**nhaliday + mental-math**
32

exponential function - Feynman's Trick for Approximating $e^x$ - Mathematics Stack Exchange

25 days ago by nhaliday

1. e^2.3 ~ 10

2. e^.7 ~ 2

3. e^x ~ 1+x

e = 2.71828...

errors (absolute, relative):

1. +0.0258, 0.26%

2. -0.0138, -0.68%

3. 1 + x approximates e^x on [-.3, .3] with absolute error < .05, and relative error < 5.6% (3.7% for [0, .3]).

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2. e^.7 ~ 2

3. e^x ~ 1+x

e = 2.71828...

errors (absolute, relative):

1. +0.0258, 0.26%

2. -0.0138, -0.68%

3. 1 + x approximates e^x on [-.3, .3] with absolute error < .05, and relative error < 5.6% (3.7% for [0, .3]).

25 days ago by nhaliday

Zero-based numbering - Wikipedia

july 2019 by nhaliday

https://www.cs.utexas.edu/users/EWD/transcriptions/EWD08xx/EWD831.html

https://softwareengineering.stackexchange.com/questions/110804/why-are-zero-based-arrays-the-norm

idk about this guy's competence. he seems to want to sensationalize the historical reasons while ignoring or being ignorant of the legitimate mathematical reasons:

http://exple.tive.org/blarg/2013/10/22/citation-needed/

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https://softwareengineering.stackexchange.com/questions/110804/why-are-zero-based-arrays-the-norm

idk about this guy's competence. he seems to want to sensationalize the historical reasons while ignoring or being ignorant of the legitimate mathematical reasons:

http://exple.tive.org/blarg/2013/10/22/citation-needed/

july 2019 by nhaliday

Introduction to Scaling Laws

august 2017 by nhaliday

https://betadecay.wordpress.com/2009/10/02/the-physics-of-scaling-laws-and-dimensional-analysis/

http://galileo.phys.virginia.edu/classes/304/scaling.pdf

Galileo’s Discovery of Scaling Laws: https://www.mtholyoke.edu/~mpeterso/classes/galileo/scaling8.pdf

Days 1 and 2 of Two New Sciences

An example of such an insight is “the surface of a small solid is comparatively greater than that of a large one” because the surface goes like the square of a linear dimension, but the volume goes like the cube.5 Thus as one scales down macroscopic objects, forces on their surfaces like viscous drag become relatively more important, and bulk forces like weight become relatively less important. Galileo uses this idea on the First Day in the context of resistance in free fall, as an explanation for why similar objects of different size do not fall exactly together, but the smaller one lags behind.

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http://galileo.phys.virginia.edu/classes/304/scaling.pdf

Galileo’s Discovery of Scaling Laws: https://www.mtholyoke.edu/~mpeterso/classes/galileo/scaling8.pdf

Days 1 and 2 of Two New Sciences

An example of such an insight is “the surface of a small solid is comparatively greater than that of a large one” because the surface goes like the square of a linear dimension, but the volume goes like the cube.5 Thus as one scales down macroscopic objects, forces on their surfaces like viscous drag become relatively more important, and bulk forces like weight become relatively less important. Galileo uses this idea on the First Day in the context of resistance in free fall, as an explanation for why similar objects of different size do not fall exactly together, but the smaller one lags behind.

august 2017 by nhaliday

Kelly criterion - Wikipedia

august 2017 by nhaliday

In probability theory and intertemporal portfolio choice, the Kelly criterion, Kelly strategy, Kelly formula, or Kelly bet, is a formula used to determine the optimal size of a series of bets. In most gambling scenarios, and some investing scenarios under some simplifying assumptions, the Kelly strategy will do better than any essentially different strategy in the long run (that is, over a span of time in which the observed fraction of bets that are successful equals the probability that any given bet will be successful). It was described by J. L. Kelly, Jr, a researcher at Bell Labs, in 1956.[1] The practical use of the formula has been demonstrated.[2][3][4]

The Kelly Criterion is to bet a predetermined fraction of assets and can be counterintuitive. In one study,[5][6] each participant was given $25 and asked to bet on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at $250. Behavior was far from optimal. "Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment." Using the Kelly criterion and based on the odds in the experiment, the right approach would be to bet 20% of the pot on each throw (see first example in Statement below). If losing, the size of the bet gets cut; if winning, the stake increases.

