Just watched a YouTube video from Stand-Up Maths about “troll math,” or mathematical occurrences that seem to imply a general principal but are in fact singular coincidences.
He cites the example of log(1+2+3) = log(1) + log(2) + log(3) as one. It seems to imply that the log of any sum of numbers is equal to the sum of the logs of those same numbers, but this isn’t true. In reality, log(1) = 0, and what you’re really seeing is that the sums of the logs of numbers is equal to the log of the product of those numbers: log(2) + log(3) = log(6). This is why logarithms proved so useful before the invention of calculators: they allowed one to skip the annoying process of multiplication and use the slightly less annoying process of addition.
There are a lot of such points, at least one at ever longitude. But there is also at least one pair of points that are simultaneously at the same temperature and barometric pressure.
@Tabco, it also works for log(1) + log(2) + log(4) + log(7) + log(14).
Even integers are actually much more prominent and prevalent than odd integers. The sum of any two odd integers is an even integer (as noted above). BUT also, the sum of any two even integers is also an even integer! Odd integers are much more difficult to construct, as there seem to be many fewer of them. (Your method, of adding 1 to an even integer, requires that you first acquire an even integer, so that doesn’t actually get you anywhere.)
Here’s an oddity that you can use as a mathematical mind-reading parlor trick:
Think of any whole number from 1 to 10. (This will actually work with any number, but sticking with an integer from 1 to 10 makes the mental arithmetic easier.)
Double it.
Add 10.
Divide that by 2.
Subtract the original number. (Uh… You do still remember your original number, right? If not, see a neurologist for cognitive testing.)
Now think hard on your resulting number, and I will try to read your mind, even over the vast distances of netspace.
If you didn’t get >>>5<<< then all bets are off.
It’s amazing!
For the perplexed, I came up with a simplified version of the same trick:
Think of a number (as before).
Add 5.
Subtract the original number.
Did you get 5?
I knew it!
A lot of people are amazed when they shop and the cash register rings up an exact dollar amount. But such an event has a probability the same as having an amount that ends in .57, or any other two digits. People find strange coincidences everywhere they look.
There aren’t enough small numbers to meet the many demands made of them.
It’s something of a joke, but there’s nevertheless a deep truth there. There really aren’t that many small numbers, but there is a nearly infinite number of things that can be said about them. Hence, we should expect to see many surprising coincidences involving them. If the coincidences were spread out evenly over the number line, they would be so infrequent that they would not be surprising.
My favorite coincidence is probably the fact that eπ√163 is very nearly an integer (262537412640768743.999999999999250072). As it happens, the coincidence is justified by some deep math relating to string theory and the Monster group.
You all know that 3^2 + 4^2 = 5^2, but I bet not many know that
3^3 + 4^3 + 5^3 = 6^3. Does this pattern continue? Alas, no. But here is one that does. The sum of all the cubes from 1 to n^3 is a square and in fact the square of the sum of the numbers from 1 to n. As far as is known, every even number is either prime or the sum of two primes, but this has never been proved.
Pick any number between 1-10. You picked 7. Most of the time that’s true because you don’t pick 1 or 10 as they are the ends, you don’t pick 5 because it’s in the middle, you don’t pick any even number because they’re too round and you don’t pick three because it’s too low and you don’t pick 9 because it’s square.
When ever the number 1488 exist in any sequence, computation or whatnot some nuit job tells your that that’s racist.
Any sequence generated from two integers following the rule that the next one is the sum of previous two has the property that the limit of dividing a number on the sequence with the previous number approaches Phi. Fibonacci numbers is not special in this sense.
Lucas numbers (generated from 2,1) have the property of being phi^n rounded.
When you ask people to prove that 1+1=2 they start to sweat. It’s not easy as there is no proof but the whole thing is the basis of how natural numbers and addition are defined: 1+1=2 because a+1 is the follower of a and by definition 2 is the follower of 1.
Sorry, but scanning trough those pages I didn’t see how 2 was defined. Using undefined objects in a proof makes the proof not valid.
And as I stated the problem of proving that 1+1=2 is that in the essence that is the definition what 2 means. And a definition of an object is not a proof of the existence of that object but a mutual understanding that the object exists.
Without even trying. It only took a few minutes. I came up with a simple formula using only single digit integers. Only 0,1,2. Easy to derive 5 from those. That gives the answer for the length of a tropical year. Specifically the number of years between leap years. Out to 7 digits of precision. Can do the same for 9 or 10 digits. The length of the formula is much less than 20 characters.
I challenge you to do this for this number, or any random LOOKING irrational number. 365.24… 128.2048. 2^7.2*7
Reminds me of the Richard Feynman quote: “You know, the most amazing thing happened to me tonight… I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!”
Not the same as my best friends phone number using only 2 constants that can be represented by 3 characters total. And it’s symmetrical. I only had one best friend in my life. One constant is a single digit. The other is PI. Can you find a better example?
Nice will read it thanks! 3,4,5 are special because they are the only integers that make a right triangle. Or multiples of them. They come up in another strange way.
In the section on cardinal arithmetic, 2 is defined on page 376.
I assume R+W had an intuitive understanding of what it means for there to be 2 things, not 1, 5 is right out before they formulated that definition. What, if anything, do you feel is invalid about their approach?
There is a lot of algebraic machinery mentioned but you can read about it here; the amazing coincidence is that the Monster Group just happens to be the group of automorphisms of a certain vertex algebra.
