Thanks!

]]>Instead of talking about my feelings here, I will present you with some quantitative food for thought.

There are 31 states that have enacted constitutional gay marriage bans of one sort or another. I've taken the data on that link and compiled it into a publicly accessible Google Doc so that you can mess around with the data if you wish.

I don't have it in me right now to do any fancy modeling (I am still too angry) but I did upload the data into Wolfram Alpha. I've blogged before about Alpha, which is a great tool. If you upgrade to a pro subscription (I am not a salesman... I do find the upgrade to be a little pricey) you can access even more convenient features, like uploading data sets and such. This is what I did with the marriage amendment data. I didn't even know what I wanted to do with the data, so I simply typed the name of my data set as the query, and Alpha decided on all kinds of analyses to spit back out at me. Here is some of it. I guess that if there is a pedagogical point here, it's that the way a quantitative modeler often gets started is just by plotting things.

Here's a geographic heat map where the shading corresponds to the fraction of votes supporting a discriminatory measure in a given state. The reason Nevada shows up as over 100% is that there were two votes in Nevada, one in 2000 and one in 2002 (each one receiving in the high 60%'s range). At any rate, I don't see too much structure in this map.

Here's a histogram of the years in which the discriminatory amendments passed.

You can see that 2004/05 were busy years. Here's what I think are the most interesting plots, namely a histogram of the discriminatory vote percentage and a distribution fit (which you can get a feel for via the quantile plot on the right). Alpha finds that the best fit distribution is uniform.

I am not yet sure what I want to do with any of this, but it at least lets me start trying (perhaps in vain) to make sense of the hate in this country.

]]>Two posts ago, I talked about "the unreasonable effectiveness of mathematics" -- the surprising and amazing success that mathematics has had describing phenomena in the real world. I also showed this graph I put together of the number of Facebook users over time:

Then, last time I went on a big long digression to introduce you to differential equations. I introduced the differential equation

[frac{dx}{dt} = rx ]

which describes a particular scenario of growth in continuous time. More specifically, the left side of the equation is just formal mathematical notation for change, known as a derivative. If is the amount of money in your savings account, what this equation says is that at every moment in time, the way your bank account changes is that it accrues times the amount of money currently in the account. This is called continuous compounding. The solution to the differential equation -- meaning the formula that actually follows -- is

[ x(t) = x_0 mathrm{e}^{r t} ]

where is the amount of money you start out with. This is exponential growth. If you don't withdraw anything, and if your bank doesn't fold, then your money grows forever.

And now we are ready for the new stuff. In reality, nothing can grow forever. All growth in the world is bounded by some constraints. Think about population. The famous *Malthusian* idea of growth, proposed in the late 18th century, is the one we have already discussed: unbounded exponential growth. Thomas Malthus was thinking about growth in the context of the human population on Earth. But in 1838 a guy named Pierre François Verhulst wrote a paper whose French title translates as "A notice on the Law that Population Growth Follows" in which he pointed out that populations really can't grow forever because of constraints such as limited food and space. This same idea was highlighted in a 1920 paper by Pearl and Reed called "On the Rate of Growth of the Population of the United States Since 1790 and its Mathematical Representation." The differential equation that Verhulst, Pearl, and Reed had in mind to describe constrained growth is known as the *logistic model*, which can be written as

[frac{dx}{dt} = rx(1 - x/K) ]

where describes the population size over time. The constant is a growth rate, and plays the same role as in our bank account problem, except than instead of dollars reproducing, we are now thinking of people reproducing. However, in our money problem, the growth percentage was always . In the logistic model, it's not. To see this, think of rearranging the right hand side of the equation. You could write it as

[frac{dx}{dt} = [r(1 - x/K)]x. ]

Instead of multiplying by as in the bank account problem, we are multiplying it by , which depends on the actual value of ! If is very small, this prefactor is essentially equal to , and growth will be (nearly) exponential, like in the bank account problem. But then, as grows, the prefactor gets smaller and smaller, meaning that growth slows down. This is how we capture limited resources. In fact, when , the prefactor is zero! The parameter is known as the *carrying capacity* -- it is meant to capture the maximum population that an environment with limited resourced can support. Also notice that if happened to be bigger than the carrying capacity , the prefactor would be negative, and so the rate of change of the population would be negative, which means the population would shrink. People would die off because of the unavailability of resources.

