## The Summer 2017 Mathematics A To Z: Diophantine Equations

I have another request from Gaurish, of the For The Love Of Mathematics blog, today. It’s another change of pace.

# Diophantine Equations

A Diophantine equation is a polynomial. Well, of course it is. It’s an equation, or a set of equations, setting one polynomial equal to another. Possibly equal to a constant. What makes this different from “any old equation” is the coefficients. These are the constant numbers that you multiply the variables, your x and y and x2 and z8 and so on, by. To make a Diophantine equation all these coefficients have to be integers. You know one well, because it’s that $x^n + y^n = z^n$ thing that Fermat’s Last Theorem is all about. And you’ve probably seen $ax + by = 1$. It turns up a lot because that’s a line, and we do a lot of stuff with lines.

Diophantine equations are interesting. There are a couple of cases that are easy to solve. I mean, at least that we can find solutions for. $ax + by = 1$, for example, that’s easy to solve. $x^n + y^n = z^n$ it turns out we can’t solve. Well, we can if n is equal to 1 or 2. Or if x or y or z are zero. These are obvious, that is, they’re quite boring. That one took about four hundred years to solve, and the solution was “there aren’t any solutions”. This may convince you of how interesting these problems are. What, from looking at it, tells you that $ax + by = 1$ is simple while $x^n + y^n = z^n$ is (most of the time) impossible?

I don’t know. Nobody really does. There are many kinds of Diophantine equation, all different-looking polynomials. Some of them are special one-off cases, like $x^n + y^n = z^n$. For example, there’s $x^4 + y^4 + z^4 = w^4$ for some integers x, y, z, and w. Leonhard Euler conjectured this equation had only boring solutions. You’ll remember Euler. He wrote the foundational work for every field of mathematics. It turns out he was wrong. It has infinitely many interesting solutions. But the smallest one is $2,682,440^4 + 15,365,639^4 + 18,796,760^4 = 20,615,673^4$ and that one took a computer search to find. We can forgive Euler not noticing it.

Some are groups of equations that have similar shapes. There’s the Fermat’s Last Theorem formula, for example, which is a different equation for every different integer n. Then there’s what we call Pell’s Equation. This one is $x^2 - D y^2 = 1$ (or equals -1), for some counting number D. It’s named for the English mathematician John Pell, who did not discover the equation (even in the Western European tradition; Indian mathematicians were familiar with it for a millennium), did not solve the equation, and did not do anything particularly noteworthy in advancing human understanding of the solution. Pell owes his fame in this regard to Leonhard Euler, who misunderstood Pell’s revising a translation of a book discussing a solution for Pell’s authoring a solution. I confess Euler isn’t looking very good on Diophantine equations.

But nobody looks very good on Diophantine equations. Make up a Diophantine equation of your own. Use whatever whole numbers, positive or negative, that you like for your equation. Use whatever powers of however many variables you like for your equation. So you get something that looks maybe like this:

$7x^2 - 20y + 18y^2 - 38z = 9$

Does it have any solutions? I don’t know. Nobody does. There isn’t a general all-around solution. You know how with a quadratic equation we have this formula where you recite some incantation about “b squared minus four a c” and get any roots that exist? Nothing like that exists for Diophantine equations in general. Specific ones, yes. But they’re all specialties, crafted to fit the equation that has just that shape.

So for each equation we have to ask: is there a solution? Is there any solution that isn’t obvious? Are there finitely many solutions? Are there infinitely many? Either way, can we find all the solutions? And we have to answer them anew. What answers these have? Whether answers are known to exist? Whether answers can exist? We have to discover anew for each kind of equation. Knowing answers for one kind doesn’t help us for any others, except as inspiration. If some trick worked before, maybe it will work this time.

There are a couple usually reliable tricks. Can the equation be rewritten in some way that it becomes the equation for a line? If it can we probably have a good handle on any solutions. Can we apply modulo arithmetic to the equation? If it is, we might be able to reduce the number of possible solutions that the equation has. In particular we might be able to reduce the number of possible solutions until we can just check every case. Can we use induction? That is, can we show there’s some parameter for the equations, and that knowing the solutions for one value of that parameter implies knowing solutions for larger values? And then find some small enough value we can test it out by hand? Or can we show that if there is a solution, then there must be a smaller solution, and smaller yet, until we can either find an answer or show there aren’t any? Sometimes. Not always. The field blends seamlessly into number theory. And number theory is all sorts of problems easy to pose and hard or impossible to solve.

We name these equation after Diophantus of Alexandria, a 3rd century Greek mathematician. His writings, what we have of them, discuss how to solve equations. Not general solutions, the way we might want to solve $ax^2 + bx + c = 0$, but specific ones, like $1x^2 - 5x + 6 = 0$. His books are among those whose rediscovery shaped the rebirth of mathematics. Pierre de Fermat’s scribbled his famous note in the too-small margins of Diophantus’s Arithmetica. (Well, a popular translation.)

But the field predates Diophantus, at least if we look at specific problems. Of course it does. In mathematics, as in life, any search for a source ends in a vast, marshy ambiguity. The field stays vital. If we loosen ourselves to looking at inequalities — $x - Dy^2 < A$, let's say — then we start seeing optimization problems. What values of x and y will make this equation most nearly true? What values will come closest to satisfying this bunch of equations? The questions are about how to find the best possible fit to whatever our complicated sets of needs are. We can't always answer. We keep searching.

## Theorem Thursday: What Is Cramer’s Rule?

KnotTheorist asked for this one during my appeal for theorems to discuss. And I’m taking an open interpretation of what a “theorem” is. I can do a rule.

# Cramer’s Rule

I first learned of Cramer’s Rule in the way I expect most people do. It was an algebra course. I mean high school algebra. By high school algebra I mean you spend roughly eight hundred years learning ways to solve for x or to plot y versus x. Then take a pause for polar coordinates and matrices. Then you go back to finding both x and y.

