800,000,000,000 Stars

 

(Image: NASA, ESA, CSA, STScI )

In the image above, the James Webb Space Telescope (JWST) captures the Sombrero galaxy in near-infrared and mid-infrared wavelengths. This galaxy is estimated to have approximately 800,000,000,000 stars.

The image above compares the galaxy in infrared and visible wavelengths. 

Read more from the JWST imaging of the Sombrero galaxy at: https://www.nasa.gov/image-article/webb-sees-sombrero-galaxy-in-near-infrared/

Prior posts on JWST:

Number of planets in the universe:

https://jamesmacmath.blogspot.com/2022/01/number-of-planets-in-universe.html

Earliest star formation:

https://jamesmacmath.blogspot.com/2025/05/jwst-detects-earliest-formed-galaxies.html

Top ten JWST photographs:

https://jamesmacmath.blogspot.com/2024/07/top-ten-photographs-of-james-webb.html

Rogue Planets

 

(Image: https://www.iconfinder.com/raminrzdh)

A rogue planet is a planet not orbiting a star. Instead of orbiting stars, rogue planets wander space between the stars. There are estimates that there may be trillions of such planets in our galaxy alone. IEEE Spectrum recently had an article about rogue planets and their detection: https://spectrum.ieee.org/rogue-planet.

Also see: Stars

Black Holes

Planets in the Universe

The James Webb Space Telescope recently found some rogue planets:

https://www.space.com/james-webb-space-telescope-rogue-planets-star-formation

The Monster: 808017424794512875886459904961710757005754368000000000

(Image: https://www.iconfinder.com/Spot)

While reviewing entries in the On-line Encyclopedia of Integer Sequences, I came across one of the larger integers that is a term of a sequence: 808017424794512875886459904961710757005754368000000000. It is the order of the 26th Sporadic Simple Groups and is also known as the Monster Group.

Grant Sanderson has a video on the 3Blue1Green series: Group theory, abstraction, and the 196,833-dimensional monster. 

The OEIS sequence with this number is found here: https://oeis.org/A001228.

Other mathematicians posted videos about their favorite number over one million at: https://www.youtube.com/results?search_query=MegaFavNumbers.

200,000,000,000,000,000,000,000 Stars

(Image: https://www.iconfinder.com/iStar_Design_Bureau)

I've seen several different estimations of the number of stars in the universe. Recently, The Conversation, an online independent news organization, published this article by Brian Jackson: How many stars are there in Space?

He estimates there are approximately 2 trillion galaxies with each having an average of 100 billion stars giving a total of 200,000,000,000,000,000,000,000 stars.

An image from the Hubble Space Telescope showing hundreds of faraway galaxies. (Image credit: NASA, ESA, CSA, STScI, J. Diego (Instituto de Física de Cantabria, Spain), J. D’Silva (U. Western Australia), A. Koekemoer (STScI), J. Summers & R. Windhorst (ASU), and H. Yan (U. Missouri))

The Very Small and the Very Big

In a prior post we considered how many digits of pi are required for a calculation, and I recently listened to a Lex Fridman podcast with guest Tim Urban, where they discussed if man is smaller than we are large or larger than we are small (Lex Fridman podcast).

Let's explore both extremes of small and large. 

Starting with the characteristic size of a human, 1 meter. Most humans are taller than a meter but are thinner and not at wide as a meter so for this discussion 1 meter will be the scale of comparison we use.

Moving up in scale:

10 m            About the size of a garage or apartment

100 m          The length of athletic field or pitch

1000 m        The span of a large bridge (The New York George Washington bridge has a span of 1067 m)

10,000 m     A 10k race in which many runners have competed

100,000 m   A distance typically traveled in an hour by automobile at highway speeds 

10⁶ m            A full day's driving on a long trip or traversing the north-south extent of California

10⁷ m            The distance from the north pole to the equator through Paris (by definition of meter)

10¹¹ m          The distance from the Earth to the Sun (One AU = 1.5 10x11m)

10¹² to 10¹³ m   The orbit of Neptune is 30 AU

10¹⁶ m          The distance to the Oort cloud (the furthest extent of our Solar system)

10²¹ m          The diameter of the Milky Way Galaxy

10²⁷ m          The radius of the observable universe

View this animation on the large scale of the universe: (1535) your mind will collapse if you try to imagine this | UNIVERSE SIZE COMPARISON - YouTube

Another good visualization is given here: https://bigthink.com/starts-with-a-bang/logarithmic-view-universe/

Moving down in scale

0.1 m            A human hand width

0.01 m          A small finger width

0.001 m        1 mm or about the thickness of a credit card

0.0001 m      A fine human hair or diameter of a human egg cell (the largest human cell)

0.00001 m    Diameter of a human capillary vessel

10^-6 m           The low end of the size of a bacterium (one micron) 

10^-9 m            Size of a molecule  (one nanometer) 

10^-10 m          Angstrom - approximate size of a hydrogen atom

10^-15 m           Size of a proton

10^-19 m           Upper limit of the theoretical size of a quark

10^-35 m          The Planck length is 1.62 x 10^-35 m (smallest possible dimension)

Now, let’s consider the ratio of the very largest thing, the observable universe, to the vary smallest, theoretical size, the Planck length. The ratio is approximately:
10²⁷ m: 10^-35 m or 10⁶² to 1. Humans are somewhat in the middle but closer to large end in the comparison given above. If we only venture down to the size of a quark, then we closer to small end.