The logistic differential equation has solution

[x(t) = frac{C mathrm{e}^{rt}}{1+frac{C}{K}mathrm{e}^{rt}}. ]

You can trust me that this is the solution. If you ever decide to learn differential equations, you'll be able to obtain it yourself. Now, remember that measures growth in the absence of constraints, and the carrying capacity that measures the constraints is . The number just depends on how much population you start out with. If the population starts under the carrying capacity, the solution has a characteristic s-like shape that looks like this:

In this example, the carrying capacity is and you can see that levels off to that value towards the right, that is, for later values of .

In case you are thinking "who cares? How could this ever describe anything in real life?" let me go ahead and make you say "whoa!" two times.

First, we have Georgy Gause, who in 1932 wrote a paper with the awesomely grandiose title "Experimental Studies on the Struggle for Existence." Gause let some yeast grow in a lab experiment, and he plotted the amount of yeast over time (shown as symbols below). Then he also plotted solutions to the logistic equation (drawn in as curves). Here's what he got:

If you are not saying "whoa!" let me assure you that you really should be. Yeast are living organisms with their own complicated biology. Even a controlled lab culture is a complicated environment. And yet somehow, a single differential equation does a good job describing yeast population growth.

And second, we have -- finally -- Facebook. When I plotted the data showing the number of active Facebook users over time, I saw the s-shape, which screamed "logistic." So I did a quick fit of the data to a logistic curve. That is to say, I tried to find values of , , and so that the logistic curve would do a good job approximating the many data points. While there are lots of fancy ways to do more precise curve fitting, I wanted to show you that it doesn't take fancy skills to do at least a heuristic job, so I just did some trial and error and came up with , , and . Here's the result, with the actual Facebook data plotted (again) as blue dots and the logistic curve in red:

Crazy, right?! Note that I have included times well beyond those where the Facebook data is available -- or in other words, I've extended the horizontal axis a bit -- to give a sense of what the future holds. From our choice of we can predict that Facebook will max out at 1100 million -- or 1.1 billion -- active users.

People are complicated creatures. Social networks are complicated. People's access to and relationship with technology is complicated. And yet one simple differential equation seems to do a good job matching the Facebook data.

Talk about the unreasonable effectiveness of mathematics.

Since we are in the midst of an economic depression, let's talk about money. Suppose you have a savings account that yields interest on whatever you current balance is. And say that you deposit 100.41. We can write this as

[100 (1+0.004074)^1 = 100.82.]

This is called compounding. After the second month, you didn't just earn interest on your original 105.00. ]

We say that the annualized percentage rate is 5%. In other words, in terms of your bottom line, earning 0.4074% per month 12 times a year on an untouched account is that same as earning 5% one time at the end of the year.

This example so far has ben set in what we call *discrete time*. Compounding happens at distinct moments (for instance, the end of each month). But what if instead of happening at distinct moments, things happen at every moment in time? In case that made your brain explode, suppose that instead of earning 0.4074% once per month, you earned half of that, namely 0.2037%, twice a month. In this case, instead of ending up with 100.05 and an additional 1/2 cent. This is close to, but not the same as, our original example. And then imagine that you earned a quarter of the original interest rate, but compounding happened four times a month. Then you'd end up with 105.01. In short, you'd earn a whopping extra cent due to the continuous, rather than discrete, compounding.

