a collection of short slideplayers about googology

all the articles on this page include 2 slide player presentations. This article was not created by me but by someone else I don't know, later I will send you the slide player link, there are several presentations here

Googol

https://slideplayer.com/slide/9571083/ created by Adela Mason 8 years ago

100000000000000000 000000000000000000 000000000000000000 000000000000000000 000000000000000000 00000000000 Googol

10¹⁰⁰

1. The number of seconds since the Universe began is 473040000000000000 15 billion years x 365 days x 24 hours x 60 minutes x 60 seconds
2. 1 googol of salt grains is… 39,909,940,000 universes of space
3. A googol 1 followed by 100 zeros
4. A googolplex 1 followed by a googol of zeros
5. A googolplexian 1 followed by a googolplex of zeros
6. Counting to: Would take this long: one million 5 days, 18 hours, 53 minutes, 20 seconds one billion15 years, 308 days, 9 hours, 41 minutes, 50 seconds one googol 1.584 * 10 90 centuries!!!
7. Googolhedron 3 dimensional shape bounded by 1 x 10 100 similar polygons. This shape would look very much like a sphere. Having this many sides or facets would make it smoother than any man made object. One could never have a googolhedron because there are not a googol particles in the universe

Seeing things differently!

https://slideplayer.com/slide/4052196/, created by Kamron Ingraham 9 years ago

ויהי בבואם וירא את־אליאב ויאמר אך נגד יהוה משיחו

But the LORD said to Samuel, "Do not look at his appearance or at the height of his stature, because I have rejected him; for God sees not as man sees, for man looks at the outward appearance, but the LORD looks at the heart."

  —1 Samuel 16:7

A better view

and raised us up with Him, and seated us with Him in the heavenly places in Christ Jesus,

  —Ephesians 2:6

Therefore if you have been raised up with Christ, keep seeking the things above, where Christ is, seated at the right hand of God.

  —Colossians 3:1

How to receive his promises

The steadfast of mind You will keep in perfect peace, Because he trusts in You.

  —Isaiah 26:3

If you ask Me anything in My name, I will do it.

  —John 14:14

In that day you will not question Me about anything. Truly, truly, I say to you, if you ask the Father for anything in My name, He will give it to you.

  —John 16:23

This is the confidence which we have before Him, that, if we ask anything according to His will, He hears us.

  —1 John 5:14

A new way to learn

that is the Spirit of truth, whom the world cannot receive, because it does not see Him or know Him, but you know Him because He abides with you and will be in you.

  —John 14:17

"But when He, the Spirit of truth, comes, He will guide you into all the truth; for He will not speak on His own initiative, but whatever He hears, He will speak; and He will disclose to you what is to come

  —John 16:13

את־הכל עשה יפה בעתו גם את־העלם נתן בלבם מבלי אשר לא־ימצא האדם את־המעשה אשר־עשה האלהים מראש ועד־סוף

He has made everything appropriate in its time. He has also set eternity in their heart, yet so that man will not find out the work which God has done from the beginning even to the end.

  —Ecclesiastes 3:11

A googol is the large number 10¹⁰⁰, that is, the digit 1 followed by one hundred zeros in decimal representation. large number digi tzeros decimal large number digi tzeros decimal

Other names for googol include ten duotrigintillion on the short scale, ten thousand sexdecillion on the long scale, or ten sexdecilliard on the Peletier long scale namesshort scalelong scalePeletier long scalenamesshort scalelong scale Peletier long scale

A googolplex is the number 10^googol, which can also be written as the number 1 followed by a googol zeros (i.e., 10¹⁰⁰ zeros). googol zerosgoogol zeros

1 googolplex = 10^googol googol =

10⁽¹⁰¹⁰⁰) = 10^{10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,00 0,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000}

Man struggles to grasp eternity!

