The Babylonians calculated the square root of 2 with 99.9999% accuracy, amazing
Note: Base-60 is referred to as 'sexagesimal'
They calculated square root of 2 as 1.41421296296 without a calculator, very impressive.
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Yitang Zhang didn't get to go to middle or high school but he is a mathematician today
As a boy in Shanghai, China, Yitang Zhang believed he would someday solve a great problem in mathematics. In 1964, at around the age of nine, he found a proof of the Pythagorean theorem, which describes the relationship between the lengths of the sides of any right triangle. He was 10 when he first learned about two famous number theory problems, Fermat’s last theorem and the Goldbach conjecture. While he was not yet aware of the centuries-old twin primes conjecture, he was already taken with prime numbers, often described as indivisible “atoms” that make up all other natural numbers.
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"How Much Will the Climate Bill Reduce Emissions? It Depends"
This is an absolutely fantastic breakdown and explanation of how climate emission predictions are made, their shortcomings, etc.
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Dissolving the Fermi Paradox (2018)
Due to the discussion I dove into around the quantum communication, I was pointed to this 2018 mathametics paper which seeks to disprove the Fermi paradox. Here's their first paragraph of the introduction:
While working at the Los Alamos National Laboratory in 1950, Enrico Fermi famously asked his colleagues: ”Where are they?” He was pointing to a discrepancy that he found puzzling: Given that there are so many stars in our galaxy, even a modest probability of extraterrestrial intelligence (ETI) arising around any given star would imply the emergence of many such civilizations within our galaxy. Further, given modest assumptions about their ability to travel, to modify their environs, or to communicate, we should see evidence of their existence, and yet we do not. This discrepancy has become known as the Fermi paradox, and we shall call the apparent lifelessness of the universe the Fermi observation.
And then from the paper's conclusion, the bracketed segment is my filling in context:
When we update [the Drake equation's point estimates with probability distributions] in light of the Fermi observation, we find a substantial probability that we are alone in our galaxy, and perhaps even in our observable universe (53%–99.6% and 39%–85% respectively). ’Where are they?’ — probably extremely far away, and quite possibly beyond the cosmological horizon and forever unreachable.
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Once in a Lifetime
Evelyn Marie Adams won $3.9 million in the New Jersey lottery in 1986. Four months later she won again, collecting another $1.4 million.
[...]
Three years later two mathematicians, Persi Diaconis and Frederick Mosteller, threw cold water on the excitement.
If one person plays the lottery, the odds of picking the winning numbers twice are indeed 1 in 17 trillion.
But if one hundred million people play the lottery week after week – which is the case in America – the odds that someone will win twice are actually quite good. Diaconis and Mosteller figured it was 1 in 30.
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How Many Decimals of Pi Do We Really Need?
For JPL's highest accuracy calculations, which are for interplanetary navigation, we use 3.141592653589793. Let's look at this a little more closely to understand why we don't use more decimal places.
Up to now, I have only ever memorized 3.14159 as that is plenty accurate for any calculation I might need to do. But knowing that NASA uses just ten more digits is a good motivation that I should memorize that far in case I ever need to calculate space travel.
Give the article a read as it goes into great depth as to why fifteen digits is more than enough for NASA.
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3x+1
3x+1 is a wonderful thing, I wasn't aware of it this morning but now I'm thinking about it. I'm certainly not going to be the person who solves it, but looking at it I was immediately seeing a relationship to binary.
With the loop end at 4, 2, 1, being powers of two, makes it a natural realization and I am quite certain I'm not the first person to consider it in this way.
The rules of 3x+1 in binary are the same, except dividing by 2 in binary is extremely simple. An even numbers in 0, to divide by 2, you drop the last 0.
So for example, I quickly jotted out the paths of 1-7 in binary:
- 1 -> 100 -> 10 -> 1 -> L (the loop)
- 10 -> 1 -> L
- 11 -> 1010 -> 101 -> 10000 -> 1000 -> 100 -> 10 -> 1 -> L
- 100 -> 10 -> 1 -> L
- 101 -> 10000 -> ... -> 1 -> L
- 110 -> 11 -> 1010 -> 101 -> 10000 -> ... -> 1 -> L
- 111 -> 101010 -> 10101 -> 100010 -> 110100 -> 11010 -> 1101 -> 101000 -> 10100 -> 1010 -> 101 -> 10000 -> ... -> 1 -> L
Mostly did this as a mental exercise and to explore the tie I noted with binary, but it is interesting. I am certain I'm not the first person to explore it this way, so I'm retreading well trodden ground, but it is still a nice stretch out of my normal day-to-day thinking.
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Pi is fascinating for any number of reasons, but this video does an excellent job summarizing the history of pi and how it was calculated. I knew some of it, in concept, but I would have struggled to explain it. I still wouldn't get a full explanation right, but I feel definitely more capable of attempting it. That said, rather than have me explain it, just watch the video.
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