A common way to describe a problem where something grows in continuous time is to use a *differential equation*. I'm about to write down a simple differential equation which I call "the most important differential equation in the world." My students know it by this name. If you ask them "what's the most important differential equation in the world," they will write down this equation. If you write down the equation and ask them the name, they'll tell it to you. The equation is

[frac{dx}{dt} = rx. ]

This equation can describe our continuously compounded interest problem. In that case, is the amount of money in your savings account, and it depends on time measured in months. The left side of the equation is just formal mathematical notation for change, known as a derivative. I'll post more on derivatives another time. For now, when you see , you should replace it in your head with "how quickly my bank balance is changing over time." Then there's an equals sign, which means that on the right hand side of the equation, we are going to put in the proper rule describing a savings account. The parameter is the rate of continuous compounding. So the equation says that at every moment in time, the way your bank account changes is that it accrues times the amount of money currently in the account. If we want to have an annualized rate of 5%, it turns out that we need to choose to be about 0.0040658, which is close to, but not the same as, the value of 0.004074 needed to earn 5% annual when compounding only takes place 12 times a year, rather than continuously. To restate: if your 105.

One advantage of having a differential equation description of something is that it is sometimes possible to solve the differential equation, that is, to write down a formula for the unknown function which in our case describes the amount of money in the bank account over time. If you start out with r=0.0040658ttt = 12x(12) = .

Believe it or not, we are now ready to move back to Facebook... tomorrow. But I swear, I am going to make you say "whoa!" and it will have been worth the wait.

]]>With the goal of making you say "whoa" in mind, let's talk about The Unreasonable Effectiveness of Mathematics in the Natural Sciences. In 1960, Nobel prize winning physicist and mathematician Eugene Wigner wrote a paper with this title. His main point was to ask the philosophical question "Why on earth does mathematics do such a good job describing stuff in the real world?" In 1980, applied mathematician Richard Hamming followed up Wigner's paper with a similarly-but-slightly-more-succinctly-titled homage, The Unreasonable Effectiveness of Mathematics.

Wigner and Hamming were trying to understand how it can be that by pushing around a bunch of abstract mathematical symbols, we can discover how electromagnetic waves behave, or what trajectory a launched missile will follow, or countless other phenomena. Wigner and Hamming were thinking mostly of physics and engineering, and indeed, these areas were the focal application areas of mathematics for quite a long time. Starting in the mid-to-late 20th century, though, there was explosive growth in the application of mathematics to biology. So in addition to describing physical and engineering systems, math does an uncanny job describing the human immune system, the spread of disease within a population (which I will post about sometime soon), the march of evolution, and much more. Even more recently, math has crossed over to the realm of describing social systems. Indeed, much of my own research is about aggregations formed by social organisms: think fish schools, bird flocks, and swarms of locusts.

Speaking of social phenomena, let's talk about Facebook. Since this is a blog, and since you are reading it on a computer, I dare not even bother to explain to you what Facebook is. Chances are that you are a Facebook user, whether you are the type who logs in every day or the type who created an account but has abandoned it. Yesterday, I came across this data on the number of active Facebook users over time. Now, I am not sure how "active" is defined, but still, I was curious about the data, so I did the simplest thing possible, namely, make a graph. Here's what I got:

Hmmm, I thought, those points look like they are tracing out a pretty well-defined curve. That's amazing! But what curve is it? (The mathematicians among you can likely tell from one glance at the picture.)

In the next post, we'll answer this question very explicitly. First we will have to go on a digression that relates to money and differential equations. Please don't be put off. Even if you are a total non-mathematician, you can still learn about differential equations. Seriously. Get yourself ready, and we'll talk about it next week.

]]>Seau, who played in the NFL for parts of 20 seasons, is the eighth member of San Diego's lone Super Bowl team who has died, all before the age of 45. Lew Bush, Shawn Lee, David Griggs, Rodney Culver, Doug Miller, Curtis Whitley and Chris Mims are the others. Causes of death ranged from heart attacks to a plane crash to a lightning strike.