Cramer’s Rule came up in the context of solving simultaneous equations. You have more than one variable. So x and y. Maybe z. Maybe even a w, before whoever set up the problem gives up and renames everything x1 and x2 and x62 and all that. You also have more than one equation. In fact, you have exactly as many equations as you have variables. Are there any sets of values those variables can have which make all those variable true simultaneously? Thus the imaginative name “simultaneous equations” or the search for “simultaneous solutions”.

If all the equations are linear then we can always say whether there’s simultaneous solutions. By “linear” we mean what we always mean in mathematics, which is, “something we can handle”. But more exactly it means the equations have x and y and whatever other variables only to the first power. No x-squared or square roots of y or tangents of z or anything. (The equations are also allowed to omit a variable. That is, if you have one equation with x, y, and z, and another with just x and z, and another with just y and z, that’s fine. We pretend the missing variable is there and just multiplied by zero, and proceed as before.) One way to find these solutions is with Cramer’s Rule.

Cramer’s Rule sets up some matrices based on the system of equations. If the system has two equations, it sets up three matrices. If the system has three equations, it sets up four matrices. If the system has twelve equations, it sets up thirteen matrices. You see the pattern here. And then you can take the determinant of each of these matrices. Dividing the determinant of one of these matrices by another one tells you what value of x makes all the equations true. Dividing the determinant of another matrix by the determinant of one of these matrices tells you which values of y makes all the equations true. And so on. The Rule tells you which determinants to use. It also says what it means if the determinant you want to divide by equals zero. It means there’s either no set of simultaneous solutions or there’s infinitely many solutions.

This gets dropped on us students in the vain effort to convince us knowing how to calculate determinants is worth it. It’s not that determinants aren’t worth knowing. It’s just that they don’t seem to tell us anything we care about. Not until we get into mappings and calculus and differential equations and other mathematics-major stuff. We never see it in high school.

And the hard part of determinants is that for all the cool stuff they tell us, they take forever to calculate. The determinant for a matrix with two rows and two columns isn’t bad. Three rows and three columns is getting bad. Four rows and four columns is awful. The determinant for a matrix with five rows and five columns you only ever calculate if you’ve made your teacher extremely cross with you.

So there’s the genius and the first problem with Cramer’s Rule. It takes a lot of calculating. Many any errors along the way with the calculation and your work is wrong. And worse, it won’t be wrong in an obvious way. You can find the error only by going over every single step and hoping to catch the spot where you, somehow, got 36 times -7 minus 21 times -8 wrong.

The second problem is nobody in high school algebra mentions why systems of linear equations should be interesting to solve. Oh, maybe they’ll explain how this is the work you do to figure out where two straight lines intersect. But that just shifts the “and we care because … ?” problem back one step. Later on we might come to understand the lines represent cases where something we’re interested in is true, or where it changes from true to false.

This sort of simultaneous-solution problem turns up naturally in optimization problems. These are problems where you try to find a maximum subject to some constraints. Or find a minimum. Maximums and minimums are the same thing when you think about them long enough. If all the constraints can be satisfied at once and you get a maximum (or minimum, whatever), great! If they can’t … Well, you can study how close it’s possible to get, and what happens if you loosen one or more constraint. That’s worth knowing about.

The third problem with Cramer’s Rule is that, as a method, it kind of sucks. We can be convinced that simultaneous linear equations are worth solving, or at least that we have to solve them to get out of High School Algebra. And we have computers. They can grind away and work out thirteen determinants of twelve-row-by-twelve-column matrices. They might even get an answer back before the end of the term. (The amount of work needed for a determinant grows scary fast as the matrix gets bigger.) But all that work might be meaningless.

The trouble is that Cramer’s Rule is numerically unstable. Before I even explain what that is you already sense it’s a bad thing. Think of all the good things in your life you’ve heard described as unstable. Fair enough. But here’s what we mean by numerically unstable.

Is 1/3 equal to 0.3333333? No, and we know that. But is it close enough? Sure, most of the time. Suppose we need a third of sixty million. 0.3333333 times 60,000,000 equals 19,999,998. That’s a little off of the correct 20,000,000. But I bet you wouldn’t even notice the difference if nobody pointed it out to you. Even if you did notice it you might write off the difference. “If we must, make up the difference out of petty cash”, you might declare, as if that were quite sensible in the context.

And that’s so because this multiplication is numerically stable. Make a small error in either term and you get a proportional error in the result. A small mistake will — well, maybe it won’t stay small, necessarily. But it’ll not grow too fast too quickly.

So now you know intuitively what an unstable calculation is. This is one in which a small error doesn’t necessarily stay proportionally small. It might grow huge, arbitrarily huge, and in few calculations. So your answer might be computed just fine, but actually be meaningless.

This isn’t because of a flaw in the computer per se. That is, it’s working as designed. It’s just that we might need, effectively, infinitely many digits of precision for the result to be correct. You see where there may be problems achieving that.

Cramer’s Rule isn’t guaranteed to be nonsense, and that’s a relief. But it is vulnerable to this. You can set up problems that look harmless but which the computer can’t do. And that’s surely the worst of all worlds, since we wouldn’t bother calculating them numerically if it weren’t too hard to do by hand.

(Let me direct the reader who’s unintimidated by mathematical jargon, and who likes seeing a good Wikipedia Editors quarrel, to the Cramer’s Rule Talk Page. Specifically to the section “Cramer’s Rule is useless.”)

I don’t want to get too down on Cramer’s Rule. It’s not like the numerical instability hurts every problem you might use it on. And you can, at the cost of some more work, detect whether a particular set of equations will have instabilities. That requires a lot of calculation but if we have the computer to do the work fine. Let it. And a computer can limit its numerical instabilities if it can do symbolic manipulations. That is, if it can use the idea of “one-third” rather than 0.3333333. The software package Mathematica, for example, does symbolic manipulations very well. You can shed many numerical-instability problems, although you gain the problem of paying for a copy of Mathematica.

