Aw, Galu, don't get all morose about 1 being the loneliest number.
Actually I am quite the expert on curious and interesting numbers:
The Penguin Dictionary of Curious and Interesting Numbers (Penguin Press Science): Amazon.co.uk: Wells, David: 9780140261493: Books
Even read Volume 2.
The dullest number in teh Universe actually is quite interesting, just for being the dullest. If you follow?
But I am sticking with 1728 here. Don't push me into detail too much at this time of night, but 1728 is part of the Monster Group:
Quanta Magazine
YouTube
Morose, dull? Seek psychiatric help dear, your delusions have become a handicap.
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Oh, what a hurt ego! You are afraid of the least of competition...system7 said:
I am NOT EMBARRASSED to describe my 'competition' as the LEAST.
I am PROUD of my attainments considering the DEPRIVED social conditions I was brought up in.
No bullying from any self entitled inflated ego will silence me.
Doctor Ph.D (A..Z), expects me to be manipulated?! Keep your manipulation to yourself, dear manipulator.
Now, awaiting your next attempt at humiliation.
I refuse to be manipulated.
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No human being can translate in concrete terms the size of infinity including mathematicians. However, there are mechanisms, which allow a mathematician to handle infinitely large parameters/variables.mchambin said:
For instance:
1/x as x approaches infinity gives a quantity that approaches 0.
The reason for the above is simple if *we consider fractions. If x increases, the fraction becomes smaller as the denominator determines the number in one whole.
Similarly, if you have an infinite summation, like the following:
Summation {from r=0 to infinity} of 1/2^r = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...
Although there are an infinite number of addition terms, the result of the entire infinite series is not infinite. This seems a logical contradiction, as adding a quantity for an infinite number of times, is expected to yield an infinitely large number. However, if we consider the formula for the sum of a geometric series, we can understand why the summation yields a finite number.
For a geometric series, the sum is given by:
a(r^n - 1)/(r - 1)
Now, let us imagine the number of terms in this series is finite but very very large, say 1000000.
Let us evaluate r^n = (1/2)^1000000. The value is very very near zero. So, we conclude, if n, the number of terms, is infinite the value of this term is zero.
Evaluating the formula for an infinite n:
1(0 - 1)/(1/2 - 1) = 2
The summation never exceeds 2, no matter how big n is.
As anyone can see, mathematicians use common reasoning available to anyone naturally to arrive at their seemingly weird conclusions.
*we: refers to myself and the reader.
For a geometric series, the sum is given by:
a(r^n - 1)/(r - 1)
We used that one to prove something about 20 pages back.
On infinity, I seem to recall that the real numbers are an infinite but countable set. But some sets of numbers don't map to them, being an order of infinity bigger. Cantor isn't it.
All good fun. The Universe exdpanding into something has mostly got us stumped though.
You can just take a break from it though. I like swimming and sport at the moment. It's quite hard to stay sane in these weird times.
a(r^n - 1)/(r - 1)
We used that one to prove something about 20 pages back.
On infinity, I seem to recall that the real numbers are an infinite but countable set. But some sets of numbers don't map to them, being an order of infinity bigger. Cantor isn't it.
All good fun. The Universe exdpanding into something has mostly got us stumped though.
You can just take a break from it though. I like swimming and sport at the moment. It's quite hard to stay sane in these weird times.
You are tired of being challenged?! You are sane?! You are irritated by the least of competition to your self inflated ego! You call that sanity: that is a delusion fit for psychiatric presciptions.system7 said:
You are so self possessed, that a normal post, is highly irritating to you.
Go and burn incense to yourself and your paraoid ego in the sanctuary of delusion and paranoia.
Derivation for the sum of a Geometric Series:
Let S = a + ar^1 + ar^2 + ar^3 + ... + ar^(n-1) [S is the sum, a the first term, and r the multiplicative factor of terms]
Now, let us multiply both sides by r:
rS = ar + ar^2 + ar^3 + ... + ar^n
Inspecting the Right Hand Side, we find that, it is equivalent to:
S - a + ar^n
This implies:
rS = S - a + ar^n
Rearranging:
rS - S = ar^n - a
Factorising:
S(r - 1) = a(r^n - 1)
Rearranging to make S subject of the formula:
S = a(r^n - 1)/(r - 1)
Let S = a + ar^1 + ar^2 + ar^3 + ... + ar^(n-1) [S is the sum, a the first term, and r the multiplicative factor of terms]
Now, let us multiply both sides by r:
rS = ar + ar^2 + ar^3 + ... + ar^n
Inspecting the Right Hand Side, we find that, it is equivalent to:
S - a + ar^n
This implies:
rS = S - a + ar^n
Rearranging:
rS - S = ar^n - a
Factorising:
S(r - 1) = a(r^n - 1)
Rearranging to make S subject of the formula:
S = a(r^n - 1)/(r - 1)
Planet Nine
The clustering of the orbits of small planetoids far beyond the orbit of Neptune has been thought to be due to the presence of a massive but undetected planet - Planet Nine.
