Month: February 2014

A signal processing snippet.  Let $latex x(t)$ be a bandlimited signal restricted to the interval $latex [T_1, T_2]$, so that in particular

$latex x(t) = \displaystyle{\sum_{n=0}^{N-1} x(n \Delta t) \text{sinc}\left(\frac{t – n\Delta t}{\Delta t}\right)} 1_{T_1 \leq t \leq T_2}$

Here, as opposed to previous entries, we have defined $latex \text{sinc}(t) = \sin(\pi t)/(\pi t)$ (I have had a change of heart). Then the Fourier transform of $latex x(t)$ is

$latex X(f) = \displaystyle{\sum_{n=0}^{N-1}} x(n\Delta t) e^{-i 2\pi f n \Delta t} R(n \Delta t – T_2, n \Delta t – T_1, f – 1/2\Delta t, f + 1/2 \Delta t) \,\Delta t$

where we define

$latex R(t_1, t_2, f_1, f_2) = \dfrac{(\text{Ei}(i 2\pi f_2 t_2) – \text{Ei}(i 2\pi f_2 t_1)) – (\text{Ei}(i 2\pi f_1 t_2) – \text{Ei}(i 2\pi f_1 t_1))}{i 2\pi}$

and $latex \text{Ei}$ is the exponential integral, which for imaginary arguments is

$latex \text{Ei}(it) = i \dfrac{\pi}{2} -\displaystyle{\int_{t}^{\infty}} \dfrac{e^{i…

Why is this called 2=1, one might ask. Here’s “proof” for something that’s wrong. If you’re able to appreciate these things, try finding the mistake.

a  =  b

a²  =  a b

a² + a²  =  a² + a b

2 a²  =  a² + a b

2 a² – 2 a b  =  a² + a b – 2 a b

2 a² – 2 a b  =  a² – a b

2 ( a² – a b )  =  1 ( a² – a b )

2  =  1

This will be the first and only geeky post, I promise.

Symbols and conventions are both ubiquitous and necessary in our every day life.  They enable our culture, on the one hand. On the other hand, they limit our imagination and we need to challenge them sometimes. This is a conflict I find fascinating and would like to base…

I absolutely love this idea. I’ve done two other worksheets with this same idea (click here and click here to see them) and I’ve found it to be really successful. It’s super important to get kids familiar with the hundreds chart. Hopefully children will be so familiar with it that they can make their own hundreds chart on a piece of scratch paper during a test (since kids don’t get a printed hundreds chart on their standardized tests).Click here to see my third 10 more that/10 less than hundreds chart worksheet: hundreds chart- more than- less than OWL

So a friend of mine got a little riled up about this video:

You know, I write and read poetry, and there’s this thing that happens when I talk about poetry, a thing that I know also happens all the time when people who write and read math talk about math. People say, “I don’t like poetry.” Or sometimes, more charitably, “I don’t understand poetry.” Sometimes — if they like me — they think my interest in poetry is adorable. But they don’t want to talk about it with me. And meanwhile, I’m thinking, what do you mean you don’t like poetry? Poetry is a big thing! It’s like saying you don’t like music! Or food! There’s so much of it, I’m sure we could find something you would like.

Well. That’s poetry. Because you know what my reaction was to my friend’s curiosity about that video up there?…

… For most people anyway. Because I missed my video post yesterday, here’s a very short TED talk from Arthur Benjamin on why we should teach statistics before calculus.

Of course, calculus has it’s place and is extremely important, but statistics, probability, and data analysis are so much more useful in everyday life. It would be a tremendous leap forward in promoting scientific literacy in the US. After all, if more people knew what a p-value was or what a confidence interval actually is, society as a whole would be a lot better equipped to understand the numbers that are constantly thrown at us. Reporters might realize that if one study finds a statistically significant result which is unable to be replicated, most likely they made a type I error and the null was probably rejected by random chance error.

I hope that in my lifetime, all students will…

This week Year 10 (UK age 14-15) have been exploring different graph types and also transformations and graphs.

For homework I asked them to draw just a small number of graphs by hand but wanted them to check their work and explore further graphs using the Desmos graphing calculator. Early in the week I made sure they could all use Desmos including the use of tables so in an IT room they used the slideshow here and created several graphs of their own.

Once all the students were confident to use Desmos to create various lines and curves I asked them to explore a series of graphs so that this coming week we can discuss transformations and graphs. Using Desmos allowed them to explore many graphs in a short space of time and several students chose to take screenshots and make notes for themselves.

Having used sliders they were able…

http://www.math.niu.edu/%7Ebeachy/abstract_algebra/study_guide/contents.html

The Study Notes on 600 problems and solutions:

http://www.math.niu.edu/~beachy/abstract_algebra/guide/contents.html