Perfect Number Generator

 

A perfect number is a positive integer which equals the sum of all its divisors, excluding itself. The first perfect number is 6 since 6 = 1 + 2 + 3. The next perfect number is 28 since 28 = 1 + 2 + 4 + 7 + 14.

Over two thousand years ago, Euclid found a formula for generating perfect numbers. The product of
(2^p-1) and (2^p-1) is a perfect number when p is a prime and 2^p-1 is also a prime number. Many high prime numbers can be found using the formula 2^p-1, but the formula doesn’t always produce prime numbers. When the formula does produce a prime number, it is called a Mersenne prime named after the French Friar, Marin Mersenne (Marin Mersenne - Wikipedia).

The ancients knew of the first four perfect numbers: 6, 28, 496 and 8128. Through trial division, the list was confirmed with three additional perfect numbers: 33550336, 8589869056 and 137438691328. In 1772 Euler found the eighth: 2305843008139952128 (with no modern computing aids). Euler also proved the converse of Euclid’s original proof, that is even numbers are perfect if and only if they can be expressed in the form (2^p-1) (2^p-1).

It is known that there are an infinite number of primes; however, it is not known if there are an infinite number of Mersenne prime numbers. As of this posting, there have been 51 confirmed Mersenne primes (the largest has over 24 million digits). If one were able to prove the Mersenne primes continue infinitely, then so would perfect numbers. The size of the perfect number associated with the largest known Mersenne prime has over 49 million digits.

Here is a link to a spreadsheet for calculating the first eight perfect numbers.

How Many Black Holes are there in the Universe?

 

In a recent post, we explored the estimated number of planets in the universe which could be as high as 10²⁸ including rogue planets (not orbiting a star). The existence of black holes has been predicted for over a hundred years, but it wasn’t until 1964 when a black hole was actually detected (Cygnus X-1).

A black hole cannot be directly observed, but clear evidence for its presence can be detected as material is accelerated toward the black hole and is heated or by a black hole’s large mass can deflect light from stars. In the Astrophysical Journal this month, a paper provided an estimate of the black holes in the universe to be 40 x 10¹⁸ (40 quintillion) and they may make up 1% of the matter of the universe.

Image: NASA Goddard Media Studio - GMS: Black Hole Accretion Disk Visualization (nasa.gov)

Also see possible black hole image at M87.

Also see: Rogue Planets.

Number of Planets in the Universe

The first exoplanets were detected in 1992 and since then nearly 5000 been detected. The pace of these discoveries will accelerate greatly once the James Webb Space Telescope (JWST) becomes operational later this year (as of the time of this writing, the JWST was just three days from reaching its final position at the Lagrange Point L2. (Related post from book with profile of 20 exoplanets).

The site Big Think has a nice estimate of planets in the universe. It estimates that there are over a trillion galaxies and each galaxy has a trillion to ten trillion planets orbiting its starts putting the total at 10²⁵. The article brings up the fact that there are also rogue planets that do not orbit starts which can put the total number of planets 10 to 10,000 times higher. These rogue planets are also described in the book, Imagined Life: Math Vacation: Book Review: Imagined Life by James Trefil and Michael Summers (jamesmacmath.blogspot.com).

Also see this post regarding number of black holes in the universe: Math Vacation: How Many Black Holes are there in the Universe? (jamesmacmath.blogspot.com)

Update 1/24/2022: The JWST successfully entered into its halo orbit at L2 - Where Is Webb? NASA/Webb.

Update 2/12/2022: The JWST took its first image of a star. This part of the long process to align the 18 primary mirrors and the secondary mirror to produce a single image. The image (below) and the alignment process are described at the NASA JWST site.

Image Credit: NASA - Post: Edit (blogger.com)

Update 2/22/2022
The alignment of the JWST mirrors continues as the image above has been adjusted to the hexagonal arrangement of the individual mirrors: JWST alignment continues, light-points now in hexagonal formation - Space Explored

Furthermore, considering a prior post about other civilizations in our galaxy, the JWST may be able to gain evidence of remote civilizations by detecting their pollution or by detecting evidence of photosynthesis on distant planets: Could Nasa's James Webb Space Telescope detect alien life? - BBC News.

Update 2/26/2022

The JWST made another step toward its mirror alignment. It stacked the eighteen individual images into one. Before and after images below (from Webb Mirror Alignment Continues Successfully – James Webb Space Telescope (nasa.gov)). 

