## Real Numbers Base…Factorials! And A By-product

PROPOSITION 1:  For a real number  x  there exists a sequence $x_1, x_2, x_3,…$ of integers such that

$\hspace{4cm} x=x_1 +\frac{x_2}{2!}+\frac{x_3}{3!} + \cdots + \frac{x_n}{n!} + \cdots, \hspace{2cm} (*)$

where $x_1$ can be any integer, but for $n \geq 2$, $x_n \in \{ 0,1,…,n-1 \}.$ Furthermore, if we require that the partial sums be strictly smaller than  x, then such a representation is unique.

Remark: One cannot help recalling decimal or binary expansion of numbers. Notice that $\frac{n}{n!}=\frac{1}{(n-1)!}$ (drops back to previous digit), so the bound on $x_n$ is logical. Continue reading “Real Numbers Base…Factorials! And A By-product” »

## Daily Quizzes: the Good, the Bad, and the Ugly—Part 2

You may recall that quite some time ago, I tried to convince you that giving your students a one- or two-question quiz every single day had a myriad of good aspects. You can check out why I loved this method in Part 1. As a quick refresher, I taught Calculus I four days a week the semester that I employed this method. Now, we’re going to discuss the bad (easily fixable) and ugly (not so easily fixable) issues which I ran into that semester. To keep this post from being a total downer, we are also going to talk about a new experiment I tried the next semester that I taught.  Continue reading “Daily Quizzes: the Good, the Bad, and the Ugly—Part 2” »

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## Shedding light on AI’s black boxes

A recent special issue in Science highlights the increasingly important role that artificial intelligence (AI) plays in science and society. Providing a small but compelling sample of the types of challenges AI is equipped to tackle—from aiding chemical synthesis efforts to detecting strong gravitational lenses—the issue captures the palpable excitement about AI’s potential in a world saturated with data.

But one article in particular, “The AI detectives,” captured my attention. Rather than highlighting a specific application of AI, as the other articles do, this piece draws attention to the lack of transparency in certain machine learning algorithms, particularly neural networks. The inner workings of such algorithms remain almost entirely opaque, and they are accordingly termed “black boxes”: though they may generate accurate results, it’s still unclear how and why they make the decisions they do.

Researchers have recently turned their attention to this problem, seeking to understand the way these algorithms operate. “The AI detectives” introduces us to these researchers, and to their approaches to unlocking AI’s black boxes. Continue reading “Shedding light on AI’s black boxes” »