If you just care about, or just need, one of the variables then what the heck. Cramer’s Rule lets you solve for just one or just some of the variables. That seems like a niche application to me, but it is there.

And the Rule re-emerges in pure analysis, where numerical instability doesn’t matter. When we look to differential equations, for example, we often find solutions are combinations of several independent component functions. Bases, in fact. Testing whether we have found independent bases can be done through a thing called the Wronskian. That’s a way that Cramer’s Rule appears in differential equations.

Wikipedia also asserts the use of Cramer’s Rule in differential geometry. I believe that’s a true statement, and that it will be reflected in many mechanics problems. In these we can use our knowledge that, say, energy and angular momentum of a system are constant values to tell us something of how positions and velocities depend on each other. But I admit I’m not well-read in differential geometry. That’s something which has indeed caused me pain in my scholarly life. I don’t know whether differential geometers thank Cramer’s Rule for this insight or whether they’re just glad to have got all that out of the way. (See the above Wikipedia Editors quarrel.)

I admit for all this talk about Cramer’s Rule I haven’t said what it is. Not in enough detail to pass your high school algebra class. That’s all right. It’s easy to find. MathWorld has the rule in pretty simple form. Mathworld does forget to define what it means by the vector d. (It’s the vector with components d1, d2, et cetera.) But that’s enough technical detail. If you need to calculate something using it, you can probably look closer at the problem and see if you can do it another way instead. Or you’re in high school algebra and just have to slog through it. It’s all right. Eventually you can put x and y aside and do geometry.

• #### KnotTheorist 3:44 pm on Thursday, 9 June, 2016 Permalink | Reply

Thanks for the post! It’s good to know I’m not the only one who wondered about the usefulness of Cramer’s Rule for computation. That was part of my motivation for asking about it, actually; I was curious about what, if anything, it was good for.

Also, thanks for linking to that Wikipedia article. It was an interesting read.

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• #### Joseph Nebus 3:21 am on Saturday, 11 June, 2016 Permalink | Reply

I’m happy to be of service. And, as I say, it’s not like the rule is ever wrong. The worst you can hold against it is that it’s not the quickest or most stable way of doing a lot of problems. But if you can control that, then it’s a tool you have.

But I admit not using it except as the bit that justifies some work in later proofs since I got out of high school algebra. It’s so beautiful a thing it seems like it ought to be useful.

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• #### xianyouhoule 12:08 pm on Friday, 10 June, 2016 Permalink | Reply

Can we understand Cramer’s Rule

in a geometrical way??

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• #### xianyouhoule 12:16 pm on Friday, 10 June, 2016 Permalink | Reply

Can we understand Cramer’s Rule in a geometrical way??

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• #### Joseph Nebus 3:32 am on Saturday, 11 June, 2016 Permalink | Reply

Happy to help.

We can work out geometric interpretations of Cramer’s Rule. But I’m not sure how compelling they are. They come about from looking at sets of linear equations as a linear transformation. That is, they’re stretching out and rotating and adding together directions in space. Then the determinant of the matrix corresponding to a set of equations has a good geometric interpretation. It’s how much a unit square gets expanded, or shrunk, by the projection the matrix represents.

For Cramer’s Rule we look at the determinants of two matrices. One of them is the matrix of the original set of equations. And the other is a similar matrix that has, in the place of (say) constants-times-x, the constant numbers from the right-hand-sides of the original equations. The constants with no variables on them. This matrix projects space in a slightly different way.

So Cramer’s Rule tells us that the value of x (say) which makes all the equations true is equal to how much the modified matrix with constants instead of x-coefficients expands space, divided by how much the original matrix expands space. And similarly for y and for z and whatever other coordinates you have. And as I say, Wikipedia’s entry on Cramer’s Rule has some fair pictures showing this.

I admit I’m not sure that’s compelling, though. I don’t have a good answer offhand for why we should expect these ratios to be important, or why these particular modified matrices should enter into it. But it is there and it might help someone at least remember how this rule works.

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• #### howardat58 3:48 pm on Thursday, 16 June, 2016 Permalink | Reply

I am of the opinion that cramers rule sucks. What is wrong with Gaussian Elimination ????????

(apart from the relatively enormous speed !!!)

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• #### Joseph Nebus 4:34 am on Friday, 17 June, 2016 Permalink | Reply

Well, speed is the big strike against Gaussian Elimination. But Gaussian Elimination is a lot better off than Cramer’s Rule on that count. Gaussian Elimination also isn’t numerically stable for every matrix. But for diagonally dominant or positive-definite matrices it is, and that’s usually good enough.

As often happens with numerical techniques, nothing’s quite right all the time. Best you can do is have some idea what’s usually all right.

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## Who Discovered Boyle’s Law?

Stigler’s Law is a half-joking principle of mathematics and scientific history. It says that scientific discoveries are never named for the person who discovered them. It’s named for the statistician Stephen Stigler, who asserted that the principle was discovered by the sociologist Robert K Merton.

If you study much scientific history you start to wonder if anything is named correctly. There are reasons why. Often it’s very hard to say exactly what the discovery is, especially if it’s something fundamental. Often the earliest reports of something are unclear, at least to later eyes. People’s attention falls on a person who did very well describing or who effectively publicized the discovery. Sometimes a discovery is just in the air, and many people have important pieces of it nearly simultaneously. And sometimes history just seems perverse. Pell’s Equation, for example, is named for John Pell, who did not discover it, did not solve it, and did not particularly advance our understanding of it. We seem to name it Pell’s because Pell had translated a book which included a solution of the problem into English, and Leonhard Euler mistakenly thought Pell had solved it.

The Carnot Cycle blog for this month is about a fine example of naming confusion. In this case it’s about Boyle’s Law. That’s one of the rules describing how gases work. It says that, if a gas is held at a constant temperature, and the amount of gas doesn’t change, then the pressure of the gas times its volume stays constant. Squeeze the gas into a smaller volume and it exerts more pressure on the container it’s in. Stretch it into a larger volume and it presses more weakly on the container.

