Posts from the ‘Math’ Category

Book Review: “Proofiness” by Charles Seife

I used to devour books when I was younger. I literally read all the time–in the car, under the covers, in restaurants (to my parents’ chagrin). However, as I’ve gotten older and found other pursuits (or they have found me), I find it increasingly difficult to stop and physically sit down and hold a book. As the consummate multitasker, having both my body and mind engaged with just one activity seems so inefficient! However, I’m slowly starting to re-appreciate taking the time to be still and engage my brain, especially now that I’m on the other side of my thesis. I’ve been reading a bit more lately and thought it might be fun to share some reviews of books I find particularly interesting.

The first book I wanted to share is called Proofiness: The Dark Arts of Mathematical Deception by Charles Seife. You might also find the same text under the alternative, slightly less maniacal paperback subtitle Proofiness: How you’re being fooled by the numbers. This was recommended to me by a friend and fellow fan of books about math, so I had DH check it out for me from the library.

The book is about how we generally trust numbers as being fundamentally pure and truthful; however, malevolent people with agendas can purposefully twist–or even just flat make up–numbers in order to manipulate our beliefs to their own ends. This is “proofiness.” This might be done in order to promote a social or political agenda, to scare people, or to make oneself rich. Only if we educate ourselves on how numbers might be misused can we learn to be smart and cautious before swallowing a phony statistic or number given to us.

There are a few basic forms of proofiness. One basic technique is using Potemkin numbers. Grigory Potemkin, a Russian politician in the late 1700s, allegedly built fake towns on the banks of the Dnieper River in Crimea to fool Empress Catharine II into thinking the area was thriving as she toured the area by boat when in fact it was desolate and depressed. Whether this story is true or not, a Potemkin now stands to represent some complete falsehood that is constructed to fool the observer; therefore, a Potemkin number is one that it totally made up to mislead someone. Another approach is presenting numbers in that of presenting not-so appealing numbers in such a way that they become much more palatable. This is called fruit polishing. A couple of examples of this are cherry-picking, where you only pick the one or two numbers that suit you and ignore the more inconvenient and less appealing results, or comparing apples to oranges, where two things not at all alike are compared.

Of the numerous situations in which various forms of proofiness have significant implications, Seife devotes quite a bit of time to the consequences of proofines in polling, elections, and vote counting. He details the fiascos in 2000 presidential election in Florida and the 2008 senate race in Minnesota and explains how proofiness was used to make these much more difficult situations than really needed. While his assessment of “one person, one vote” is somewhat depressing, I still don’t feel dissuaded from my civic duty. 🙂

Also, as a scientist I can definitely relate to the presentation of numbers. My job is to carefully measure something and present it to be as close to the actual truth as can be measured. Seife talks in detail about the numbers that we can trust, actual measurements that are as close true as uncertainty in measurement allows. As a metrologist whose job is to measure things to seventeen digits, I definitely appreciated this section of the book!  However, every scientist understands that when presenting data, whether or not you mean to, it’s almost impossible to present it in the most unbiased, open-handed way possible. Even your best intentions for how to plot something, how to scale axes, what to include and what not to include injects some inherent bias. While my measurements don’t always come out the way I want, I ultimately feel obligated to present the reality of my results. However, I do admit that “fruit polishing” does happen, whether intended or not. Some scientists or researchers may do this with full knowledge and purposefully, but I have to say that at least my personal experience says it is largely not so malevolent as made out in the worst cases in the book. Plus, having faced these issues myself, I am able to spot and identify these forms of proofiness a little more readily.

Both sides of the aisle are guilty of using proofiness for their own devices, and while he chastises both parties, I feel that I can kind of tell which side he feels is the most egregious culprit of the deed. Nevertheless, Seif presents this unappreciated topic in an interesting manner. Much of what he explains is stuff you probably have sensed from time to time but haven’t ever quite articulated before. Once you are informed and paying attention, you’ll be able to spot proofiness everywhere!