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The Kelly Criterion is to bet a predetermined fraction of assets and can be counterintuitive. In one study,[5][6] each participant was given $25 and asked to bet on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at $250. Behavior was far from optimal. "Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment." Using the Kelly criterion and based on the odds in the experiment, the right approach would be to bet 20% of the pot on each throw (see first example in Statement below). If losing, the size of the bet gets cut; if winning, the stake increases.

august 2017 by nhaliday

The Flynn effect for verbal and visuospatial short-term and working memory: A cross-temporal meta-analysis

august 2017 by nhaliday

Specifically, the Flynn effect was found for forward digit span (r = 0.12, p < 0.01) and forward Corsi block span (r = 0.10, p < 0.01). Moreover, an anti-Flynn effect was found for backward digit span (r = − 0.06, p < 0.01) and for backward Corsi block span (r = − 0.17, p < 0.01). Overall, the results support co-occurrence theories that predict simultaneous secular gains in specialized abilities and declines in g. The causes of the differential trajectories are further discussed.

http://www.unz.com/jthompson/working-memory-bombshell/

https://www.newscientist.com/article/2146752-we-seem-to-be-getting-stupider-and-population-ageing-may-be-why/

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http://www.unz.com/jthompson/working-memory-bombshell/

https://www.newscientist.com/article/2146752-we-seem-to-be-getting-stupider-and-population-ageing-may-be-why/

august 2017 by nhaliday

Interstellar travel - Wikipedia

july 2017 by nhaliday

https://en.wikipedia.org/wiki/Space_travel_using_constant_acceleration

https://twitter.com/gcochran99/status/886766460365832192

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https://twitter.com/gcochran99/status/886766460365832192

july 2017 by nhaliday

A sense of where you are | West Hunter

may 2017 by nhaliday

Nobody at the Times noticed it at first. I don’t know that they ever did notice it by themselves- likely some reader brought it to their attention. But this happens all the time, because very few people have a picture of the world in their head that includes any numbers. Mostly they don’t even have a rough idea of relative size.

In much the same way, back in the 1980s,lots of writers were talking about 90,000 women a year dying of anorexia nervosa, another impossible number. Then there was the great scare about 1,000,000 kids being kidnapped in the US each year – also impossible and wrong. Practically all the talking classes bought into it.

You might think that the people at the top are different – but with a few exceptions, they’re just as clueless.

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In much the same way, back in the 1980s,lots of writers were talking about 90,000 women a year dying of anorexia nervosa, another impossible number. Then there was the great scare about 1,000,000 kids being kidnapped in the US each year – also impossible and wrong. Practically all the talking classes bought into it.

You might think that the people at the top are different – but with a few exceptions, they’re just as clueless.

may 2017 by nhaliday

Pearson correlation coefficient - Wikipedia

may 2017 by nhaliday

https://en.wikipedia.org/wiki/Coefficient_of_determination

what does this mean?: https://twitter.com/GarettJones/status/863546692724858880

deleted but it was about the Pearson correlation distance: 1-r

I guess it's a metric

https://en.wikipedia.org/wiki/Explained_variation

http://infoproc.blogspot.com/2014/02/correlation-and-variance.html

A less misleading way to think about the correlation R is as follows: given X,Y from a standardized bivariate distribution with correlation R, an increase in X leads to an expected increase in Y: dY = R dX. In other words, students with +1 SD SAT score have, on average, roughly +0.4 SD college GPAs. Similarly, students with +1 SD college GPAs have on average +0.4 SAT.

this reminds me of the breeder's equation (but it uses r instead of h^2, so it can't actually be the same)

https://www.reddit.com/r/slatestarcodex/comments/631haf/on_the_commentariat_here_and_why_i_dont_think_i/dfx4e2s/