Obvious? Perhaps. But it is a thing that had to be discovered. There’s a story behind that. Peter Mander explains some of its tale.

• #### laboroflike 1:46 pm on Monday, 11 April, 2016 Permalink | Reply

Sadly, upon his death, Robert Boyle was buried in Grant’s Tomb, depriving Ulysses S. Grant of a proper funeral. President Grant, accompanied by equation translator John Pell, continues to roam the earth, squeezing gasses into ever smaller volumes, hoping that one day he will find a container with enough extra room to hold his rotting corpse.

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• #### Joseph Nebus 3:34 am on Friday, 15 April, 2016 Permalink | Reply

I am amused, and yet, Robert Boyle’s actual burial spot is no less wondrous. According to Wikipedia, he’s buried in Saint Martin-in-the-Fields, a church which is in Trafalgar Square and not any field in particular. More: The crypt houses a café which hosts jazz concerts whose profits support the programs of the church.'' Fair enough. But then,the crypt is also home to the London Brass Rubbing Centre.” And “A life-sized marble statue of Henry Croft, London’s first pearly king, was moved to the crypt in 2002 from its original site at St Pancras Cemetery”, in what I suppose was a really giddy prank. Furthermore, twelve of its bells are now in the Swan Bells tower in Perth, Western Australia. And doesn’t that improve your day to hear?

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• #### laboroflike 7:56 pm on Friday, 15 April, 2016 Permalink | Reply

It does. It makes me realize that no matter how ridiculous the stuff I make up is, reality has the power to be even more insane. No, wait, not “improve”. What’s the other thing?

On a related note, it may interest you to know that St. Pancras, who lived in the 3rd century, is the patron saint of teenagers, having been martyred at age 15 for talking back to an Imperial centurion. Consequently, pranks such as moving statues, being pearly, having jazz concerts in a crypt, and rubbing brass are considered devotions to St. Pancras, asking him to intercede so that God will give them \$150 sneakers, or possibly the latest iPhone.

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• #### Joseph Nebus 2:02 am on Friday, 22 April, 2016 Permalink | Reply

Thank you so. And this is all so wonderful I hesitate to bring it up but …

Saint Pancras is regarded as the second of the Ice Saints, a group of saints given that curious name because their feast days fall on the 11th through 13th of May.

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## Xor.

Xor comes to us from logic. In this field we look at propositions, which can be be either true or false. Propositions serve the same rule here that variables like “x” and “y” serve in algebra. They have some value. We might know what the value is to start with. We might be hoping to deduce what the value is. We might not actually care what the value is, but need a placeholder for it while we do other work.

A variable, or a proposition, can carry some meaning. The variable “x” may represent “the longest straight board we can fit around this corner”. The proposition “A” may represent “The blue house is the one for sale”. (Logic has a couple of conventions. In one we use capital letters from the start of the alphabet for propositions. In other we use lowercase p’s and q’s and r’s and letters from that patch of the alphabet. This is a difference in dialect, not in content.) That’s convenient, since it can help us understand the meaning of a problem we’re working on, but it’s not essential. The process of solving an equation is the same whether or not the equation represents anything in the real world. So it is with logic.

We can combine propositions to make more interesting statements. If we know what whether the propositions are true or false we know whether the statements are true. If we know starting out only that the statements are true (or false) we might be able to work out whether the propositions are true or false.

Xor, the exclusive or, is one of the common combinations. Start with the propositions A and B, both of which may be true or may be false. A Xor B is a true statement when A is true while B is false, or when A is false while B is true. It’s false when A and B are simultaneously false. It’s also false when A and B are simultaneously true.

It’s the logic of whether a light bulb on a two-way switch is on. If one switch it on and the other off, the bulb is on. If both switches are on, or both switches off, the bulb is off. This is also the logic of what’s offered when the menu says you can have french fries or onion rings with your sandwich. You can get both, but it’ll cost an extra 95 cents.

• #### sheldonk2014 3:20 pm on Friday, 17 July, 2015 Permalink | Reply

Your telling me there an mathematical theory to McDonald’s craziness
I can believe it
In my mind it is a stretch
Sheldon

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• #### Joseph Nebus 5:02 am on Tuesday, 21 July, 2015 Permalink | Reply

There’s a surprising amount of mathematics behind stuff McDonald’s does, actually. Have you ever encountered the McNugget problem?

The idea dates to the early days when Chicken McNuggets were sold in packs of 6, 9, or 20. If you wanted to get 12 McNuggets, that’s easy enough: buy two packs of 6. If you want 15, buy a pack of 9 and a pack of 6. If you want 18, buy three packs of 6 or two packs of 9. If you want 26, buy a pack of 20 and a pack of 6. And so on.

But you can’t get exactly seven McNuggets. And you can’t get exactly ten. You can’t get exactly 19, either.

What’s the largest number of McNuggets you can’t buy, then?

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## Reading the Comics, April 27, 2015: Anthropomorphic Mathematics Edition

They’re not running at the frantic pace of April 21st, but there’s still been a fair clip of comic strips that mention some kind of mathematical topic. I imagine Comic Strip Master Command wants to be sure to use as many of these jokes up as possible before the (United States) summer vacation sets in.

Dan Thompson’s Brevity (April 23) is a straightforward pun strip. It also shows a correct understanding of how to draw a proper Venn Diagram. And after all why shouldn’t an anthropomorphized Venn Diagram star in movies too?

