I just learned of this cognitive experiment that dates back to 1966.
In brief, it consists of showing someone the following cards:
3 8 A B
and asking which cards need to be flipped to see if they follow this rule:
"if a card shows an even number on one face, the other face shows a vowel".
Less than 10% in the original study answered correctly. But in 1983, it was found people tend to get it correct if the problem is presented with cards like these:
15 25 beer coke
and are asked which cards need to be flipped to determine if they follow this rule:
"if a card shows alcohol on one face, the other face shows a number 18 or higher".
Thursday, January 12, 2017
Saturday, October 22, 2016
most irrational number
Today I was first exposed to the idea that, in a sense, we can say a number is "more irrational" than another... and even that there is a "most irrational number". The idea is to ask, "how well can we approximate the irrational with a simple rational?" We can approximate as close as we want, if we're willing to have huge numerators and denominators. But I can see that some irrationals will be approximated very well with a "small" fraction, and others won't. The ones that aren't are "more irrational".
This gets quantified by analyzing continuous fractions, and it turns out that ϕ (the golden ratio, or (1 + √5)/2) is the "most irrational". For more, see David Rutter's answer here.
This gets quantified by analyzing continuous fractions, and it turns out that ϕ (the golden ratio, or (1 + √5)/2) is the "most irrational". For more, see David Rutter's answer here.
Sunday, October 9, 2016
kinematic vector patterns for circular movement
When we consider a point moving in a circle at constant speed, we usually limit the discussion to position, velocity, and acceleration. But if we keep going (that is, consider the derivative of acceleration, and the derivative of that, etc) a simple pattern emerges.
A quick review:
The position vector points from the circle's center to the moving point.
The velocity vector (which is the derivative of the position vector) is at a right angle to the position vector, pointing in the direction of motion.
The acceleration vector (which is the derivative of the velocity vector) is at a right angle to the velocity vector, pointing toward the circle's center.
Note that the position and velocity vectors have constant magnitude but changing direction. The acceleration vector is no different; as the point moves in the circle, the acceleration vector changes direction to keep pointing to the circle's center. So we can ask about the jerk vector (which is the derivative of the acceleration vector). The jerk vector is at a right angle to the acceleration vector, and points opposite the velocity vector.
Notice that each successive derivative vector is rotated 90° from the preceding vector. If the moving point is rotating, say, clockwise, each successive derivative vector is rotated 90° clockwise. We can keep taking derivatives as much we want, and each one will be rotated another 90° from the last one.
A little analysis reveals that this falls out from the derivatives of sine and cosine. Just like taking the derivative of sine 4 times brings you back to sine, rotating a vector 90° either clockwise or clockwise 4 times will bring the vector back to its original direction.
----
For reference, the successive derivatives of position are:
velocity
acceleration
jerk (or: jolt (UK), surge, lurch)
jounce (or snap)
[no concensus]
Sunday, June 12, 2016
radioactive dating
When we know all of these:
I should have gotten a handle on this a long time ago.
- a rock contains A
- A decays into B
- A's half-life
- and both of these:
- how much A was in the rock when it formed
- how much A is in it now
- or one from the above list, and both of these:
- how much B was in the rock when it formed
- how much B is in it now
I should have gotten a handle on this a long time ago.
- some minerals strongly reject B, so we can be reasonably sure that all B found in such a mineral is from decay.
- in some materials, A and B would separate when forming (eg, from settling), so that all B found at A would be from decay.
- In the case of A being $C^{14}$, A is continuously generated by cosmic rays. So when we know all the carbon found in a rock, we know that a certain fraction of it was A ($C^{14}$).
Thursday, June 9, 2016
Derivative and Integral: the play
[Background: Given a graph, I can visualize its derivative much more easily than I can visualize any of its antiderivatives. Time for that to change, even if I have to resort to something as ridiculous as giving those operations personalities.]
Suppose a graph represents how good the day is going versus time. Positive values mean a good day; negative values mean a bad day. A derivative is like a person who only cares about whether things are getting better or worse, and whose mood is based solely on that direction. It doesn't care about the past. It doesn't care about how good things actually are right now. The integral is like a person whose mood gets gradually better or worse depending solely on how good things are in the moment. It's affected very much by the past. It doesn't care if things are improving or worsening.
Let's graph these characters' moods over a day that has its up and downs, as represented by the graph of cosine. Because Integral's mood depends on his past mood, I pick an arbitrary starting mood for him at 0.
Derivative: If you think so, why aren't you happier? Your happiness is at 0.
Integral: Oh, I'm getting happy very quickly! Look how quickly my mood is climbing! Besides, you're at 0 also.
Derivative: But I'm at 0 because the day has stagnated. Who knows what the future holds...
Integral: Pessimist!