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what does this mean?: https://twitter.com/GarettJones/status/863546692724858880

deleted but it was about the Pearson correlation distance: 1-r

I guess it's a metric

https://en.wikipedia.org/wiki/Explained_variation

http://infoproc.blogspot.com/2014/02/correlation-and-variance.html

A less misleading way to think about the correlation R is as follows: given X,Y from a standardized bivariate distribution with correlation R, an increase in X leads to an expected increase in Y: dY = R dX. In other words, students with +1 SD SAT score have, on average, roughly +0.4 SD college GPAs. Similarly, students with +1 SD college GPAs have on average +0.4 SAT.

this reminds me of the breeder's equation (but it uses r instead of h^2, so it can't actually be the same)

https://www.reddit.com/r/slatestarcodex/comments/631haf/on_the_commentariat_here_and_why_i_dont_think_i/dfx4e2s/

may 2017 by nhaliday

Learning mathematics in a visuospatial format: A randomized, controlled trial of mental abacus instruction

march 2017 by nhaliday

We asked whether MA improves students’ mathematical abilities, and whether expertise – which requires sustained practice of mental imagery – is driven by changes to basic cognitive capacities like working memory. MA students improved on arithmetic tasks relative to controls, but training was not associated with changes to basic cognitive abilities. Instead, differences in spatial working memory at the beginning of the study mediated MA learning. We conclude that MA expertise can be achieved by many children in standard classrooms and results from efficient use of pre-existing abilities.

Cohen’s d = .60 (95% CI: .30 - .89) for arithmetic, .24 (-.05 - .52) for WJ-III, and .28 (.00 - .57) for place value

This finding suggests that the development of MA expertise is mediated by children’s pre-existing cognitive abilities, and thus that MA may not be suitable for all K-12 classroom environments, especially in groups of children who have low spatial working memory or attentional capacities (which may have been the case in our study). Critically, this does not mean that MA expertise depends on unusually strong cognitive abilities. Perhaps because we studied children from relatively disadvantaged backgrounds, few Mental Abacus 21 children in our sample had SWM capacities comparable to those seen among typical children in the United States.

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Cohen’s d = .60 (95% CI: .30 - .89) for arithmetic, .24 (-.05 - .52) for WJ-III, and .28 (.00 - .57) for place value

This finding suggests that the development of MA expertise is mediated by children’s pre-existing cognitive abilities, and thus that MA may not be suitable for all K-12 classroom environments, especially in groups of children who have low spatial working memory or attentional capacities (which may have been the case in our study). Critically, this does not mean that MA expertise depends on unusually strong cognitive abilities. Perhaps because we studied children from relatively disadvantaged backgrounds, few Mental Abacus 21 children in our sample had SWM capacities comparable to those seen among typical children in the United States.

march 2017 by nhaliday

Sir Ronald Aylmer Fisher | West Hunter

january 2017 by nhaliday

In 1930 he published The Genetical Theory of Natural Selection, which completed the fusion of Darwinian natural selection with Mendelian inheritance. James Crow said that it was ‘arguably the deepest and most influential book on evolution since Darwin’. In it, Fisher analyzed sexual selection, mimicry, and sex ratios, where he made some of the first arguments using game theory. The book touches on many other topics. As was the case with his other works, The Genetical Theory is a dense book, not easy for most people to understand. Fisher’s tendency to leave out mathematical steps that he deemed obvious (a leftover from his early training in mental mathematics) frustrates many readers.

The Genetical Theory is of particular interest to us because Fisher there lays out his ideas on how population size can speed up evolution. As we explain elsewhere, more individuals mean there will be more mutations, including favorable mutations, and so Fisher expected more rapid evolution in larger populations. This idea was originally suggested, in a nonmathematical way, in Darwin’s Origin of Species.