John Atkinson’sWrong Hands (April 23) gets into more comfortable territory with plain old numbers being anthropomorphized. The 1 is fair to call this a problem. What kind of problem depends on whether you read the x as a multiplication sign or as a variable x. If it’s a multiplication sign then I can’t think of any true statement that can be made from that bundle of symbols. If it’s the variable x then there are surprisingly many problems which could be made, particularly if you’re willing to count something like “x = 718” as a problem. I think that it works out to 24 problems but would accept contrary views. This one ended up being the most interesting to me once I started working out how many problems you could make with just those symbols. There’s a fun question for your combinatorics exam in that.

• #### sheldonk2014 10:12 pm on Monday, 27 April, 2015 Permalink | Reply

I tried to give props to all my comic people on the weekend by writing a poem about the movie Pink Flamingos no one got it, it’s called Eddy,please tell me you have heard of this movie,otherwise I will crawl back in the corner and suck my dust

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• #### Joseph Nebus 2:48 am on Wednesday, 29 April, 2015 Permalink | Reply

I’m honestly surprised. I would have thought that even if one hadn’t seen Pink Flamingos at least the title would be familiar as a movie. Possibly it’s a generational thing.

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• #### abyssbrain 2:05 am on Tuesday, 28 April, 2015 Permalink | Reply

Yes, I agree. Quickly simplifying “3(x + 1) – 2” to “3x + 1” without showing the steps can confuse the students, especially if they are just being introduced to algebra.

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• #### Joseph Nebus 2:50 am on Wednesday, 29 April, 2015 Permalink | Reply

The double simplification is a problem, but I think it’s especially a problem that a 1 appears inside the parenthesis and then on the next line. That is, I think it’d be less confusing if they went from (say) “3(x + 3) – 2” directly to “3x + 7” since there’d be no suggestive-but-false connection between the number in parentheses and the number in the second line.

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• #### ivasallay 8:20 am on Thursday, 30 April, 2015 Permalink | Reply

“x” looks a lot like a “+” if you roll it a little.
Students might like Brevity’s Venn diagram strip, so it could be a fun way to refresh their memories.

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• #### Joseph Nebus 6:03 am on Tuesday, 5 May, 2015 Permalink | Reply

You know, I kept wondering whether the x should be considered a + in this case. It makes forming an equation a lot easier. I just feel like if it were meant to be a plus sign, then the character wouldn’t have feet coming out between two legs of the figure. (I hope you follow what I mean.) But the characters could probably roll over, if they wanted.

I think they use the term “cow tools” to describe the reaction the strip’s set off in me.

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## When 2 plus 2 Equals 5, plus Another Unsettling Equation

I just wanted to note for folks who don’t read The Straight Dope — the first two books of which were unimaginably important to the teenage me, hundreds of pages of neat stuff to know delivered in a powerful style, that overwhelmed even The People’s Almanac 2 if you can imagine — that the Straight Dope Science Advisory board tried to take on the question of Does 2 + 2 equal 5 for very large values of 2?

Straight Dope Staffer Dex takes the question a bit more literally than I have ever interpreted the joke to be. I’ve basically read it as just justifying a nonsense result with a nonsense explanation, fitting in the spectrum of comic answers somewhere between King Lear’s understanding of why there are seven stars in the Pleiades and classic 1940s style double-talk. But Dex uses the equation to point out how rounding and estimation, essential steps in translating between the real world and the mathematical representation of the world, can produce results which are correct at every step but wrong in the whole, which is worth considering.

Also, in a bit of reading I’m doing and which I might rip off^W^W use as inspiration for some posts around here the (British) author dropped in an equation meant to be unsettling and, yeah, this unsettles me. Let me know what you think:

$3 \mbox{ feet } + 2 \mbox{ tons } = 36 \mbox{ inches } + 2440 \mbox{ pounds }$

I should say it’s not like I’m going to have nightmares about that, but it feels off anyway.

• #### abyssbrain 1:42 am on Sunday, 8 March, 2015 Permalink | Reply

Then there’s also the classic of Abbott and Castello “proving” that 13 x 7 = 28 :)

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• #### Joseph Nebus 11:37 pm on Monday, 9 March, 2015 Permalink | Reply

You know, I’m not familiar with that sketch offhand.

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• #### abyssbrain 12:24 am on Tuesday, 10 March, 2015 Permalink | Reply

Actually, it’s a part of a film.

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• #### Joseph Nebus 7:50 pm on Thursday, 12 March, 2015 Permalink | Reply

Oh, that’s a great routine, and I hadn’t seen it before. Thank you. (It’s surely from their TV show, though?)

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• #### abyssbrain 1:07 am on Friday, 13 March, 2015 Permalink | Reply

If I remembered correctly, that particular video was from one of their films, but they had also done this sketch once on their tv show.

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• #### Joseph Nebus 3:25 am on Saturday, 14 March, 2015 Permalink | Reply

That makes sense. It’d be uncharacteristic for them to use a good bit only the once, especially since it could be years between anyone in the audience seeing a movie, a radio program where they did it, and a TV show using the same bit.

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• #### Matthew Wright 1:55 am on Sunday, 8 March, 2015 Permalink | Reply

To me British Imperial measures like the long ton (which is only half a smoot longer than a short ton) pretty much sum up the problem the British also have making reliable cars. And landing things on Mars, when one half of the team is using Imperial and the other half metric.

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• #### Joseph Nebus 11:48 pm on Monday, 9 March, 2015 Permalink | Reply

As best I can tell the short ton is an invention of the Americans, so the British aren’t directly at fault for the long ton/short ton divide. Granted that 2240 pounds is a superficially weird number of pounds to put into any unit, but that is at least a nice convenient twenty hundredweights, which admittedly moves the problem back to why a hundredweight is a hundred pounds. In that case it’s because a hundredweight was a nice convenient eight stone, which had been twelve and a half pounds avoirdupois, until King Edward III yielded to the convenience of the wool trade and increase the stone to fourteen pounds (making a sack of cloth, 28 stone, more conveniently measured without cheating on available scales and also a nice (nearly) round 500 Florentian libbrae, and the rest followed from there.) Which is to admit that it’s daft, but every step made sense at the time, which is the best we can ever hope for.