My rating: 3.5/5 stars. While it is technically about math, I can recommend this book as an interesting read for a general audience, particularly those who keep up with and are interested in current affairs. 🙂

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Christmas-itis

It’s Wednesday afternoon, and I can already feel Christmas-itis kicking in. That’s a strain of mental illness genetically related to senior-itis, when the end is nearing and motivation fades. As I mentioned last Friday, the end of this week is filled with special events at work; while none of them necessarily take up the whole day, having to stop in the middle of the day to attend them will wreck any momentum I gain during the morning. Plus, knowing that break is mere hours away now, I’ll be sorely tempted to just hang out at my desk and slide into vacation rather than exerting any effort in the lab. I’m going to try really hard to not do that, but it’s going to take supernatural effort to maintain my momentum. Even now late this afternoon I’m feeling a little tired and droopy, not at all wanting to go back to the lab to finish something up for the day. Arg.

ETA: I just found out there is yet another meeting scheduled for tomorrow morning. That really throws yet another kink in my already difficult-to-stay-motivated day!! Oh well, I’ll figure it out. 🙂

After a frustrating Monday afternoon and Tuesday morning in the lab, I finally made the modification to my optical setup. I even did it all by myself, so I am quite proud! However, as quickly as things fell into place after hours of spinning my wheels, I am firmly convinced that I had divine assistance making it all work. I was certainly thanking Him for it as I finished up yesterday. Today, a check of the measurement now shows all numbers corresponding to previous levels, so it all seems legitimate. However, I am not reproducing all the ups and downs in the data that I saw before. Since those are very important results, I still don’t feel like I’m ready to forge ahead without resolving that issue first. But at least the things I worried most about have been accounted for.

ETA: Ok, so if one decides to use the right scaling for the x-axis to correspond to the previously-used scaling to which one is trying to compare, then one might actually find that the ups and downs in the data were really just dangling off the right side of the plot. This means my new measurement system is 100% verified! I’m glad I decided to double-check my Excel spreadsheet before finishing this post and heading back to the lab!

It’s been a little distracting in the lab, too. Yesterday, my colleague asked if I would help him set up the (very involved) measurement system with the photodiodes we got from our guest researcher a few months ago. I know I said that I hate my current measurement more than anything else I’ve done, but I decided that using these diodes is up there as well, due to the complicated nature of connecting to them. But since he helps me with a lot of things, I certainly owed him some assistance in return. It hasn’t been onerous, just slightly distracting. And my office mate/lab mate is also back today, and we are working literally back-to-back on our respective systems…also a bit distracting.

I’m due for an update today, but this blog post is definitely enabling my Christmas-itis. Before I get too comfortable in my office chair, I should drag myself back to the lab to do something before I leave, especially now that I’m totally confident about my measurement!

I Excel at Spreadsheets

There are many fancy software packages to perform even the most grueling computational tasks out there. Need to do an integral? Mathematica is the way to go! Need to write a little algorithm to solve that complex formula for various data sets? Whip one up in Matlab! I have used these and some others during my time as a scientist, however, my favorite tool of them all is the somewhat basic, subtly powerful, and incredibly versatile Excel.

My desire to choose Excel over another data processing platform most likely stems from the proficiency I have gained after hours upon hours of using it, as I always have access to it yet cannot afford my own copies of other, more “professional” software. Most programs do have a significant learning curve to operate efficiently; some of them can be worth the extra time to learn, giving a lifetime of benefit. However, for many of my purposes of analyzing and plotting huge (1,000+ point) data sets, I find Excel to be simple yet flexible enough to produce the desired results without having to relearn the same skills on a different program.

Despite what you might say about Microsoft, Excel does allow for making lovely plots. I’m quite obsessed with making the perfect graphics in any papers or presentations that I need to draw up, and I find Excel to be sufficiently suitable for such. I’m sure there are more graphical programs out there, and many a time I’ve yearned for Photoshop, but with minimum effort I can produce plots that satisfy my desires. I get immense satisfaction from color and texture, and I incorporate this pleasure in my graphics by making plots the colors of the rainbow, in descending ROYGBIV order, of course, and varying the line style in solids, dashes, and dots. I’ve learned to make my fonts big so they are readable in a single-column paper when the rest of the graphic is shrunken. And I can also add arrows, wonderful arrows for further clarity.