Although Fisher was fiercely loyal to friends and could be very charming, he had a quick temper and was a fine hater. The same uncompromising spirit that fostered his originality led to constant conflict with authority. He had a long conflict with Karl Pearson, who had also played an important part in the development of mathematical statistics. In this case, Pearson was more at fault, resisting the advent of a more talented competitor, as well as being an eminently hateable person in general. Over time Fisher also became increasing angry at Sewall Wright (another one of the founders of population genetics) due to scientific disagreements – and this was just wrong, because Wright was a sweetheart.

Fisher’s personality decreased his potential influence. He was not a school-builder, and was impatient with administrators. He expected to find some form of war-work in the Second World War, but his characteristics had alienated too many people, and thus his team dispersed to other jobs during the war. He returned to Rothamsted for the duration. This was a difficult time for him: his marriage disintegrated and his oldest son, an RAF pilot, was killed in the war.

...

Fisher’s ideas in genetics have taken an odd path. The Genetical Theory was not widely read, sold few copies, and has never been translated. Only gradually did its ideas find an audience. Of course, that audience included people like Bill Hamilton, the greatest mathematical biologist of the last half of the 20th century, who was strongly influenced by Fisher’s work. Hamilton said “By the time of my ultimate graduation,will I have understood all that is true in this book and will I get a First? I doubt it. In some ways some of us have overtaken Fisher; in many, however, this brilliant, daring man is still far in front.“

In fact, over the past generation, much of Fisher’s work has been neglected – in the sense that interest in population genetics has decreased (particularly interest in selection) and fewer students are exposed to his work in genetics in any way. Ernst Mayr didn’t even mention Fisher in his 1991 book One Long Argument: Charles Darwin and the Genesis of Modern Evolutionary Thought, while Stephen Jay Gould, in The Structure of Evolutionary Theory, gave Fisher 6 pages out of 1433. Of course Mayr and Gould were both complete chuckleheads.

Fisher’s work affords continuing insight, including important implications concerning human evolution that have emerged more than 50 years after his death. We strongly discourage other professionals from learning anything about his ideas.

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The Genetical Theory is of particular interest to us because Fisher there lays out his ideas on how population size can speed up evolution. As we explain elsewhere, more individuals mean there will be more mutations, including favorable mutations, and so Fisher expected more rapid evolution in larger populations. This idea was originally suggested, in a nonmathematical way, in Darwin’s Origin of Species.

Although Fisher was fiercely loyal to friends and could be very charming, he had a quick temper and was a fine hater. The same uncompromising spirit that fostered his originality led to constant conflict with authority. He had a long conflict with Karl Pearson, who had also played an important part in the development of mathematical statistics. In this case, Pearson was more at fault, resisting the advent of a more talented competitor, as well as being an eminently hateable person in general. Over time Fisher also became increasing angry at Sewall Wright (another one of the founders of population genetics) due to scientific disagreements – and this was just wrong, because Wright was a sweetheart.

Fisher’s personality decreased his potential influence. He was not a school-builder, and was impatient with administrators. He expected to find some form of war-work in the Second World War, but his characteristics had alienated too many people, and thus his team dispersed to other jobs during the war. He returned to Rothamsted for the duration. This was a difficult time for him: his marriage disintegrated and his oldest son, an RAF pilot, was killed in the war.

...

Fisher’s ideas in genetics have taken an odd path. The Genetical Theory was not widely read, sold few copies, and has never been translated. Only gradually did its ideas find an audience. Of course, that audience included people like Bill Hamilton, the greatest mathematical biologist of the last half of the 20th century, who was strongly influenced by Fisher’s work. Hamilton said “By the time of my ultimate graduation,will I have understood all that is true in this book and will I get a First? I doubt it. In some ways some of us have overtaken Fisher; in many, however, this brilliant, daring man is still far in front.“

In fact, over the past generation, much of Fisher’s work has been neglected – in the sense that interest in population genetics has decreased (particularly interest in selection) and fewer students are exposed to his work in genetics in any way. Ernst Mayr didn’t even mention Fisher in his 1991 book One Long Argument: Charles Darwin and the Genesis of Modern Evolutionary Thought, while Stephen Jay Gould, in The Structure of Evolutionary Theory, gave Fisher 6 pages out of 1433. Of course Mayr and Gould were both complete chuckleheads.