Before:

After:

Images Credit: NASA/STScI

Also see: Rogue Planets

The specialty of the JWST is imaging in the infrared spectrum. NASA has a good article here: https://webb.nasa.gov/content/science/firstLight.html.

The James Webb Space Telescope recently found some rogue planets:

https://www.space.com/james-webb-space-telescope-rogue-planets-star-formation

One Hundred Trillion

My last post wrote about the approximate one trillion platelets in my blood donation. Another post described the power of compound interest in which a very modest investment could grow substantially over four hundred years at just a modest interest rate.

There is another example of when interest rates are very high. The enormous multiplying effect doesn't take hundreds of years - only months. That it what led the country of Zimbabwe to issue a one hundred trillion dollar note. At the peak of this period of hyper-inflation around 2007-2009, the cash in your pocket could be worth half, or less, than what it was the day before. In a single year, prior to the issuance of the note shown above, the country issued notes ranging from 10 dollars to 100 billion dollars.

When Zimbabwe gain independence in 1980, their dollar was about worth one United States dollar (USD). When my friend gave me the 100 trillion dollar note shown, it was worth about one USD.

Update 8-17-2021 - see this post for how pi has been calculated to over 62 trillion digits.

One Trillion!

I'm a regular blood donor and in the photo I'm shown with the local blood bank staff as I surpassed the 10-gallon milestone (we're all in masks since this donation was made during the Covid crisis of 2020). This particular session, I donated both blood and platelets. As I finished, I reviewed the monitor on the apheresis machine that summarized my donation: 980,000,000,000 platelets collected (just 2% shy of one trillion).

1/15/2022 Update - I recently reached my 15-gallon milestone. I checked my last platelet donation, and the count was 1.1 trillion.

Pi - How Many Digits are Really Needed?

A prior post showed how to estimate pi using random numbers. The spreadsheet linked to the post can estimate pi correctly to about 3 places; however, since each set of random numbers is different, the results will vary. Readers are encouraged to extend the spreadsheet beyond 1000 random numbers to improve the accuracy of the estimation. 

Pi taken out to 50 decimal places is:

3.14159265358979323846264338327950288419716939937510...

Princeton University has a post where the digits of pi are listed to 10 million places. Beyond 100 places does not provide much immediate value. Just using pi to the 50 places listed above, one could calculate the circumference of the observable university to an accuracy smaller than the diameter of a single hydrogen atom. Readers are challenged to verify given that the estimate of the diameter of the observable universe is 8.8 x 10^26 m and the size of a hydrogen atom is about 1 angstrom (1/10,000,000,000 m).

For simple estimations, consider the following approximations and their relative error:

Approximation     Relative Error

3                            .045

22/7                       .0004

355/113                 .00000008

Update (8-17-2021): A Swiss team recently established a new record for calculating pi to the most digits: 62.8 trillion.

Compound Interest

I recently read a debate between two colleagues who disagreed on the math topics that should be taught to high school students. The essence of the disagreement boiled down to classical topics versus practical. I'll offer one topic of practical importance that should be emphasized, compound interest. Understanding the power of compound interest over long period of times makes the difference of wealth accumulation versus debt accumulation. 

Interest rates for savers are currently very low, but just two or three years ago, I bought a 5-year CD with 3.3% interest. The power of compound interest is that one earns interest on the principal and on the earnings. Simplified at bit, $100 becomes $103.30 after one year. After 5 years, the initial investment becomes: $100 x (1.033) x (1.033) x (1.033) x (1.033) x (1.033) = $117.63. That is a bit more than the $116.50 that would have been earned under a simple interest arrangement ($100 x 5 x 1.033). 

On the debt-accumulation side, things are much worse as credit card companies are not generous in their terms to consumers. Consider a debt of $100 carried for 5 years at the 18%. This debt now becomes, $228.78.

The amount by which the debt or the asset grows depends on two key things, the interest rate and time. Above we saw how the grow varies by interest rate, but time becomes a big factor especially when long periods of time are involved.

For a final example, let's look at the purchase of Manhattan. This week marks the 396th anniversary of when Dutch settlers traded $24 dollars worth of trinkets in exchange for the now, very valuable, New York real estate. Had the sellers been able to invest in a 396-year certificate of deposit, what would their proceeds be worth now? That would be $24 times (1.033) x (1.033) ... x (1.033) or $24 x (1.033) raised to the power of 396 which now would be $9,203,194. The investments didn't exist at the time, but one with a very long-term time frame would be told by financial advisors to invest in something with more risk, like stocks. The last sixty years or so, the S&P500 index has yielded about 8%. An investment at 8% over four centuries becomes astronomical so for the final calculation, let's back that average to 6.5%. The $24, if invested in 1624, would now be worth $24 x (1.065) ^ 396 = $1.6 trillion. A search for the current value of all Manhattan real estate revealed the current value to be approximately $1.74 trillion.