Without diminishing the tragedy of Seau's death -- or the deaths of his teammates -- let's analyze how coincidental this situation really is. On the way to sussing out the question of coincidence, I'll describe in layperson's terms some basic notions of probability, counting, and survival analysis.

I proceed with caution here, as I am neither a probabilist, a statistician, an actuary, a demographer... or even, really, a sports fan. I am confident that someone with more relevant mathematical training could do a better job than I do below. I welcome comments, corrections, and improvements.

Let's get going. As is often necessary with real-world problems, we will make some simplifying assumptions for tractability. According to Business Week, the average age of an NFL player in 2011 was 27 years old. We'll assume this was true in 1994. And to simplify further, we'll assume that *all* of the Chargers were 27 that year. Right now, it is 2012, so those players would be 45 years old.

Now we turn to life tables, clever little tools used by actuaries and demographers to calculate survival probabilities and life expectancies. There are two kinds of life tables. A *period* life table give the probability of death for people of different ages in a given calendar year. So for instance, a 2012 period life table would give the probability of a 90 year old dying this year, an 89 year old dying this year, and so on. A *cohort* life table shows the probability of death of people born in a particular year over the course of future years. So for instance, for people born in 1974 (my birth year), the cohort table would give the probability that they would die in 1975, 1976, and so on.

Since we are interested in people who were 27 in 1994, the best tool for us to use would be a cohort table for 1967 (since 1994 - 27 = 1967) but I couldn't find one. Instead, I looked at this historical period table for 1967 from the Centers for Disease Control. A period table and a cohort table are not the same thing, but nonetheless, we'll use the period table data to get a rough solution to our problem.

Take a peek at Table 5-3 (page 5-9) and focus on the first and third columns. The first column contains ages. The third column tells the story of 100,000 males. Specifically, it gives the number of the original 100,000 surviving until the age given in the first column. Of the original 100,000, 94,600 survive until age 27. These 94,600 live 27 year olds are our base population. Now we want to know the probability of living (at least) to age 45. Looking in the third column for the entry corresponding to 45, we find 89,456. This means that for those alive at 27, the probability of living to 45 would be 89456/94600 = 0.94562. In case you are unfamiliar with calculating probabilities this way, here's another example. Say a bag contains 10 balls, 3 of which are red. We reach in without looking and grab one. The probability that we choose a red ball is 3/10.

If the probability of an individual alive at age 27 in 1994 living to age 45 in 2012 is 0.94562, the probability of them dying before age 45 would be 1 - 0.94562 = 0.05438. That's because the probabilities have to sum to one (you can either live or die).

Now we have to consider a team. An NFL team has a roster limit of 53, so we'll take that as our team size. What is the probability that 8 players alive at age 27 in 1994, would have died by age 45 in 2012? This is a binomial probability question.

Let's work up to this idea. One basic idea used is that to get the probability of multiple things happening (different players living/dying) you multiply together their probabilities, so long as whether each person lives/dies is independent of the other people. For example, suppose that each day, the weather can be either rainy or sunny. And also suppose that the weather on one day is independent of weather on other days. If the probability of rain each day is 1/3, the probability that Monday and Tuesday will be rainy but Wednesday will be sunny next week would be 1/3 * 1/3 * 2/3 = 2/27. The 2/3 value is the probability of sunny weather, which is 1 - the probability of rain, or 1 - 1/3 = 2/3.

Back to our football problem. We want to know the probability of 8 players dying and the other 53 - 8 = 45 players living. With being the probability of one of the 1994 Chargers living until 45, this is something like

[p^{45} (1-p)^8 = 0.94562^{45} cdot 0.04538^8 = 6.1745 times 10^{-12}].

That's minuscule!