Derivative: It's all downhill from here. At this rate, we'll start having a bad day.
Integral: Just enjoy the present! It's still a great day!
Derivative: Well, I notice you're not getting happier quite as fast.
Integral: Because it's not quite as nice a day. But it's still pretty great!
Integral: You're so worried about the future that you can't enjoy the present. I'm getting happier all the time!
Derivative: Yes, but not nearly as quickly.
Integral: Who cares? The day's still good, and I'm still getting happier.
Derivative: You do realize you're coasting, right? The day's no longer good, and you're not getting happier.
Integral: I'm high from all the good stuff that just happened. I can't understand why you're so depressed.
Derivative: Isn't it obvious? This is the fastest rate at which the day has declined!
Integral: Why do I even talk to you?
Derivative: Well, I don't feel quite as bad. The day's not declining as quickly.
Integral: Yeah, well the last part of this day sucked so much, it wiped out all my happiness from the first part. I'm no longer happy. This could start to irritate me.
Integral: Shut up. It sucks. I'm feeling worse and worse.
Derivative: But look at the trend! It's improving.
Integral: You're even more irritating when you're happy.
Integral: We just came through so much suck. Twice as much suck as good stuff, in fact. This is misery.
Integral: Yes, so well that it's pulling me out of this bad mood. But you're not as happy.
Derivative: I'm still happy, just not quite as much, because things aren't looking up quite as much.
Suppose a graph represents how good the day is going versus time. Positive values mean a good day; negative values mean a bad day. A derivative is like a person who only cares about whether things are getting better or worse, and whose mood is based solely on that direction. It doesn't care about the past. It doesn't care about how good things actually are right now. The integral is like a person whose mood gets gradually better or worse depending solely on how good things are in the moment. It's affected very much by the past. It doesn't care if things are improving or worsening.
Let's graph these characters' moods over a day that has its up and downs, as represented by the graph of cosine. Because Integral's mood depends on his past mood, I pick an arbitrary starting mood for him at 0.
t=0
Integral: Wow, what a day, huh? This is tops!Derivative: If you think so, why aren't you happier? Your happiness is at 0.
Integral: Oh, I'm getting happy very quickly! Look how quickly my mood is climbing! Besides, you're at 0 also.
Derivative: But I'm at 0 because the day has stagnated. Who knows what the future holds...
Integral: Pessimist!
t=0.6
Integral: Why so negative?Derivative: It's all downhill from here. At this rate, we'll start having a bad day.
Integral: Just enjoy the present! It's still a great day!
Derivative: Well, I notice you're not getting happier quite as fast.
Integral: Because it's not quite as nice a day. But it's still pretty great!
t=1.2
Derivative: Wow, the day's really taken a turn for the worst, hasn't it? I'm so bummed.Integral: You're so worried about the future that you can't enjoy the present. I'm getting happier all the time!
Derivative: Yes, but not nearly as quickly.
Integral: Who cares? The day's still good, and I'm still getting happier.
t=π/2 (about 1.6)
Integral: Well, it doesn't get better than this! I'm so happy!Derivative: You do realize you're coasting, right? The day's no longer good, and you're not getting happier.
Integral: I'm high from all the good stuff that just happened. I can't understand why you're so depressed.
Derivative: Isn't it obvious? This is the fastest rate at which the day has declined!
Integral: Why do I even talk to you?
t=2.5
Integral: I'm still happy, but I have to admit the day's sucked lately. It's bringing me down.Derivative: Well, I don't feel quite as bad. The day's not declining as quickly.
t=π (about 3.14)
Derivative: Did you notice? The day stopped getting worse. I'm no longer sad. If things start looking up, I'll actually be happy.Integral: Yeah, well the last part of this day sucked so much, it wiped out all my happiness from the first part. I'm no longer happy. This could start to irritate me.
t=4
Derivative: Hey, the day's finally headed in the right direction. This is great!Integral: Shut up. It sucks. I'm feeling worse and worse.
Derivative: But look at the trend! It's improving.
Integral: You're even more irritating when you're happy.
t=3π/2 (about 4.7)
Derivative: Wow, things are really looking up! I couldn't be happier!Integral: We just came through so much suck. Twice as much suck as good stuff, in fact. This is misery.
t=5.5
Derivative: You have to admit, things have been good lately.Integral: Yes, so well that it's pulling me out of this bad mood. But you're not as happy.
Derivative: I'm still happy, just not quite as much, because things aren't looking up quite as much.
t=2π: repeat
Wednesday, June 8, 2016
GPS error
Previously posted in a Facebook discussion. I thought I'd archive it here.