I have spent the entire day plotting up the last week’s results in Excel. As I created the spreadsheets, set up the formulas, and plotted the graphs, I noted with interest that my fingers have learned to operate keyboard shortcuts and quick mouse strokes like playing an instrument. I’ve developed such muscle memory for cutting, pasting, and selecting that I feel like I’m playing the piano at my computer keyboard. It’s also quite amusing that when I stop to think about a particular keyboard shortcut that I’m using repeatedly, I tend to immediately forget what I was doing. It’s kind of like singing a song; you sing along with all the words until you have to stop and write down the words…then you cannot recall them without singing the whole song through!

Excel has evidently become my scientific instrument of creation. Instead of a beautiful strain of music, I get lousy, boring plots about photodiodes, but by golly they are going to be beautiful!

 

Packing my thesis-writing toolbox

I hope all my American readers had a nice Labor Day yesterday. I enjoyed mine, although, as is always the case it seems with long weekends, I didn’t quite get as much done as I planned to do! DH and I still had a nice time hanging out.

At work this week, my goal is hammer out some technical items related to thesis writing. If you are not familiar with LaTex (pronounced “lah-tek”), it is basically a computer language that generates exceptionally formatted text documents from basic commands. Normally, if I want to write something on a computer, all I need is a basic text editor that can take keyboard input and make words show up on the screen. If I wanted to write something that has some formatting to it, like indents and margins and titles and headings and such, then I would use a program like Microsoft Word to generate such a document. Word has built into it the functionality to address these formatting things along with making your words show up.

LaTex goes a step further than that. It has predefined commands for all types of document formatting. You type in formatting commands and insert your text in the appropriate places, and when you “run” it, it sees your formatting commands and words and builds a document that looks exactly like you want it. In a way, it’s like a printing press for a computer.

One of the biggest benefit of using LaTex is its handling of mathematical symbols, which is lacking in other basic text editors and document formatting software. Each symbol or special character, like all Greek letters used in physics, are pre-programmed. All you have to do is build an expression or equation symbol by symbol, and the program converts them into the actual visual shape at the final output. This makes it a very powerful tool in the scientific community, and that’s why I intend to use it for writing my thesis.

The Graduate School has guidelines for how to set up LaTex to produce a thesis in accordance with its formatting requirements. My goal for this week is to get all of this set up and working so that I am ready to write my thesis. While I’m quite proficient in typing up a document and equations in LaTex format, I’m not a computer programmer and I’m a little bit hazy on the gritty internal workings of the compiler and stuff like that. So it very well might take all week to get it going.

I will probably start off my writing in Word, actually, focusing on separate sections at a time. When I’ve done a few iterations of words and I feel like it’s getting close to the final format, I intend to transfer my words and any equations into LaTex and, piece by piece, build my thesis from these smaller blocks. This won’t be as overwhelming as sitting down on page one and trying to write the whole thing through. That’s a perfect way to end up with writer’s block! Small sections will allow me to focus on one idea at a time and also spread out my writing over a number of months. In principle, this should be less stressful. 😉

So that’s my goal at work this week. At home, my current goal is actually one for the entire month of September. I’m doing a clean-out-my-pantry-stash challenge where I throw away all the old stuff lurking in my pantry and use up everything else. This challenge deserves some dedicated blog space of its own, so I will describe it more in detail for you tomorrow.

So that’s my outlook for this short week at work. After writing a thesis outline and preparing my LaTex software for writing, I might actually feel like I’m seriously on the home stretch!

Frequency

It’s time for another quick lesson in the basics. Today I’d like to talk about frequency and how it is related to time. First of all, frequency is a rate; that is, how many “somethings” happen in a given period of time (units of  “per (unit of time)”).

Take a watch with a second hand. The second hand “ticks” sixty times during every minute. Its rate of ticking is sixty ticks per minute. Since there are sixty seconds in one minute, that also means that the second hand ticks at a rate of one tick per second. The standard SI measurement of frequency is “per second,” and the official name for this unit is the Hertz [Hz]. The namesake, Heinrich Hertz, was a 19th century German physicist who made significant contributions to the field of electricity and magnetism (no pun intended ;)), and we honor him with this unit.

Since a rate describes something happening repeatedly, it can often be rightly assumed that the action in question is identical every time. When the identical action happens repeatedly, we call that a periodic action. Period is the inverse of frequency; that is, how many seconds happen between actions. For example, if you get paid two times per month, then the period of your paychecks is roughly two weeks. The whole course of action from when something occurs to when it occurs again is often called one cycle (by the way, ladies, these words aren’t coincidental!).