Now, the Imperial/Metric problem with the space probe is interesting because while the difference in units is the proximate cause of the vehicle’s loss, it’s not the real cause. There were hints, from earlier maneuvers, that something was wrong in the way thrusts were being calculated or executed, but those weren’t followed up on. Had they been, a correction would’ve been straightforward. It’s a lesson in the importance of having good project management, and that project management has to include people signaling clearly when they suspect there’s problems and exploring adequately whether these suspicions are well-founded.

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• #### elkement 2:55 pm on Sunday, 8 March, 2015 Permalink | Reply

It was not until recently that I learned how ‘ton’ is used in engineering (related to air conditioning). I learned a lot of – maybe a ton of – new units when trying to respond to questions in the comments section on my blog :-)

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• #### Joseph Nebus 11:54 pm on Monday, 9 March, 2015 Permalink | Reply

I did not know there were custom uses of the ‘ton’ for engineering purposes until just now, and I’m fascinated to see how many different “big mass of the thing we’re measuring” get called tons, now. (Panama Canal Net Ton? Who ordered that?)

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• #### elkement 10:57 am on Tuesday, 10 March, 2015 Permalink | Reply

I hope I understood it correctly finally – but I was baffled about “ton” being used as a unit for (heating or cooling) *power*, rather than weight.

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• #### Joseph Nebus 7:55 pm on Thursday, 12 March, 2015 Permalink | Reply

Yeah, thinking of ton as a unit of power is weird, although I suppose it’s not inherently stranger than describing a distance by the amount of time it’d take to get there. It’s just less familiar.

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## Denominated Mischief

I’ve finally got around to reading one of my Christmas presents, Alfred S Posamentier and Ingmar Lehman’s Magnificent Mistakes in Mathematics, which is about ways that mathematical reasoning can be led astray. A lot, at least in the early pages, is about the ways a calculation can be fowled by a bit of carelessness, especially things like dividing by zero, which seems like such an obvious mistake that who could make it once they’ve passed Algebra II?

They got to a most neat little erroneous calculation, though, and I wanted to share it since the flaw is not immediately obvious although the absurdity of the conclusion drives you to look for it. We begin with a straightforward problem that I think of as Algebra I-grade, though I admit my memories of taking Algebra I are pretty vague these days, so maybe I missed the target grade level by a year or two.

$\frac{3x - 30}{11 - x} = \frac{x + 2}{x - 7} - 4$

Multiply that 4 on the right-hand side by 1 — in this case, by $\frac{x - 7}{x - 7}$ — and combine that into the numerator:

$\frac{3x - 30}{11 - x} = \frac{x + 2 - 4(x - 7)}{x - 7}$

Expand that parentheses and simplify the numerator on the right-hand side:

$\frac{3x - 30}{11 - x} = \frac{3x - 30}{7 - x}$

Since the fractions are equal, and the numerators are equal, therefore their denominators must be equal. Thus, $11 - x = 7 - x$ and therefore, 11 = 7.

Did you spot where the card got palmed there?

• #### Little Monster Girl 11:13 pm on Monday, 2 February, 2015 Permalink | Reply

You didn’t do anything to the left side of the equation?

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• #### Joseph Nebus 10:20 am on Tuesday, 3 February, 2015 Permalink | Reply

It’s true nothing’s done on the left-hand side, but that isn’t by itself an error. If we start from the assumption that the original equation is true we can manipulate one side, or the other, or both, into a form that’s more convenient without changing whether or not the whole equation is true. The catch is that somewhere in this is a manipulation that doesn’t preserve the truth of the whole thing.

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• #### howardat58 11:30 pm on Monday, 2 February, 2015 Permalink | Reply

Formally, cross multiply is in order. Of course, they don’t call it that these days.

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• #### Joseph Nebus 10:21 am on Tuesday, 3 February, 2015 Permalink | Reply

Cross-multiplying ought to give a fair shot at avoiding the error, yeah. But I couldn’t blame someone for seeing an equation of the form a/b = a/d and going right to b = d directly.

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• #### ivasallay 7:41 am on Tuesday, 3 February, 2015 Permalink | Reply

Where did (3x – 30)/ (11 – x) = (x + 2)/(x – 7) – 4 come from? It certainly isn’t true for all x.

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• #### Joseph Nebus 10:23 am on Tuesday, 3 February, 2015 Permalink | Reply

Well, that’s just a problem to be solved, to find values of x which make it true. It’s just that along the way to finding those x’s, we end up with a conclusion that 11 equals 7.

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• #### elkement 7:18 pm on Tuesday, 3 February, 2015 Permalink | Reply

I think the trick is to keep in mind that when the numerator is zero then it does not matter if both denominators are different (as long as they are not equal to zero as well).

So if x is equal to 10 the equation is true as both sides are equal to zero although the denominators are 1 and -3, respectively.

The short version is: You must not divide both sides of an equation of zero.

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• #### Joseph Nebus 11:31 pm on Wednesday, 4 February, 2015 Permalink | Reply

That’s it exactly, and I’m delighted by the problem since it is one in which the ever-forbidden division by zero is made nicely non-obvious.

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## What Is True Almost Everywhere?

I was reading a thermodynamics book (C Truesdell and S Bharatha’s The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, which is a fascinating read, for the field, and includes a number of entertaining, for the field, snipes at the stuff textbook writers put in because they’re just passing on stuff without rethinking it carefully), and ran across a couple proofs which mentioned equations that were true “almost everywhere”. That’s a construction it might be surprising to know even exists in mathematics, so, let me take a couple hundred words to talk about it.