Article: Due to statistical bias, GPS distance estimates tend to always be about 10-20% higher than is correct. (http://www.i-programmer.info/news/145-mapping-a-gis/9164-gps-always-over-estimates-distances.html)
My comments:
This is surprising and counter-intuitive. I didn't follow the details of the explanation, so I did my own test: take points A and B, and place each in the center of a 3x3 grid. This grid represents possible error (each cell representing a place where the point might be measured), so there are 81 possible ways to measure the distance between A and B. The result: 27/81 (or 1/3) chance of under-measuring the distance; 9/81 (or 1/9) chance of measuring an equivalent distance; 45/81 (or 5/9) chance of over-measuring distance. How about that!
--------------------
As I worked through it, I could see the above falls out of the fact that at small angles, sin(θ) changes more quickly than cos(θ).
--------------------
I thought of a way to think about this for a continuous range of error. Say the true distance between A and B is D. B will be erroneously taken to be at a random point on a circle of radius R centered on B. Call this circle B-false.
Draw the circle of radius D centered at A. This divides B-false into 2 arcs. The smaller arc is where B can be falsely placed and result in a smaller measured distance between A and B. The larger arc, where B is more likely to be taken to be, results in a larger measured distance between A and B.
Article: Due to statistical bias, GPS distance estimates tend to always be about 10-20% higher than is correct. (http://www.i-programmer.info/news/145-mapping-a-gis/9164-gps-always-over-estimates-distances.html)
My comments:
This is surprising and counter-intuitive. I didn't follow the details of the explanation, so I did my own test: take points A and B, and place each in the center of a 3x3 grid. This grid represents possible error (each cell representing a place where the point might be measured), so there are 81 possible ways to measure the distance between A and B. The result: 27/81 (or 1/3) chance of under-measuring the distance; 9/81 (or 1/9) chance of measuring an equivalent distance; 45/81 (or 5/9) chance of over-measuring distance. How about that!
--------------------
As I worked through it, I could see the above falls out of the fact that at small angles, sin(θ) changes more quickly than cos(θ).
--------------------
I thought of a way to think about this for a continuous range of error. Say the true distance between A and B is D. B will be erroneously taken to be at a random point on a circle of radius R centered on B. Call this circle B-false.
Draw the circle of radius D centered at A. This divides B-false into 2 arcs. The smaller arc is where B can be falsely placed and result in a smaller measured distance between A and B. The larger arc, where B is more likely to be taken to be, results in a larger measured distance between A and B.
Tuesday, June 7, 2016
calculus really likes radians
I remember being introduced to radians by my teacher: "If we met an alien civilization, they wouldn't be using degrees. Why would they divide a circle into 360 parts? There's no mathematical reason to do that. But everyone would know to use an arc length equal to the radius. The radius is universal." That made sense to me. There's nothing arbitrary about the radius!
Or is there? Why not use a unit that's the same length as the diameter? I'll call it the "diametan". So whereas 1 radian is the angle needed to make an arc whose length is as long as the radius, 1 diametan is the angle needed to make an arc whose length is as long as the diameter. (An angle of 1 diametan = 2 radians.)
So... radians, diametans... one sounded as good as the other to me. Until I graphed them, and thought about their derivatives.
Compared to the graph of sin(radians) (which peaks at π/2), the graph of sin(diametans) will be squashed (it peaks at π/4). This will mess up its derivative (and its anti-derivative, and its derivative's derivative, etc), instead of keeping the nice alternations between ±sin and ±cos.
Perhaps this should have been obvious to me; the same problem is to be seen with the graph of sin(degrees) (but instead of being squashed, it's stretched).
What makes radians special is the same thing that makes e special: they balance things just right such that derivatives (or anti-derivatives) find their way back to the starting function.
I got on this line of thought by working through the derivation of the Maclaurin series for sin x, and wondering what would change if the trig functions expected diametans instead of radians.
Or is there? Why not use a unit that's the same length as the diameter? I'll call it the "diametan". So whereas 1 radian is the angle needed to make an arc whose length is as long as the radius, 1 diametan is the angle needed to make an arc whose length is as long as the diameter. (An angle of 1 diametan = 2 radians.)
So... radians, diametans... one sounded as good as the other to me. Until I graphed them, and thought about their derivatives.
Compared to the graph of sin(radians) (which peaks at π/2), the graph of sin(diametans) will be squashed (it peaks at π/4). This will mess up its derivative (and its anti-derivative, and its derivative's derivative, etc), instead of keeping the nice alternations between ±sin and ±cos.
Perhaps this should have been obvious to me; the same problem is to be seen with the graph of sin(degrees) (but instead of being squashed, it's stretched).
What makes radians special is the same thing that makes e special: they balance things just right such that derivatives (or anti-derivatives) find their way back to the starting function.
I got on this line of thought by working through the derivation of the Maclaurin series for sin x, and wondering what would change if the trig functions expected diametans instead of radians.
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