1. Magnitude of Grahams number
2. To convey the difficulty of appreciating the enormous size of Graham's number, it may be helpful to express— in terms of exponentiation alone—just the first term (g1) of the rapidly growing 64-term sequence. First, in terms of tetration ( ) alone
3. tetration   where the number of 3s in the expression on the right is
4. $g(1)=3\uparrow \uparrow \uparrow \uparrow 3=3\uparrow \uparrow \uparrow (3\uparrow \uparrow \uparrow 3)=3\uparrow \uparrow (3\uparrow \uparrow (3\uparrow \uparrow (3\uparrow \uparrow ...(3\uparrow \uparrow 3))))$
5. Now each tetration ( ) operation reduces to a "tower" of exponentiations ( ) according to the definition
6. and where the number of 3s in each tower, starting from the leftmost tower, is specified by the value of the next tower to the right
7. In other words, g1 is computed by first calculating the number of towers, n = 3^3^3^...^3 (where the number of 3s is 3^3^3 = 7625597484987), and then computing the nth tower in the following sequence:

We need spiritual ability to see eternal stuff.

ויתפלל אלישע ויאמר יהוה פקח־נא את־עיניו ויראה ויפקח יהוה את־עיני הנער וירא והנה ההר מלא סוסים ורכב אש סביבת אלישע

Then Elisha prayed and said, "O LORD, I pray, open his eyes that he may see." And the LORD opened the servant's eyes and he saw; and behold, the mountain was full of horses and chariots of fire all around Elisha.

  —2 Kings 6:17

ויאמר אל־תירא כי רבים אשר אתנו מאשר אותם

Do not fear, for those who are with us are more than those who are with them

  —2 Kings 6:16

You are from God, little children, and have overcome them; because greater is He who is in you than he who is in the world.

  —1 John 4:4

We are in the best relationship

For you have not received a spirit of slavery leading to fear again, but you have received a spirit of adoption as sons by which we cry out, "Abba! Father!

  —Romans 8:15

There is no fear in love; but perfect love casts out fear, because fear involves punishment, and the one who fears is not perfected in love.

  —1 John 4:18

והוא ישפט־תבל בצדק ידין לאמים במישרים ויהי יהוה משגב לדך משגב לעתות בצרה ויבטחו בך יודעי שמך כי לא־עזבת דרשיך יהוה

The LORD also will be a stronghold for the oppressed, A stronghold in times of trouble; and those who know Your name will put their trust in You, For You, O LORD, have not forsaken those who seek You. Sing praises to the LORD, who dwells in Zion; Declare among the peoples His deeds.

  — Psalms 9:9-11

Take time to listen to Him

So faith comes from hearing, and hearing by the word of Christ. 

  —Romans 10:17

This is the only thing I want to find out from you: did you receive the Spirit by the works of the Law, or by hearing with faith?

  —Galatians 3:2

So then, does He who provides you with the Spirit and works miracles among you, do it by the works of the Law, or by hearing with faith?

