the same type, then $f(g(x))=g(f(x))$. The types are: exponentiation (including roots), multiplication or division, and addition or subtraction. Below are some examples, which you can easily see are interchangeable this way:

Exponentiation
$f(x) = x^2$, $g(x) = \sqrt[3]{x}=x^{\frac{1}{3}}$:
$(x^{\frac{1}{3}})^2=x^{\frac{2}{3}}=(x^2)^{\frac{1}{3}}$

Multiplication and division
$f(x) = 3x$, $g(x) = \frac{x}{5}$:
$3(\frac{x}{5})=\frac{3x}{5}=\frac{(3x)}{5}$

Addition and subtraction
$f(x) = x-5$, $g(x) = x+3$:
$(x+3)-5=x-2=(x-5)+3$

In general, this does not work with algebraic operations of different types:

Exponentiation and multiplication
$f(x) = x^2$, $g(x) = 3x$:
$(3x)^2=9x^2\neq 3(x^2)=3x^2$

I’ve also easily found examples that even simple trigonometric, logarithmic, and exponential functions are not generally interchangeable in composition.