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987654321 / 123456789
The ratio 987654321 / 123456789 is very nearly 8. What about the analogous ratio in other bases?
John D. Cook
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Visit johndcook.com → RSS feedOn Hacker News Jan 2010 – Sep 2026
Math & Algorithms 44Software Engineering 24Life, Learning & Writing 11Career & Management 11Science & Space 9Programming Languages 6
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Estimating the chances of something that hasn’t happened yet
A handy rule of thumb for creating a confidence interval for the probability of an event you haven't seen happen yet.
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Queueing theory: The science of waiting in line
The math behind how people wait in line
4
Golden powers are nearly integers
As n increases, the distance from phi^n to the nearest integer decreases exponentially where phi is the golden ratio.
5
Why isn't everything normally distributed?
Heights are normally distributed because height is determined by many factors. But why are other things with many inputs not normally distributed?
6
Test for Divisibility by 13
How to manually test whether a large number is divisible by 7, 11, and 13 all at the same time.
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Kalman filters and functional programming
A paper on Kalman filtering and functional folds that I alluded to previously has been split into several papers and posted online.
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How Lewis Carroll computed determinants (2023)
Charles Dodgson (a.k.a. Lewis Carroll) developed a method of computing determinants which has some practical advantages.
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US Army applying new areas of math
Many times on this blog I've argued that the difference between pure and applied math is motivation. As my graduate advisor used to say, “Applied mathematics is not a subject classification. It's an attitude.” Traditionally there was general agreement regarding what is pure math and what is applied. Number theory and…
10
Planets evenly spaced on log scale
The distances to the planets are fairly evenly distributed on a logarithmic scale, not only in our solar system but in extrasolar systems as well.
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Reminder of Complexity
An impressive poster summarizing biochemical pathways reminds us how complex life is, and how interconnected the world is.
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Banned math book
The applied math classic Courant & Hilbert was banned in Nazi Germany. The US seized the copyright and published an English edition.
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How to make a Smith chart
Deriving the basic mathematical properties of the Smith chart from electrical engineering
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The Soviet license plate game and Kolmogorov complexity
Physicist Lev Landau used to play a mental game with Soviet license plates [1]. The plates had the form of two digits, a dash, two more digits, and some letters. Rules of the game His game was to apply high school math operators to the numbers on both side of the dash so that the
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Every factorial approximates a power
For every n, there is a base b such that n! = b^n. The relation between n and b(n) is surprisingly close to linear.
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Evaluating a class of infinite sums in closed form
A family of infinite sums that evaluate to rational numbers, sometimes integers, and and effective algorithm for finding the sum.
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Surprises from numerical linear algebra (2010)
Here are ten things about numerical linear algebra that you may find surprising if you're not familiar with the field. Numerical linear algebra applies very advanced mathematics to solve problems that can be stated with high school mathematics. Practical applications often require solving enormous systems of…
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The quadratic formula and low-precision arithmetic
The familiar quadratic formula can give the wrong answer in a program due to numerical precision issues. This post shows what can go wrong and how to fix it. Especially important in low-precision arithmetic.
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Math library functions that seem unnecessary
Functions include in the standard C math library that seem unnecessary at first glance and why they are necessary.
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Why Mathematica does not simplify sinh(arccosh(x))
When Mathematica seems to not to know how to simplify an expression, it may be doing the right thing, being more sophisticated than you expect.
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What is an elliptic curve? (2019)
The informal definition of an elliptic curve is simple. The formal definition is more involved.
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A triangle whose interior angles sum to zero
In hyperbolic geometry, the sum of a the interior angles of a triangle can be any value less than π. If the sum is zero the triangle has area π.
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Gravitational attraction of stars and cows
One attempt at rationalizing astrology is to say that the gravitational effects of celestial bodies impact our bodies. To get an idea how hard the stars and planets pull on us, let's compare their gravitational attraction to that of cows at various distances. Newton's law of gravity says that the gravitational…
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Circulant Matrices, Eigenvectors, and the FFT
The eigenvectors of a circulant matrix do not depend on the matrix itself, only on the size, and the eigenvectors are columns of the FFT matrix.
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Don’t invert that matrix (2010)
There is hardly ever a good reason to invert a matrix. What do you do if you need to solve Ax = b where A is an n x n matrix? Isn't the solution A−1 b? Yes, theoretically. But that doesn't mean you need to actually find A−1. Solving the equation Ax = b is
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Ramanujan Factorial Approximation (2012)
Ramanujan came up with an approximation for factorial that resembles Stirling's famous approximation but is much more accurate. As with Stirling's approximation, the relative error in Ramanujan's approximation decreases as n gets larger. Typically these approximations are not useful for small values of n. For n = 5…
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Evolution of Random Number Generators
Prime powers have a kinda chaotic pattern in their bits. We bootstrap this idea into a good random number generator, recapitulating history.
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Tofolli gates are all you need
Any Boolean function can be computed with Tofolli gates, and Tofolli gates are reversible, and so any Boolean function can be computed reversibly.
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The Buenos Aires Constant
Generating primes using the Buenos Aires constant.
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How medieval astronomers made trig tables
Clever tricks used by medieval astronomers to tabulate sines at every degree, before calculators and before calculus.