However, failure to detect light from Planet Nine has now led astronomers to consider that it may be a grapefruit sized black hole with a mass five to ten times that of the Earth.
Astrophysicists now hope that the Legacy Survey of Space and Time (LSST) mission About LSST | Rubin Observatory will confirm the latest hypothesis by detecting the accretion flares which would result when small bodies in the outskirts of the solar system approach the black hole.
The clustering of the orbits of small planetoids far beyond the orbit of Neptune has been thought to be due to the presence of a massive but undetected planet - Planet Nine.
However, failure to detect light from Planet Nine has now led astronomers to consider that it may be a grapefruit sized black hole with a mass five to ten times that of the Earth.
Astrophysicists now hope that the Legacy Survey of Space and Time (LSST) mission About LSST | Rubin Observatory will confirm the latest hypothesis by detecting the accretion flares which would result when small bodies in the outskirts of the solar system approach the black hole.
Attachments
Do they know the exact location of this presumed black hole ?
Is the telescope imager able to detect distortions from the gravitational lens made by this mass, heavy enough to bend light from background stars ?
Could we lauch a space probe to appraise gravity nearby ? How long would it take to go there ?
When is the telescope ready ?
Is the telescope imager able to detect distortions from the gravitational lens made by this mass, heavy enough to bend light from background stars ?
Could we lauch a space probe to appraise gravity nearby ? How long would it take to go there ?
When is the telescope ready ?
- The predicted orbit is about 20 times further on average than Neptune which orbits 2.8 billion miles from the Sun, making it hard to find. If Planet Nine is indeed a planet, it will be exceedingly dim if it is presently at the furthest point in its highly elliptical orbit and is even more difficult to spot because the brightness of the Milky Way galaxy is in the background.
- If Planet Nine is a black hole, it is only the size of a grapefruit and only 5 to 10 earth masses. Observing gravitational lensing effects would be extremely unlikely even if we knew exactly where to look.
- Yes, if we knew where to send it! Look at the time taken for the New Horizon probe to reach Pluto for a lower limit to the journey time.
- Construction began in 2015. The 10 year survey was due to commence in 2022, but Covid-19 has delayed progress.
And here is the million dollar question:
Is the number of objects in the Universe countable or uncountable? Theoretically, of course. For the first approximation, let's restrict the term "objects" to stars.
I think you can calculate the number of stars easily enough. This is what astrophysicists do.
There's a black hole calculator here:
Viktor T. Toth - Hawking radiation calculator
5 Earth Masses put it 8cms across as Galu says. Lasts a long time 7E52 years. That means 52 zeroes after 7, doesn't it?
Temperature 0.04 Kelvin.
Problem is how do you make a black hole so relatively small? Doesn't form from stars collapsing.
There's a black hole calculator here:
Viktor T. Toth - Hawking radiation calculator
5 Earth Masses put it 8cms across as Galu says. Lasts a long time 7E52 years. That means 52 zeroes after 7, doesn't it?
Temperature 0.04 Kelvin.
Problem is how do you make a black hole so relatively small? Doesn't form from stars collapsing.
Quite calculatable. The mass of the observable universe is c 5 * 10^54 Kgs and you can assume 10^20 atoms per Kg. So the number of atoms in the universe is about 5*10^74. Estimates vary by3 or 4 orders of magnitude either way.
Dark matter and dark energy of course a different discussion.
@Galu - the black hole (BH) hypothesis for planet 9 is indeed interesting. If it is a planet, it’s only going to be a few degrees (30-50) above the CMB so it will Indeed be exceedingly difficult to locate. Does anyone know what the lower temperature detection limit is for ground based telescopes?
I understand it is hypothesized that there are many wandering planets out there that are captured by stars - might be this is a similar case if not a BH.
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It would have to be a primordial black hole formed in the first moments of the universe, when everything was hot and dense - chunks of early matter crushed together so tightly that they condensed into singularities.
A primordial black hole wouldn't look like much of anything, its event horizon would be tiny. However, if one wandered into our solar system it would orbit the Sun just like a planet would.
I was trying that calculator for relatively small black holes of, say, 100 Million tons. Mass of a mountain.
There was a story by Larry Niven "The Hole Man" about such things. Incredibly hot. Incredibly tiny, like 1/10 of the size of a proton. Last a few thousand years.
Anyhow, some excitement this month. We have a really big comet:
Stunning images of Comet Neowise that won’t return for 7,000 YEARS – and you can still see it for two weeks
Comet Neowise. Don't often see one that bright. Near the Great Bear in the Northern evening sky. Visible for the next two weeks. Hope the skies in Portsmouth let me see something. Terrible Light pollution here.
There was a story by Larry Niven "The Hole Man" about such things. Incredibly hot. Incredibly tiny, like 1/10 of the size of a proton. Last a few thousand years.
Anyhow, some excitement this month. We have a really big comet:
Stunning images of Comet Neowise that won’t return for 7,000 YEARS – and you can still see it for two weeks
Comet Neowise. Don't often see one that bright. Near the Great Bear in the Northern evening sky. Visible for the next two weeks. Hope the skies in Portsmouth let me see something. Terrible Light pollution here.
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