The idea isn’t really exotic. You’ve seen a kind of version of it when you see an equation containing the note that there’s an exception, such as, $\frac{\left(x - 1\right)^2}{\left(x - 1\right)} = x \mbox{ for } x \neq 1$. If the exceptions are tedious to list — because there are many of them to write down, or because they’re wordy to describe (the thermodynamics book mentioned the exceptions were where a particular set of conditions on several differential equations happened simultaneously, if it ever happened) — and if they’re unlikely to come up, then, we might just write whatever it is we want to say and add an “almost everywhere”, or for shorthand, put an “ae” after the line. This “almost everywhere” will, except in freak cases, propagate through the rest of the proof, but I only see people writing that when they’re students working through the concept. In publications, the “almost everywhere” gets put in where the condition first stops being true everywhere-everywhere and becomes only almost-everywhere, and taken as read after that.

I introduced this with an equation, but it can apply to any relationship: something is greater than something else, something is less than or equal to something else, even something is not equal to something else. (After all, “$x \neq -x$ is true almost everywhere, but there is that nagging exception.) A mathematical proof is normally about things which are true. Whether one thing is equal to another is often incidental to that.

What’s meant by “unlikely to come up” is actually rigorously defined, which is why we can get away with this. It’s otherwise a bit daft to think we can just talk about things that are true except where they aren’t and not even post warnings about where they’re not true. If we say something is true “almost everywhere” on the real number line, for example, that means that the set of exceptions has a total length of zero. So if the only exception is where x equals 1, sure enough, that’s a set with no length. Similarly if the exceptions are where x equals positive 1 or negative 1, that’s still a total length of zero. But if the set of exceptions were all values of x from 0 to 4, well, that’s a set of total length 4 and we can’t say “almost everywhere” for that.

This is all quite like saying that it can’t happen that if you flip a fair coin infinitely many times it will come up tails every single time. It won’t, even though properly speaking there’s no reason that it couldn’t. If something is true almost everywhere, then your chance of picking an exception out of all the possibilities is about like your chance of flipping that fair coin and getting tails infinitely many times over.

• #### ivasallay 9:16 pm on Wednesday, 23 April, 2014 Permalink | Reply

I don’t think I’ve read the term “true almost everywhere” before? Is it a term used in the sciences more than in mathematics? I know it annoys me when people assert that ANY number raised to the zero power is 1 and completely ignore that zero to the zero power isn’t defined. Thank you for an enlightening post!

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• #### Joseph Nebus 11:23 pm on Wednesday, 23 April, 2014 Permalink | Reply

I’d encountered “almost everywhere” first in functional analysis, if I remember rightly, and at least my impression is that it has its greatest domain there. (The concept is tied to measure theory — you can be true almost everywhere if the set of exceptions has measure zero — which is probably why it doesn’t seem to get seen much before grad school.) Wikipedia suggests the same concept is in older books abbreviated pp, for “presque partout”, but that just means the same “almost everywhere”.

I was actually a bit surprised to encounter it explicitly mentioned in a thermodynamics book, but the authors were going for a rigor they felt wasn’t present in most books of the kind.

And, thank you for writing back.

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• #### rwithgrosstopology 9:01 am on Thursday, 24 April, 2014 Permalink | Reply

Nice post about one nice concept in Mathematics. “Almost everywhere” is a notion often encountered within Measure Theory, meaning that something happens except for a set of zero measure. My first meeting with it was in my Probability courses, where I was told that it mean a set of possible results of an experiment whose elements would virtually not appear on practice though. Like, say, you pick the temperature of a point at random at wonder what is the probability it will be 23.5 ºC exactly. This is certainly a possible temperature, but the probability of being that exact temperature is exactly zero, since you can only get positive probabilities for ranges of temperatures.

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• #### Joseph Nebus 8:33 pm on Tuesday, 29 April, 2014 Permalink | Reply

Thank you kindly.

I’ve been thinking to write a bit more about measure theory, since it seems to be at a golden intersection for popular mathematics: the ideas are accessible without any advanced mathematics and possibly any equations other than “the sum of 1/2 plus 1/4 plus 1/8 plus 1/16 plus (etc) is 1”, and it gets pretty quickly and easily to some results contrary to intuition, and it underlies a lot of important higher mathematics, and for all that it doesn’t seem to be over-grazed territory for pop math writers, the way the different cardinalities of infinity are.

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## Why A Line Doesn’t Have An Equation

[ To resume after some interruptions — it’s been quite a busy few weeks — the linear interpolations that I had been talking about, I will need equations describing a line. ]

To say something is the equation representing a line is to lie in the article. It’s little one, of the same order as pretending there’s just one answer to the question, “Who are you?” Who you are depends on context: you’re the person with this first-middle-last name combination. You’re the person with this first name. You’re the person with this nickname. You’re the third person in the phone queue for tech support. You’re the person with this taxpayer identification number. You’re the world’s fourth-leading expert on the Marvel “New Universe” line of comic books, and sorry for that. You’re the person who ordered two large-size fries at Five Guys Burgers And Fries and will soon learn you’ll never live long enough to eat them all. You’re the person who knows how to get the sink in the break room at work to stop dripping. These may all be correct, but depending on the context some of these answers are irrelevant, and maybe one or two of them is useful, or at least convenient. So it is with equations for a line: there are many possible equations. Some of them are just more useful, or even convenient.

## Hopefully, Saying Something True

I wanted to talk about drawing graphs that represent something, and to get there have to say what kinds of things I mean to represent. The quick and expected answer is that I mean to represent some kind of equation, such as “y = 3*x – 2” or “x2 + y2 = 4”, and that probably does come up the most often. We might also be interested in representing an inequality, something like “x2 – 2 y2 ≤ 1”. On occasion we’re interested just in the region where something is not true, saying something like “y ≠ 3 – x”. (I’ve used nice small counting numbers here not out of any interest in these numbers, or because larger ones or non-whole numbers or even irrational numbers don’t work, but because there is something pleasantly reassuring about seeing a “1” or a “2” in an equation. We strongly believe we know what we mean by “1”.)

