Totally OT - Centigrade vs Farenheit

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Nice little puzzle!!!

Wow nice work Saul!

6, 28 and 496 are known as “perfect numbers,” defined as numbers that equal the sum of all of their factors, excluding the number itself.

So, as you noted, 6 = 1+2+3, which happen to be the factors of 6 excluding 6 itself.

Likewise, 28 = 1+2+4+7+14, and 496 = 1+2+4+8+16+31+62+124+248-- in each case, the sum of all factors excluding the number itself.

I know this is not what you said, but the reason I am so impressed by your answer is that you discovered a very cool thing on your own, which is that there is a deep connection between perfect numbers and Mersenne primes – which are prime numbers of the form 2^n -1 (2 to the nth power minus 1).

So, 3 is 2^2 -1, 7 is 2^3-1, and 31 is 2^5-1 – each is a Mersenne prime, and each thus is associated with a perfect number!

How?

Exactly as you specified – the sum of the integers leading up to a Mersenne prime yields a perfect number. I am absolutely astonished (and, I admit, even a bit impressed!) that you came up with this on your own just noodling around – it is a fairly deep concept of number theory.

The connection between Mersenne primes and perfect numbers is usually expressed somewhat differently (as the Mersenne prime 2^n -1 multiplied by 2^(n-1) – e.g. if n is 5, the prime is 2^n-1 = 31 and the multiplier is 2^(5-1) =2^4 = 16),

see http://www.millersville.edu/~bikenaga/number-theory/perfect/…

but it is easy to show that 2^(n-1) equals the sum of the numbers from 1 to 2^n - 1 – matching your approach above. (The proof is pretty simple – use Gauss’s trick to add all the integers from 1 to 2^n-1 – i.e., it equals the number you are adding up to (2^n-1) times the next higher integer (2^n) divided by 2. (For example the sum of the numbers from 1 to 10 equals 10*11/2 = 55.) When you divide the 2 into the 2^n factor, you get 2^(n-1), showing the equivalence between your sum and the more common formulation.

Anyway, I bored the tears off of everyone with this post, but a very sincere well done to you, Saul

Rich

Chance Elder Dancer and A Drumlin Daisy (depending on which MF service I logged into).

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Hi Rich,

Fascinating! Thanks for your appreciation. Using your comments, I figure that 120 and 2016 are also numbers in the series I described, although 15 is and 63 are not primes. I also see now that the 3, 7 15, etc, are powers of 2 minus 1.

6 is 2^1 x 3
28 is 2^2 x 7
120 is 2^3 x 15
496 is 2^4 x 31
2016 is 2^5 x 63

AND

6 is the sum of 1 to 3,
28 is the sum of 1 to 7
120 is the sum of 1 to 15
492 is the sum of 1 to 31
2016 is the sum of 1 to 63

Is there any name for the series of numbers, 6, 28, 120, 492, 2016, etc even when they don’t involve primes?

Love this stuff. In my real life I was a psychiatrist, but in high school I was into math (60 years ago or so).

Thanks,

Saul

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Hmm, it would be interesting to know how many math geeks follow this board (well maybe not - but of interest to a few of us here).

I started out in college as an EE major but switched to a liberal arts degree (from IIT which did not award BA degrees, I got a BSLA, a BS in Liberal Arts, whatever that’s supposed to mean). In any case, my major was the ill-defined LLP (language, literature and philosophy). I also had minors in math, physics and chem.

Ended up in an IT job because in the 70s when I went to work for Boeing there was no such thing as computer science. I demonstrated an aptitude for the computie stuff while also possessing the rare skill of being able to write coherent English. They said son, “you’re an analyst if ever there was one.”

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Is there any name for the series of numbers, 6, 28, 120, 492, 2016, etc even when they don’t involve primes?

Hi Saul,

This is Rich using his other identity.

