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

Sublime Numbers

 

(Image: Grok)

The numbers 12 and 6086555670238378989670371734243169622657830773351885970528324860512791691264 are the two entries in the OEIS sequence: A081357, Sublime Numbers. 

In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.

The first entry, 12, has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.

The second entry, above, is the product: (2¹²⁶)(2⁶¹ − 1)(2³¹ − 1)(2¹⁹ − 1)(2⁷ − 1)(2⁵ − 1)(2³ − 1). 

30,000

 

(Image: Gemini Generated)

As of 7/27/2024, this blog had its 30,000th view. That is certainly not Tik Tok influencer volume, but I'm happy that thousands of visitors have had the opportunity to learn a few new things.

I did a preliminary search for 30,000 to see if there are any interesting facts. Wikipedia has listings for many numbers and it led me to its page where it lists interesting numbers in ranges. For the range 30,000 to 30,999, it listed the following with a few referencing sequences from On-Line Encyclopedia of Integer Sequences (OEIS) - a favorite site of this blog.

• 30029 = primorial prime
• 30030 = primorial^[1]
• 30031 = smallest composite number which is one more than a primorial
• 30203 = safe prime
• 30240 = harmonic divisor number^[2]
• 30323 = Sophie Germain prime and safe prime
• 30420 = pentagonal pyramidal number^[3]
• 30537 = Riordan number
• 30694 = open meandric number
• 30941 = first base 13 repunit prime

https://en.wikipedia.org/wiki/30,000

 1. Sloane, N. J. A. (ed.). "Sequence A002110 (Primorial numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

2. Sloane, N. J. A. (ed.). "Sequence A001599 (Harmonic or Ore numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

 3. Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Large Numbers

In advance of Pi Day, March 14, New Scientist recently published an article describing some very large numbers: These 7 mathematical facts will blow your mind.  The article includes some familiar numbers, such as Tree(3) and the Dedekind Numbers, and some additional curiosities including Penrose Tiles, mazes, and octonions (the 8-dimension cousin of quaternions. 

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 9 Most Massive Numbers in Existence (Tia Ghose article in Live Science)

The website Live Science  recently published an article by Tia Ghose, The 9 most massive numbers in existence. It covers many subjects that have been discussed in this blog, including Graham's Number, the scale of the universe, and prime numbers. Below is a link to the article:

The 9 most massive numbers in existence

Count to 9,192,631,770 in One Second

Counting to a very large number, such as 9,192,631,770, in on second may seem like an impossible task, but it is done on a very regular basis by atomic clocks. The International System of Units defines one second as 9,192,631,770 vibrations of the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom.

More on atomic clocks can be found here: https://en.wikipedia.org/wiki/Atomic_clock.

Official time in the United States is maintained by the U.S. Naval Observatory. A link to the official time is below:

https://www.time.gov/

3,628,800

This post is inspired by a recent puzzle published by Alex Bellos in the Guardian: Did you solve it? Puzzles you can do in the pub | Mathematics | The Guardian

The title of the post is 10! = 1x2x3x4x5x6x7x8x9x10 = 3,628,800. Interestingly, this value is the same number of seconds in six weeks. The challenge in Bellos's puzzle was to make the comparison without using a calculator.

One can confirm this without a calculator. Write out the number of seconds in six weeks as:

6 (weeks) x 7 (days) x 24 (hours) x (60 minutes) x (60 seconds)

Now one can cancel factors in the above equation by matching up with terms in the equation for 10!.

Cancel the 6 in both equations.

Cancel the 7 in both equations.

Cancel the 24 by matching with the 3 and 8 in the 10! equation

Reduce the 60 to 6 in the seconds equation by cancelling the 10 in the 10! equation.

Reduce the other 60 to 6 in the seconds equation by canceling the the 2 and the 5 in the 10! equation.

These canceling operations leaves us with 6x6=36 in the seconds equations and 4x9=36 in the factorial equation, therefore the number of seconds in six weeks equals 10!.

Alex Bellos is also the author of one of my favorite math books: Math Vacation: My Favorite Math Websites (jamesmacmath.blogspot.com)

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.

Tribute to Ronald Graham's Largest Number ...262464195387

What is the largest number? Any number that is offered can be bettered by that number plus 1. A common answer is infinity, although infinity isn't a specific number. "Infinite" describes something that is without bounds. Something could be infinitely large (set of integers) or something could be into infinitesimally small (as done in calculus). 

Another approach to this question of the largest number is to ask what is the largest number used in a proof. In 1977 the mathematician Ronald Graham established the world record for the largest specific integer used in a mathematical proof. Graham's number is so large that if all the atoms in the universe were made into ink, the number could not be written. That is certainly disappointed for the readers of this post who wanted to see the number.

However, the last digits of Graham's number are: ...262464195387.

The actual expression of Graham's number uses hyperoperators which are higher order forms of exponentiation. 

Since 1977, Graham's number has been exceeded by larger super-numbers used in proofs. One example is TREE(3).

Ronald Graham passed away in 2020 and was honored in a Numberphile podcast. An earlier podcast specifically described Graham's number. While the end digits of Graham's number have been determined, the first digit is unknown. Graham was asked what digit he would like it to be and he said he actually knew the first digit but only when the number is expressed in base-2 and then it is 1.

The icon I chose for this post is juggling because Ron Graham was an accomplished juggler. For a photo of Graham, see New York Times Obituary Link with photo.

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.