Fisher’s work affords continuing insight, including important implications concerning human evolution that have emerged more than 50 years after his death. We strongly discourage other professionals from learning anything about his ideas.

january 2017 by nhaliday

soft question - Thinking and Explaining - MathOverflow

january 2017 by nhaliday

- good question from Bill Thurston

- great answers by Terry Tao, fedja, Minhyong Kim, gowers, etc.

Terry Tao:

- symmetry as blurring/vibrating/wobbling, scale invariance

- anthropomorphization, adversarial perspective for estimates/inequalities/quantifiers, spending/economy

fedja walks through his though-process from another answer

Minhyong Kim: anthropology of mathematical philosophizing

Per Vognsen: normality as isotropy

comment: conjugate subgroup gHg^-1 ~ "H but somewhere else in G"

gowers: hidden things in basic mathematics/arithmetic

comment by Ryan Budney: x sin(x) via x -> (x, sin(x)), (x, y) -> xy

I kinda get what he's talking about but needed to use Mathematica to get the initial visualization down.

To remind myself later:

- xy can be easily visualized by juxtaposing the two parabolae x^2 and -x^2 diagonally

- x sin(x) can be visualized along that surface by moving your finger along the line (x, 0) but adding some oscillations in y direction according to sin(x)

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cool
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- great answers by Terry Tao, fedja, Minhyong Kim, gowers, etc.

Terry Tao:

- symmetry as blurring/vibrating/wobbling, scale invariance

- anthropomorphization, adversarial perspective for estimates/inequalities/quantifiers, spending/economy

fedja walks through his though-process from another answer

Minhyong Kim: anthropology of mathematical philosophizing

Per Vognsen: normality as isotropy

comment: conjugate subgroup gHg^-1 ~ "H but somewhere else in G"

gowers: hidden things in basic mathematics/arithmetic

comment by Ryan Budney: x sin(x) via x -> (x, sin(x)), (x, y) -> xy

I kinda get what he's talking about but needed to use Mathematica to get the initial visualization down.

To remind myself later:

- xy can be easily visualized by juxtaposing the two parabolae x^2 and -x^2 diagonally

- x sin(x) can be visualized along that surface by moving your finger along the line (x, 0) but adding some oscillations in y direction according to sin(x)

january 2017 by nhaliday

Important z-scores

november 2016 by nhaliday

hmm:

https://twitter.com/davidshor/status/888441370247090176

https://archive.is/BiPxf

Friends don't let friends plot 95% confidence intervals

There is a 98.3% chance that b>a, even through their 95% oonfsdenoe intervals overlap

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https://twitter.com/davidshor/status/888441370247090176

https://archive.is/BiPxf

Friends don't let friends plot 95% confidence intervals

There is a 98.3% chance that b>a, even through their 95% oonfsdenoe intervals overlap

november 2016 by nhaliday

Shut Up And Guess - Less Wrong

september 2016 by nhaliday

At what confidence level do you guess? At what confidence level do you answer "don't know"?

I took several of these tests last month, and the first thing I did was some quick mental calculations. If I have zero knowledge of a question, my expected gain from answering is 50% probability of earning one point and 50% probability of losing one half point. Therefore, my expected gain from answering a question is .5(1)-.5(.5)= +.25 points. Compare this to an expected gain of zero from not answering the question at all. Therefore, I ought to guess on every question, even if I have zero knowledge. If I have some inkling, well, that's even better.

You look disappointed. This isn't a very exciting application of arcane Less Wrong knowledge. Anyone with basic math skills should be able to calculate that out, right?