But wait, there's something we haven't accounted for. This is where the second idea related to binomial probabilities comes in. It's not a *specific* 45 players that we require to live, it's *any* 45 players. And similarly, it could be *any* 8 players who die. To account for this, we need to multiply by the number of different ways there are to select 45 live players and 8 dead players from a team of 53. This is

[frac{53!}{45!8!} = 886322710]

where the exclamation point is the *factorial* function. In case you don't know this function,

begin{eqnarray*}

1! & = & 1 \

2! & = & 2 cdot 1 = 2\

3! & = & 3 cdot 2 cdot 1 = 3

end{eqnarray*}

and so on. So to get the overall probability that of the team of 53 Chargers in 1994, exactly 8 of them would die by 2012, we take

[6.1745 times 10^{-12} times 886322710 = 5.4726 times 10^{-3}]

or about one-half of one percent. This is small, but not negligible.

We can go even one step further. We've thus far been focused on the Chargers Superbowl team. Instead, we could ask about the probability that there would be exactly 8 deaths on *any* NFL team. We can figure this out with another binomial calculation, first noting that in 1994 there were 28 NFL teams. The probability that one of them would have exactly 8 deaths, as we have said, is which we'll call . The binomial calculation for exactly one team of 28 having exactly 8 deaths is

[r^1 (1-r)^{27} frac{28!}{1!27!} = 0.13213.]

To recap, that's a 13% probability that of all the NFL teams in 1994, exactly one of them would have exactly 8 players die by 2012 (with lots of simplifying assumptions thrown in). 13% is no guarantee, but neither is it so remote that we should be really surprised by it. By the way, note that when we say that one team has exactly 8 players die, this says nothing about the other teams. They could have fewer or more than 8 players die.

Let me finish with some inspirational words. As disclaimed above, I am not a probabilist. My only training in probability came when I was 11 years old. The calculation above is a very rough (an assuredly flawed) estimate based on data gathered from Googling and a few elementary notions of life tables that I picked up from the Internet. No fancy math was involved. I encourage you -- like I encourage all my students -- not to be afraid of trying to estimate things. It's do-able, it's interesting, and it's an excellent way to apply your powers of critical thinking to what's going on in the world.

**Addendum**: My excellent colleague Victor Addona has chimed in on this. He is a real pro who a) has a degree in statistics, b) publishes in the field of survival analysis, and c) does interesting and innovative work related to sports statistics. Victor sagely prefers a refined phrasing of my question, namely "of the 28 teams, what is the probability that the one with the most deaths would have at least 8 deaths?" He answered this question with a simulation conducted in the statistical package R, and his answer is about 19%. My very rough estimate is of 13% more like a lower bound to the answer to Victor's question. In short, the eight deaths are even slightly less coincidental than my post suggests.

[This post has been modified. Thanks to Louisa Bradtmiller for pointing out a factual error in the original post, in which I stated that the Chargers had *won* the Superbowl. Thanks to Victor Addona for pointing out a typo when I originally wrote down 1 - p incorrectly.]

On Meet the Press this past weekend, commentator Rachel Maddow gets into a debate with Republican political operative Alex Castellanos about pay inequality between men and women. (The clip linked above has a flashback of the debate.) As Maddow is trying to discuss policy tools that might be used to fix the gender gap, Castellanos interrupts to say that there is, in fact, no gender gap. Maddow insists that "On average, a woman gets paid 77 cents for every dollar that a man gets paid." Castellanos says that "Actually, if you start looking at the numbers, Rachel, there are lots of reasons for that." His stated reasons include:

- "Men work an average of 44 hours a week. Women work 41 hours a week."
- "Men go into professions like engineering, science and math that earn more."
- "Women want more flexibility."
- "When you look at, for example, single women working in America today between the ages of, I think, 40 and 64, who makes more? Men or women, on average? Men make $40,000 a year. Women make $47,000."

In short, Castellanos is trying to argue that there is no gender gap. Back on her own show, Maddow dives into the details. And this is when the communication about numbers gets really excellent. Maddow first cites her source: a Bloomberg News report that crunches data from the U.S. Census Bureau. Maddow reports a few slices of the data:

- In aggregate (over all data) the average woman makes 77% of what the average man does.
- Restricted to the 20 occupations that are most common for
**men**in this country, men get paid more than women in 19 of them. - Restricted to the 20 occupations that are the most common for
**women**in this country, men get paid more than women in 19 of them. - Of the 265 different occupation categories in the study, men make more than women in 264 of them.