I do not believe the series you identified has a name, although it is undoubtedly interesting - basically, it has a 1-1 correspondence with the powers of two; the really interesting feature is, I think, the nature of that correspondence.

BTW, every known perfect number (which I defined earlier in this thread) is even, but there is no known theoretical reason why there could not be an odd perfect number. Anyone who solves this ancient question – either by finding an odd perfect number or by proving they cannot exist – would literally be remembered for thousands of years.

Love this stuff. In my real life I was a psychiatrist, but in high school I was into math (60 years ago or so).

Well, I can only regret the waste of a fine mind – frittered away on things like medical school and making gazillions of dollars as an investor when you could have made a real contribution in the math world . . . .

(just kidding, in case that is not obvious : ) )

Rich

CED/ADD

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I started university in Engineering Physics and 13 majors later ended up with a degree in Anthropology with a minor in math. There was a grad student who was ex-Physics who taught a programming class for anthropologists and by the time I got my bachelors, most of what I was doing in anthro was math and computer based. My doctorate was a new form of multivariate statistics. But, it has been a long time since I pursued math for the sake of math, but rather it has been a tool I needed to solve a problem. After a stint in academic anthropology I moved to an R&D project teaching scientific problem solving and went from there to the private sector and have been doing enterprise business computing since.

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The temperature at which books burn will always be remembered in Fahrenheit and Lord Kelvin is a real cool guy.

https://www.google.com/search?client=safari&rls=en&q…
http://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelv…

Where does that leave Celsius/Centigrade? One hundred, boiling water; zero, dense water, how mundane.

Denny Schlesinger

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http://www.vox.com/2015/2/16/8031177/america-farenheit

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One would have thought that this thread should have died a while ago, yet it lingers on . . .

The Vox article link shared by democracynow rather glibly establishes the rationale for the Farenheit scale based on experimental conclusions derived from freezing salt water and inaccurate body temperature readings. However, if you dig a bit deeper, you’ll discover that this reasoning is just one of several speculations. The fact is that Daniel Farenheit simply did not record his reasoning for the peculiar scale with water freezing at 32 and boiling as 212. We do not know why he came up with these odd numbers.

However, the article correctly records the Mr. Farenheit was the first person to invent a reliable thermometer (two, actually, one with alcohol and the other with mercury). In that he was the first to create a reliable, functional thermometer, he also got to invent the scale.

And that’s where it came from. Why is the US still stuck on it while the rest of the world’s nations have moved on to the more practical Celsius scale. Probably for the same reason that we are the only nation with a very strong anti-climate change contingency. We tend to stubbornly cling to irrational modes of thinking.

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Freezing at 32 degrees? boiling at 212 degrees? Where did they get those random numbers? 0 degrees is just a random number too, with no significance, as is 100.

The Fahrenheit temperature scale is based on the freezing temperature of salt water and the freezing temperature of fresh water. 0 to 32 in denary numbers and 0 to 100000 in binary numbers. The numbers 0 and 32 were chosen so by using compasses you could bisect the line segment to get 16 and by further bisecting get the remaining numbers.

jkm929

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jkm929
To further confuse this, salt water actually freezes at 28.4
And although I have never licked an iceberg Jules Vern has said there is no salt in one.

BGM

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Sorry jkm, it’s just not so. Simply because the Fahrenheit scale is so bizarre and I tend to be kind of a geek, I’ve researched this.

As it turns out nobody actually knows for certain why Fahrenheit came up with 32 for freezing and 212 for boiling. There are several different explanations, but they are all speculations. Fact is Fahrenheit did not leave notes on why selected this peculiar scale so we don’t really know why these numbers were chosen.

However, Fahrenheit was the first guy to build a reliable thermometer (two in fact, one with mercury and the other with died alcohol). In that he invented a thermometer that just about anyone could build, use and get reliable, transferable measurements, his scale was widely adopted.