I attend a pretty good university, and I'm in a postgraduate class where most of us have at least a bachelor's degree in a hard science, and a few have master's degrees. And yet, talking to my classmates in the cafeteria after the first test was finished, I started to realize I was the only person in the class who hadn't answered "don't know" to any questions.

even more interesting stories in the comments

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I took several of these tests last month, and the first thing I did was some quick mental calculations. If I have zero knowledge of a question, my expected gain from answering is 50% probability of earning one point and 50% probability of losing one half point. Therefore, my expected gain from answering a question is .5(1)-.5(.5)= +.25 points. Compare this to an expected gain of zero from not answering the question at all. Therefore, I ought to guess on every question, even if I have zero knowledge. If I have some inkling, well, that's even better.

You look disappointed. This isn't a very exciting application of arcane Less Wrong knowledge. Anyone with basic math skills should be able to calculate that out, right?

I attend a pretty good university, and I'm in a postgraduate class where most of us have at least a bachelor's degree in a hard science, and a few have master's degrees. And yet, talking to my classmates in the cafeteria after the first test was finished, I started to realize I was the only person in the class who hadn't answered "don't know" to any questions.

even more interesting stories in the comments

september 2016 by nhaliday

Doomsday rule - Wikipedia, the free encyclopedia

august 2016 by nhaliday

It takes advantage of each year having a certain day of the week, called the doomsday, upon which certain easy-to-remember dates fall; for example, 4/4, 6/6, 8/8, 10/10, 12/12, and the last day of February all occur on the same day of the week in any year. Applying the Doomsday algorithm involves three steps:

1. Determination of the anchor day for the century.

2. Calculation of the doomsday for the year from the anchor day.

3. Selection of the closest date out of those that always fall on the doomsday, e.g., 4/4 and 6/6, and count of the number of days (modulo 7) between that date and the date in question to arrive at the day of the week.

This technique applies to both the Gregorian calendar A.D. and the Julian calendar, although their doomsdays are usually different days of the week.

Easter date: https://en.wikipedia.org/wiki/Computus

https://www.tondering.dk/claus/cal/easter.php

*When is Easter? (Short answer)*

Easter Sunday is the first Sunday after the first full moon on or after the vernal equinox.

*When is Easter? (Long answer)*

The calculation of Easter is complicated because it is linked to (an inaccurate version of) the Hebrew calendar.

...

It was therefore decided to make Easter Sunday the first Sunday after the first full moon after vernal equinox. Or more precisely: Easter Sunday is the first Sunday after the “official” full moon on or after the “official” vernal equinox.

The official vernal equinox is always 21 March.

The official full moon may differ from the real full moon by one or two days.

...

The full moon that precedes Easter is called the Paschal full moon. Two concepts play an important role when calculating the Paschal full moon: The Golden Number and the Epact. They are described in the following sections.

...

*What is the Golden Number?*

Each year is associated with a Golden Number.

Considering that the relationship between the moon’s phases and the days of the year repeats itself every 19 years (as described in the section about astronomy), it is natural to associate a number between 1 and 19 with each year. This number is the so-called Golden Number. It is calculated thus:

GoldenNumber=(year mod 19) + 1

[link:

However, 19 tropical years is 234.997 synodic months, which is very close to an integer. So every 19 years the phases of the moon fall on the same dates (if it were not for the skewness introduced by leap years). 19 years is called a Metonic cycle (after Meton, an astronomer from Athens in the 5th century BC).

So, to summarise: There are three important numbers to note:

A tropical year is 365.24219 days.

A synodic month is 29.53059 days.

19 tropical years is close to an integral number of synodic months.]

In years which have the same Golden Number, the new moon will fall on (approximately) the same date. The Golden Number is sufficient to calculate the Paschal full moon in the Julian calendar.

...

Under the Gregorian calendar, things became much more complicated. One of the changes made in the Gregorian calendar reform was a modification of the way Easter was calculated. There were two reasons for this. First, the 19 year cycle of the phases of moon (the Metonic cycle) was known not to be perfect. Secondly, the Metonic cycle fitted the Gregorian calendar year worse than it fitted the Julian calendar year.

