Apr 8, 2025
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Super real math trick!
It's easy to tell the difference between rational and irrational numbers!
Rational numbers know that you both just need to take a break and come back with a level head before solving the problem.
Irrational numbers will continue to yell and argue well past the point of making any sense.
Integers like their personal space. Real numbers are clingy huggers. Rational numbers look like they're super touchy-feely but they're actually just standing really close together.
By random chance, the decimal expansion of pi briefly begins repeating itself after 6954 digits before resuming its randomness
3.1415926535...987571415957811...
That's the longest repeated string in the first million digits, so a really good rational approximation would have the first 6954 digits repeat infinitely. You can create any rational number with n repeating digits by taking a string of length n and dividing it by an equally long string of nines.
0.123123123... = 123/999
0.141591415914159... = 14159/99999
Of course, these may not be in simplest form
0.123123123... = 123/999 = 41/333
0.363636... = 36/99 = 12/33 = 4/11
0.102911029110291... = 10291/99999 = 251/2439
So the first 6954 digits of pi over 6954 nines could very likely be simplified. At that magnitude, the odds that they are co-prime (share no factors) is very slim
Scratch that. I plugged the first 6954 digits into a prime factorization calculator, and the first three factors it was able to find were 99551 × 18298401827 × 104229615391 (100k, 18.3 billion, 104.1 billion). It took 30 minutes to find those three, so it could take HOURS or DAYS to calculate the remaining factors because it still has over 6,900 digits to get through, but this makes it seem more likely that the string is co-prime with 9999999...9999999 just because there are multiple orders of magnitude between factors, so the odds of any two random 12+ digit numbers being the same are slim. I don't know, I'm not a statistician.
3 + (1.4159...98757 x 10^6953) / (10^6954 - 1) ≈ pi
A blog about mathematics.
Finished Real Analysis: Chapter 1! They are all basics of complex numbers, real numbers, exponentials, and so on!
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An eminent mathematician reveals that his advances in the study of millennia-old mathematical questions owe to concepts derived from physics.
Fractions and whole numbers break down into repeated × and ÷ by prime numbers. 2's exponent in this prime decomposition, is the number’s 2-adic valuation.
Paul VanKoughnett
Eitan’s Nebula, a 3D-printed puzzle by Eitan Cher. The puzzle features icosahedral geometry with cuts at two different depths and has a whopping 743 parts (including internal mechanism) and 920 stickers. Because of the nature of the face-turning icosahedral geometry, the puzzle can jumble. Jumbling is a somewhat difficult term to define, but it’s presence can be seen in the shape-shifting moves in the last picture (notice the corner that is not where it “should” be). “Normal” moves are rational rotations. For example, image 3 shows a 120-degree counterclockwise rotation of the top face (and clockwise on the smaller layer below it), or simply 1/3 of a full turn. At this position (”stop”), all moves that were available previously are still available. The geometry has not changed. With a jumbling move, a particular stop allows some moves but not others. Cuts line up in mysterious ways and unexpected moves are possible. In the 5th image, the pink face was turned less than 120 degrees before a move on the black face became possible (6th image). Some pieces were “not ready yet” and got moved along “upside down,” resulting in the shape-shifted appearance. In fact, the pink face was rotated by exactly arccos((-1+3√5)/8) = 44.47751219... degrees. Across all types of puzzles with all sorts of geometries, it seems that jumbling angles are always irrational. Curious. To see in action, check out a video of Eitan showing off his creation here.