Anyway, what we’ve written down is something describing a relationship which we are willing to suppose is true. We might not know what x or y are, and we might not care, but at least for the length of the problem we will suppose that the number represented by y must be equal to three times whatever number is represented by x and minus two. There might be only a single value of x we find interesting; there might be several; there might be infinitely many such values. There’ll be a corresponding number of y’s, at least, so long as the equation is true.

Sometimes we’ll turn the description in terms of an equation into a description in terms of a graph right away. Some of these descriptions are like as those of a line — the “y = 3*x – 2” equation — or a simple shape — “x2 + y2 = 4” is a circle — in that we can turn them into graphs right away without having to process them, at least not once we’re familiar and comfortable with the idea of graphing. Some of these descriptions are going to be in awkward forms. “x + 2 = – y2 / x + 2 y /x” is really just an awkward way to describe a circle (more or less), but that shape is hidden in the writing.

## Before Drawing a Graph

I want to talk about drawing graphs, specifically, drawing curves on graphs. We know roughly what’s meant by that: it’s about wiggly shapes with a faint rectangular grid, usually in grey or maybe drawn in dotted lines, behind them. Sometimes the wiggly shapes will be in bright colors, to clarify a complicated figure or to justify printing the textbook in color. Those graphs.

I clarify because there is a type of math called graph theory in which, yes, you might draw graphs, but there what’s meant by a graph is just any sort of group of points, called vertices, connected by lines or curves. It makes great sense as a name, but it’s not what what someone who talks about drawing a graph means, up until graph theory gets into consideration. Those graphs are fun, particularly because they’re insensitive to exactly where the vertices are, so you get to exercise some artistic talent instead of figuring out whatever you were trying to prove in the problem.

The ordinary kind of graphs offer some wonderful advantages. The obvious one is that they’re pictures. People can very often understand a picture of something much faster than they can understand other sorts of descriptions. This probably doesn’t need any demonstration; if it does, try looking at a map of the boundaries of South Carolina versus reading a description of its boundaries. Some problems are much easier to work out if we can approach it as a geometric problem. (And I admit feeling a particular delight when I can prove a problem geometrically; it feels cleverer.)

## In Case Of Sudden Failure Of Planet Earth

Have you ever figured out just exactly what you would do if the Earth were to suddenly disappear from the universe, leaving just you and whatever’s around to fall towards whatever the nearest heavenly bodies are? No, me neither. Asked to improvise one, I suppose I’d suffocate within minutes and then everything else becomes not so interesting to me, although possibly my heirs might be interested, if they’re somewhere.

I did double-check, though, that she meant the gravitational pull of the Moon, rather than its tidal pull. The shorthand reason for this is that arguments for astrology having some physical basis tend to run along the lines of, the Moon creates the tides (the Sun does too, but smaller ones), tides are made of water (rock moves, too, although much less), human bodies are mostly water (I don’t know what the fluid properties of cytoplasm are, but I’m almost curious enough to look them up), so there must be something tide-like in human bodies too (so there). The gravitational pull of the Moon, meanwhile, doesn’t really mean much: the Moon is going to accelerate the Earth and the people standing on it by just about the same amount. The force of gravity between two objects grows with the two objects’ masses, and the Earth is more massive than any person on it. But this means the Earth feels a greater force pulling it towards the Moon, and the acceleration works out tobe just the same. The force of gravity between two objects falls off as the square of the distance between them, and the people on the surface of the Earth are a little bit closer or a little bit farther away from the Moon than the center of the Earth is, but that’s not very different considering just how far away the Moon is. We spend all our lives falling into the Moon, as fast as we possibly can, and we are falling into the Moon as fast as the Earth is.

• #### Chiaroscuro 5:27 am on Tuesday, 22 November, 2011 Permalink | Reply

I’m curious to see how you take this. A big question would be, what dissappears when the planet vanishes? oceans? Atmosphere? are we keeping just people, or the world’s whole biomass?

–Chi

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• #### nebusresearch 6:00 am on Tuesday, 22 November, 2011 Permalink | Reply

It strikes me there are many possible ways to take this. For this particular column, at least, it’s the planet and all its surroundings except for the two people who had been in the same room who vanished; the rest of the biosphere and everything within it is elsewhere, wherever that should be.

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• #### CogitoErgoCogitoSum 1:23 pm on Wednesday, 23 November, 2011 Permalink | Reply

If the Earth suddenly disappeared, the moon would no longer be gravitationally locked to this location. Like letting go of a swinging weight tied to a string. No doubt the moon would accelerate away from us at a much faster speed than us to it.

Anyway, I didnt quite follow you at some point. Did you mean that the moon pulls on our bodies with much more force than a person standing next to us? Sorry, your choice of words and grammar seemed to capture a somewhat ambiguous meaning for me. There was no doubt in my mind that the moon pulls on us more strongly than my buddy here pulls on me. Can you compute those same numbers for Jupiter?

If these astronomical bodies do in fact pull us more strongly, then why would the proposition that astrology is valid be tossed out so readily? I wonder? People like to chalk up to genetics and upbringing all the crappy personality traits they have, but they credit themselves for strife and effort when citing virtuous ones. People scoff at “destiny” and “fate” being controlling factors in their lives, but not chemistry. Physics is okay, but the effects of the development of the brains neurological connections due to gravity while a fetus is… wait, is that physics or astrology, I cant tell?

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• #### nebusresearch 6:48 pm on Friday, 25 November, 2011 Permalink | Reply

Well, one big reason for supposing that the positions of planets don’t affect personality traits is that there’s just about no difference between any two people in the gravitational influences they feel from the outer planets. That would suggest if a planet’s position does affect development, everybody should be affected by about the same amount and in the same way … which may happen; I suppose we won’t know unless a large enough population grows up on the Moon or Mars or somewhere else, but it’s hard to see how something which affects everybody to the same amount could make people very different.

The relative gravitational acceleration from Jupiter is an interesting follow-up question and I am going to follow it up with otherworldly data soon, though. The results are neat.

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