It was therefore decided to base Easter calculations on the so-called Epact.

*What is the Epact?*

Each year is associated with an Epact.

The Epact is a measure of the age of the moon (i.e. the number of days that have passed since an “official” new moon) on a particular date.

...

In the Julian calendar, the Epact is the age of the moon on 22 March.

In the Gregorian calendar, the Epact is the age of the moon at the start of the year.

The Epact is linked to the Golden Number in the following manner:

Under the Julian calendar, 19 years were assumed to be exactly an integral number of synodic months, and the following relationship exists between the Golden Number and the Epact:

Epact=(11 × (GoldenNumber – 1)) mod 30

...

In the Gregorian calendar reform, some modifications were made to the simple relationship between the Golden Number and the Epact.

In the Gregorian calendar the Epact should be calculated thus: [long algorithm]

...

Suppose you know the Easter date of the current year, can you easily find the Easter date in the next year? No, but you can make a qualified guess.

If Easter Sunday in the current year falls on day X and the next year is not a leap year, Easter Sunday of next year will fall on one of the following days: X–15, X–8, X+13 (rare), or X+20.

...

If you combine this knowledge with the fact that Easter Sunday never falls before 22 March and never falls after 25 April, you can narrow the possibilities down to two or three dates.

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1. Determination of the anchor day for the century.

2. Calculation of the doomsday for the year from the anchor day.

3. Selection of the closest date out of those that always fall on the doomsday, e.g., 4/4 and 6/6, and count of the number of days (modulo 7) between that date and the date in question to arrive at the day of the week.

This technique applies to both the Gregorian calendar A.D. and the Julian calendar, although their doomsdays are usually different days of the week.

Easter date: https://en.wikipedia.org/wiki/Computus

https://www.tondering.dk/claus/cal/easter.php

*When is Easter? (Short answer)*

Easter Sunday is the first Sunday after the first full moon on or after the vernal equinox.

*When is Easter? (Long answer)*

The calculation of Easter is complicated because it is linked to (an inaccurate version of) the Hebrew calendar.

...

It was therefore decided to make Easter Sunday the first Sunday after the first full moon after vernal equinox. Or more precisely: Easter Sunday is the first Sunday after the “official” full moon on or after the “official” vernal equinox.

The official vernal equinox is always 21 March.

The official full moon may differ from the real full moon by one or two days.

...

The full moon that precedes Easter is called the Paschal full moon. Two concepts play an important role when calculating the Paschal full moon: The Golden Number and the Epact. They are described in the following sections.

...

*What is the Golden Number?*

Each year is associated with a Golden Number.

Considering that the relationship between the moon’s phases and the days of the year repeats itself every 19 years (as described in the section about astronomy), it is natural to associate a number between 1 and 19 with each year. This number is the so-called Golden Number. It is calculated thus:

GoldenNumber=(year mod 19) + 1

[link:

However, 19 tropical years is 234.997 synodic months, which is very close to an integer. So every 19 years the phases of the moon fall on the same dates (if it were not for the skewness introduced by leap years). 19 years is called a Metonic cycle (after Meton, an astronomer from Athens in the 5th century BC).

So, to summarise: There are three important numbers to note:

A tropical year is 365.24219 days.

A synodic month is 29.53059 days.

19 tropical years is close to an integral number of synodic months.]

In years which have the same Golden Number, the new moon will fall on (approximately) the same date. The Golden Number is sufficient to calculate the Paschal full moon in the Julian calendar.

...

Under the Gregorian calendar, things became much more complicated. One of the changes made in the Gregorian calendar reform was a modification of the way Easter was calculated. There were two reasons for this. First, the 19 year cycle of the phases of moon (the Metonic cycle) was known not to be perfect. Secondly, the Metonic cycle fitted the Gregorian calendar year worse than it fitted the Julian calendar year.

It was therefore decided to base Easter calculations on the so-called Epact.

*What is the Epact?*

Each year is associated with an Epact.

The Epact is a measure of the age of the moon (i.e. the number of days that have passed since an “official” new moon) on a particular date.