Maddow then interviews Dr. Heidi Hartmann, a MacArthur Genius, research professor at George Washington University, and President of the Institute for Women's Policy Research. She talks about the Census data as well as data from the Bureau of Labor Statistics and the U.S. Government Accountability Office (GAO). Hartmann says

[The GAO study] said that even when you put everything you can possibly think of in the regression equations, the statistical analyses to try to make [the gender] gap go away, you can't explain at least 20 percent of it. Now, most other studies place the part you can't explain as a quarter to a half. So, a large part of the gap probably is due to discrimination.

Maddow replies

In terms of just making it very clear, what you were talking about there about doing a statistical regression analysis on these things, controlling for other factors... What you're saying basically is when you control for things like the number of hours worked, you're still getting a gender based pay disparity that is not explained by working a different number of hours?

Hartmann's response?

Exactly. I mean, Alex seemed to believe if you put in working a different number of hours that would explain it. No, far from it. If you look at all workers and male and female in the economy, we know, let's say, during the childbearing years, about a third of women may be working part time. So count part time. Count how much women work. "OK. I'm working part time. Only making $400 a week." Compare it to all the men, more of whom are working full time. You still get a wage ratio of 72 percent. So that means that [the] 77 percent [figure we have been using] isn't going to move very much if you suddenly remove the people where the men are working 44 and the women are only making 40. No. The number of hours explains a very small part of it. I mean, these regression analysis, they include occupation. They include your education, number of years of experience, maybe sometimes marital status, number of children -- just about anything you can think of. And you cannot make the whole gap go away.

What do I admire about the job Maddow does here?

- She disaggregates opinion ("what is the right policy solution?") from fact ("what do the data say?").
- She cites the source of the data she calls upon.
- She interviews an expert whose credentials are strong.
- When the expert makes a statement that is likely too technical for the public (Harmann's statement about "putting everything in the regression equations"), Maddow makes her backtrack to explain more, and to do so with examples.
- The information provided specifically rebuts Castellanos' contention that the gender gap is an illusion arising from factors such as age and profession.

In short, this was a public, explicit, and understandable discussion of the statistical concept of fraction of variance unexplained. I am thrilled beyond belief. We need more of this.

]]>First, I read the inspirationally-titled "National Competition Shows that Math Smarts are Alive and Well in America." The article describes the most recent installment of Moody's Mega Math (M3) Challenge. The M3 challenge is a high school applied math competition sponsored (obviously) by Moody's, the credit rating and financial services corporation, and by the Society for Industrial and Applied Mathematics. In the competition, five-person student teams have one day "to solve an open-ended, realistic, applied math-modeling problem focused on a real-world issue." This year's challenge was to determine the best U.S. locations for establishing high-speed rail lines.

My spirits soared over the strong mathematical future soon to be built by our nation's youth. Or at least, they soared for about two seconds, until the universe smacked me back down to earth with the second article, "Parents' poor math skills may lead to medication errors." Here's the horrifying scoop. It's long been known that poor reading skills can lead to dosing errors. But a study recently presented at the Pediatric Academic Societies conference examined whether poor math skills play a role too. Investigators engaged 289 parents who had brought their children (under eight years old) to a certain pediatric emergency department. The parents were given a reading test and a math test, and were asked to dispense some medication. The results showed that

Nearly one-third of the parents had low reading skills and 83 percent had poor math skills. Twenty-seven percent had math skills at the third-grade level or below. 41 percent of parents made a dosing error. Parents' math scores, in particular, were associated with measuring mistakes, with parents who scored below the third grade level on the math test having almost a five times increased odds of making a dosing error.