Consider a kid on a swing at the playground. Actually, swinging is one of my favorite things to do, so let’s consider me on a swingset! Once I seat myself and get going, I can keep myself swinging at a pretty steady rate. Since the seat is confined on a chain, I can define a certain path that I travel and cannot deviate from. I can also define a certain point on this path as a starting point; let’s call it the very bottom of my motion. The time it takes me to swing from the bottom, up to the front, back through the bottom, up to the back, and return to the bottom is one cycle of my motion. If it takes me two seconds to complete one swing cycle, my frequency is one swing per two seconds, or one-half cycle per second, or 0.5 Hz.

This is a very basic example of what we call an oscillator. An oscillator is anything that, well, oscillates! But physically speaking, we like to limit it to oscillators that are very regular in the length of the cycle. And what better example of an oscillator than, you guessed it, a clock! Like we mentioned at the very beginning, typical clocks and watches tick at a steady rate of 1 Hz. But the clock in your living room or on your wrist will eventually slow down as the battery dies, or as the pendulum swings down, or even as the components wear out.  Some clocks are made from quartz, which is a material that has the peculiar quality of oscillating at a very steady frequency when an electrical current is applied. However, even its frequency will wander off a little due to the nature of the atoms inside. For you and me day-to-day, this isn’t a big deal. But, as I mentioned in my previous post about time, for ultra-precise measurements it can make a big difference.

So the name of the game in my field of work is developing clocks that tick at a frequency that does not vary, up to fifteen or sixteen decimal places…or more if we can! The current basis for these oscillators is the internal workings of sub-atomic particles, but that’s a story for a different day. The next step will be taking what we have learned about time and frequency and start making the connection to light and lasers…in bite-sized pieces of course! Thinking with your brain full is very impolite—and sometimes dangerous. We’ll take it slow, I promise. 🙂

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Appendix: Sine and Cosine Functions

If you want a little more math, read on! Oscillators can be modeled with the mathematical functions sine (sin) and cosine (cos). The position, x, of an object undergoing steady oscillation can be mapped out as a function of time, t, by the equation x(t ) = A sin(ωt ), if we know the amplitude, A, (the maximum distance away from the center of motion) and the frequency, given by the Greek lower-case letter omega, ω. Here’s a very basic graphic for a refresher.

Happy Pi Day!

Happy Pi Day everyone! In month-day date format, today is 3-14. These are the first three digits of the mathematical constant π. I thought we’d celebrate with a little history and fun facts.

Yes, I do own a book about pi. And no, it’s not the one about that Indian boy and the boat.

To fifty digits, π=3.14159265358979323846264338327950288419716939937510.

Pi is the ratio of a circle’s circumference (distance around the circle) to its diameter (distance straight across the circle). It is an irrational number, never repeating and never ending, meaning that no matter how many digits one might calculate, there are always more without discernible pattern. Never fear, however—eleven digits will give you a circle the size of the earth with an error of less than a millimeter, and 39 digits will yield a circle fitting inside the observable universe accurate to the size of a hydrogen atom. I guess that’s pretty good (we still measure time more precisely, though!).

Many ancient civilizations, including the Egyptians, Babylonians, Indians, and Hebrews, either understood the relationship between these parts of a circle or had at least a rough calculation of the ratio. The Greek scholar Archimedes first rigorously estimated the value. The Chinese later produced the most accurate value of pi that would prevail until sometime around the 1400’s. Today, with the advent of computers and modern algorithms based on converging series and Fourier transforms, one can calculate multiple trillions of decimal places even on a personal computer. However, since we previously discussed how this is not necessary, we conclude that these activities are merely exercises in nerdy pride.

The ratio 22/7 is a common approximation of pi; therefore, Pi Approximation Day is celebrated on 22 July (day-month date format). On Pi Day, the Pi minute can be celebrated at 1:59:26 a.m. as well as 1:59:26 p.m. if one uses the 12-hour clock convention. That gives the 7-decimal place value of 3.1415926. In the year 2015, Pi Day will be 3-14-15; thus, the pi second at 9:26:54 a.m. (and p.m.) yields the ten-digit value of 3.141592654. So please don’t forget to celebrate Pi Day four years from now; it will be eπc.

Thanks to the Wikipedia articles on Pi and Pi Day for info.