...

In the Julian calendar, the Epact is the age of the moon on 22 March.

In the Gregorian calendar, the Epact is the age of the moon at the start of the year.

The Epact is linked to the Golden Number in the following manner:

Under the Julian calendar, 19 years were assumed to be exactly an integral number of synodic months, and the following relationship exists between the Golden Number and the Epact:

Epact=(11 × (GoldenNumber – 1)) mod 30

...

In the Gregorian calendar reform, some modifications were made to the simple relationship between the Golden Number and the Epact.

In the Gregorian calendar the Epact should be calculated thus: [long algorithm]

...

Suppose you know the Easter date of the current year, can you easily find the Easter date in the next year? No, but you can make a qualified guess.

If Easter Sunday in the current year falls on day X and the next year is not a leap year, Easter Sunday of next year will fall on one of the following days: X–15, X–8, X+13 (rare), or X+20.

...

If you combine this knowledge with the fact that Easter Sunday never falls before 22 March and never falls after 25 April, you can narrow the possibilities down to two or three dates.

august 2016 by nhaliday

Information Processing: Bounded cognition

july 2016 by nhaliday

Many people lack standard cognitive tools useful for understanding the world around them. Perhaps the most egregious case: probability and statistics, which are central to understanding health, economics, risk, crime, society, evolution, global warming, etc. Very few people have any facility for calculating risk, visualizing a distribution, understanding the difference between the average, the median, variance, etc.

Risk, Uncertainty, and Heuristics: http://infoproc.blogspot.com/2018/03/risk-uncertainty-and-heuristics.html

Risk = space of outcomes and probabilities are known. Uncertainty = probabilities not known, and even space of possibilities may not be known. Heuristic rules are contrasted with algorithms like maximization of expected utility.

How do smart people make smart decisions? | Gerd Gigerenzer

Helping Doctors and Patients Make Sense of Health Statistics: http://www.ema.europa.eu/docs/en_GB/document_library/Presentation/2014/12/WC500178514.pdf

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Risk, Uncertainty, and Heuristics: http://infoproc.blogspot.com/2018/03/risk-uncertainty-and-heuristics.html

Risk = space of outcomes and probabilities are known. Uncertainty = probabilities not known, and even space of possibilities may not be known. Heuristic rules are contrasted with algorithms like maximization of expected utility.

How do smart people make smart decisions? | Gerd Gigerenzer

Helping Doctors and Patients Make Sense of Health Statistics: http://www.ema.europa.eu/docs/en_GB/document_library/Presentation/2014/12/WC500178514.pdf

july 2016 by nhaliday

Guess the Correlation

july 2016 by nhaliday

some basic rules?

- more trouble w/ high than low end (maybe because I'm just guessing slope/omitting outliers?)

- should try out w/ correlated Gaussians to get some intuition

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- more trouble w/ high than low end (maybe because I'm just guessing slope/omitting outliers?)

- should try out w/ correlated Gaussians to get some intuition

july 2016 by nhaliday

A story about Bayes, Part 1: Binary search | Andrew Critch

quantified-self health rationality biohacking nootropics stories street-fighting tetlock lesswrong reflection embodied epistemic ratty embodied-street-fighting embodied-cognition decision-theory paying-rent 2016 core-rats decision-making occam mental-math metabolic realness info-dynamics clever-rats spock experiment analytical-holistic being-right truth divide-and-conquer debugging expert-experience examples contrarianism thinking judgement

february 2016 by nhaliday

quantified-self health rationality biohacking nootropics stories street-fighting tetlock lesswrong reflection embodied epistemic ratty embodied-street-fighting embodied-cognition decision-theory paying-rent 2016 core-rats decision-making occam mental-math metabolic realness info-dynamics clever-rats spock experiment analytical-holistic being-right truth divide-and-conquer debugging expert-experience examples contrarianism thinking judgement

february 2016 by nhaliday

bundles : math ‧ ng ‧ problem-solving ‧ thinking

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