Later came the more reasonable Celsius scale which quickly gained wide-spread adoption and in time displaced the Fahrenheit scale in just about every country on the planet save the USofA which has stubbornly clung to this bizarre scale. In similar fashion, the USofA has also clung to the FPS system of measurement while most of the planet has adopted MKS. Though we do have a little bit more company here than we do with the Fahrenheit temperature scale.

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The Metric System uses the base 10 denary number system. It’s based on multiplying and dividing by ten, accomplished by adding or subtracting zeros. Computers, however, use the base two binary number system, based on multiplying and dividing by two, likewise accomplished by adding or subtracting zeros. The below link demonstrates the inappropriateness of the base 10 system for describing the bits in a computer.

http://physics.nist.gov/cuu/Units/binary.html

My question to you is: what would happen to the Metric System if the world converted to the binary number system?

jkm929

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If humanity can speak hundreds of languages why can’t it figure in dozens of number systems?

Denny Schlesinger

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If humanity can speak hundreds of languages why can’t it figure in dozens of number systems?

Denny Schlesinger

All sorts of fun facts caused by these differences (I am sure it hardly compares to misunderstandings due to different languages over the course of humans)

http://ta.twi.tudelft.nl/users/vuik/wi211/disasters.html

http://mentalfloss.com/article/25845/quick-6-six-unit-conver…

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My question to you is: what would happen to the Metric System if the world converted to the binary number system?

Computers think in binary because that is their nature. Binary would be stupid for humans because of the low information value per digit.

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My question to you is: what would happen to the Metric System if the world converted to the binary number system?

The binary system works for computers but not for humans. Just try putting a random number, like say 7463, in binary. See if there is anything, anything in it at all, that is recognizable and would distinguish it from another binary number of the same impractical length?

No danger of that conversion happening.

Saul

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My question to you is: what would happen to the Metric System if the world converted to the binary number system?

Many moons ago my partner and I were trying to get a Diablo printer to work properly. Back then you set up equipment using micro switches. We checked the settings again and again against the manual and we could not find anything wrong. After the third or tenth iteration I decided to examine the manual itself in more detail. Then I said to my partner “Eleven does not follow ten.” She replied: “Of course not.” We were in a room full of auditors and we got the strangest looks…

1A follows 10 but the guy writing the manual didn’t speak hexadecimal. After we corrected the manual we got the printer working properly.

Denny Schlesinger

Hexadecimal is human readable binary.
http://en.wikipedia.org/wiki/Hexadecimal

Diablo printers
https://www.google.com/search?q=Diablo+printer&newwindow…

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Just try putting a random number, like say 7463, in binary. See if there is anything, anything in it at all, that is recognizable and would distinguish it from another binary number of the same impractical length?

The number 7463 can be expressed as:
4096 + 2048 + 1024 + 256 + 32 + 4 + 2 + 1
So, the answer is: 1110100100111

http://acc6.its.brooklyn.cuny.edu/~gurwitz/core5/nav2tool.ht…

jkm929

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Just try putting a random number, like say 7463, in binary. See if there is anything, anything in it at all, that is recognizable and would distinguish it from another binary number of the same impractical length?

The number 7463 can be expressed as:
4096 + 2048 + 1024 + 256 + 32 + 4 + 2 + 1
So, the answer is: 1110100100111

Hi jkm, I knew it could be expressed as a binary. What I was saying was that that binary, while immediately recognizable to a computer, was meaningless to a human.

See if there is anything, anything in it at all, that is recognizable and would distinguish it from another binary number of the same impractical length?

For example: can you tell it from 1110101000111 or 1110100010111? Which are 7495 and 7447 (as I figure them in my head). What I meant was that there is no danger of binary replacing base 10 any time soon for humans.

Saul

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I was thinking about the binary number system for simple things like halving or doubling recipes or dividing a pound into 8 ounces, then 4 ounces, then two ounces and then finally one ounce. Multiplying or dividing by two makes more sense for some purposes than multiplying or dividing by ten.

jkm929