One of the investigators commented

Dosing liquid medications correctly can be especially confusing, as parents may need to understand numerical concepts such as how to convert between different units of measurement, like milliliters, teaspoons and tablespoons. Parents also must accurately use dosing cups, droppers and syringes, many of which vary in their measurement markings and the volume they hold.

I am no statistician, but from the brief description of the sampling scheme, I could certainly conjecture that any number of confounding variables could be at work here. Still, after reading this article, I was left wondering how our country has any live school-age kids to speak of given the massive ibuprofen and antibiotic overdoses that we are assuredly inflicting on them.

Reading these two math-related articles in quick succession made me wonder: do we have a one-percent problem when it comes to mathematics? We certainly have a mathematical elite, and we certainly have many who struggle. Do we have a healthy mathematical middle class?

One possible solution for raising math abilities in this country, as inadvertently suggested by yet a third math-related article I read yesterday, is simply for us to all get bashed in the head and become geniuses.

]]>The mathematicians and non-mathematicians among you will likely have heard of Wolfram. He's a physicist, mathematician, and computational scientist who developed a useful bit of software called Mathematica for numerical and symbolic computations. He's also a MacArthur Genius and the presenter of one of the most popular TED talks. Much earlier in his life, when he was 20, he became the youngest-ever recipient of a Caltech Ph.D., a status he has retained by a hair over my former undergraduate student Catherine Beni.

In 2002, Wolfram published his book A New Kind of Science, which sought to explore the nature of computation and its relationship and applicability to the real world. The scientific community received this book with mixed feelings. Most mathematicians I know are quite critical of the book, both for its apparent egotism and lack of rigor. I think it's fair to say that the academic math community considers the book to be a swing and a miss.

In contrast, Wolfram's real home run is his computational knowledge engine Wolfram Alpha, launched in 2009. I can't urge you enough: go play with this amazing tool. You can ask it a huge range of questions related to mathematics and, more broadly, data of many different sorts. While most mathematical and computational tools require input to be given in a particular syntax, Alpha uses natural language processing to try to figure out what you are asking it. You don't need to learn a programming language or any particular querying commands.

To get a sense of what Alpha can do, I challenge you to try to find:

- The derivative of with respect to
- When the next solar eclipse is
- How the U.S. climate has changed from 1900 - 2011
- What a fractal known as the Pythagoras Tree looks like over 10 iterations
- How many days you have been alive
- What the geographical distribution of life expectancy in Africa looks like
- What year Henry David Thoreau's Walden was published
- For how many years Flintstones and the Jetsons overlapped on the air
- The Scrabble score for the word exogenous
- How many medals Michael Phelps has won
- How Netflix stock is doing lately
- How much a $250,000 mortgage at 5.5% over 30 years costs
- What demographics in Egypt are like
- How long it would take a 44 year old male weighing 90 kg to lose 17 kg if he eats 1500 calories per day
- Where the first flight out of Chicago went to today

In the interest of keeping this post accessible, I've focused on general knowledge in the list above, but make no mistake about it: Alpha can do serious math.

Much like they decried the advent of calculators, some mathematicians and mathematics instructors decry Alpha for how it will allegedly ruin mathematics education. With a freely available, Internet-based tool that can differentiate, integrate, solve differential equations, compute eigenvalues, make graphs, and do so much more, what is the motivation for students to learn symbolic mathematics?

While I don't deny that symbolic skill is necessary for people who will become mathematicians, I nonetheless vehemently disagree with the old-school outlook above. Wolfram Alpha is going to ruin math education no more than the advent of calculators did, and no more than abacuses did. Computational tools are a natural, healthy, and useful part of civilization's mathematical progress. Consider this: the basic algebraic knowledge an educated person has today constituted advanced, cutting-edge research in ancient times. No one cries about the fact that we no longer do mathematics the same way as the ancient Egyptians, Greeks, Babylonians, and Mayans. Computational tools free us to think about ever-more advanced and complex ideas, and ideas are what math is really about.

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