  —Galatians 3:5

Thought Experiment

https://slideplayer.com/slide/8140446/ created by Madlyn Collins 8 years ago

1. Take a piece of paper one thousandth of an inch thick
2. Now fold it in half and then in half again
3. Do this 50 times (I know that this is not practicable)
4. How thick do you think it will be ?
5. Each time we fold the paper, it doubles in thickness So after the first fold it is 2x as thick
6. After the second fold it is 2x2 ie 4 times as thick
7. After 50 folds it is 2x2x2.......... 2x2 times as thick
8. That is 2⁵⁰
9. Divide by a thousand to get the thickness in inches
10. Divide by 63,360 to get the thickness in miles And we get......
11. $2^{50}==1125899906842624$
12. Dividing by a thousand = 1125899906842 inches
13. Dividing by 63360 = 17,769,885 miles
14. Carol recently gave a talk on the metal Bismuth.
15. One of its interesting facts was it's half-life This is currently calculated at :- 1.9 x 10¹⁹ years
16. If we assume that there are 64 generations between now and the start of the Christian Era Then you will have approximately 1.8 x 10¹⁹ ancestors
17. There is a story about an Indian temple which contains a large room with three posts in it surrounded by 64 golden disks.
18. Brahmin priests, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the immutable rules of the Brahma, since that time.
19. The puzzle is therefore also known as the Tower of Brahma puzzle. According to the legend, when the last move of the puzzle will be completed, the world will end
20. if the priests were able to move disks at a rate of one per second, using the smallest number of moves, it would take them 2⁶⁴ −1 seconds or roughly 585 billion years or 18,446,744,073,709,551,615 turns to finish ie 1.8 x 10¹⁹ moves The world will certainly have ended !!!
21. Just a little larger are the 4.3 x 10 19 different starting positions for a Rubik's Cube Any ideas of the smallest number of moves required to solve the cube from any starting position ? (Known as God's Number)
22. Any cube can be solved in 22 moves at most, but there is strong evidence that only 20 moves are required. Hence God's Number is either 20 or 22
23. Consider this 200 digit number 914743972814745128948036774162014302835642105 034332853395613272769334542296093046464719250945181147710162588965929074414263498975565041 45570960203925503679105245199142338806082494254050610000000000000 You now have 70 seconds to extract its 13th root assuming that you wish to claim the world record
24. The answer is 2,407,899,893,032,210
25. Compare this to :- Age of the universe 13.798×10⁹ years
26. or 4.354×10¹⁷ seconds Number of elementary particles in the observable universe between 10⁸⁰ and 10⁹⁷
27. But these numbers are mere beginners Googol, Googolplex, Skewe's number Gijswijt's sequence, Ramsey Numbers, Graham's Number etc
28. Googol Originally named by Edward Kasner a Googol = 10¹⁰⁰ ie 1000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 (The search engine Google copied the named but spelled it incorrectly)
29. Any idea what's special about this number ? 2^57885161 -1 this is approximately 10^10000000
30. It's the largest prime number currently know It was discovered in 2013 and is some 17.5 million digits long It is also the 48th Mersenne Prime ie of the type 2 n -1 It's the largest known number that cannot be expressed in terms of smaller numbers
31. Googolplex There isn't enough room or time to write out this number If you can write 2 digits/second, you would need 1.5 x 10⁹² years, which is some 10⁸² times greater than the age of the universe 10¹⁰¹⁰⁰ or 10^googol
32. Skewes' Number Gauss generated a formula to give the number of Primes up to any given number. It is known that it generally over-estimates the true number. Stanley Skewes showed that Gauss's formula would under- estimate when we got above the number that bears his name
33. 10¹⁰¹⁰³⁴ This result, however depends upon the Riemann Hypothesis being true If the Riemann Hypothesis is ever shown to be false, then Skewes number increases to Which is 10^1010963
34. Gijswijt's sequence: 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 1............. where does 4, 5, etc appear ?
35. The first time 3 appears is in the ninth position. 4 appears for the first time in the 221 st position. The first 5 occurs in position 10^{100000000000000000000000} ie 10¹⁰²³
36. The other numbers will eventually appear, although with no sense of urgency
37. Ramsey Number: You are going to have a party. How many people do you need to invite to be sure that at least 3 people will be mutual aquaintances, or at least 3 people will be mutual strangers ?
38. However, if we need to know the minimum number of guests such that either 5 people are mutually aquainted or mutually strangers,
39. there are some 2⁹⁰³ or 6.7 x 10²⁷¹ cases to try out using brute computer force alone !
40. We do know that between 43 and 49 guests would be required
41. Grahams number: If we now consider Ramsey's numbers but venture into multi-dimensional space the numbers get very large. So large, that we need a whole new way to represent them
42. Once again, say we have some points, but now they are the corners of an n dimensional hypercube.
43. They are still all connected by blue and red lines.
44. For any 4 points, there are 6 lines connecting them.
45. Can we find 4 points that all lie on one plane, and the 6 lines connecting them are all the same colour? Graham's Number gives an upper limit of the points needed for this to be certain
46. Knuth's up-arrow notation Invented by Donald Knuth in 1976 It uses a series of up arrows and looks like :- g 64 =3↑↑↑↑ g 63 times ↑↑↑↑3 If each digit could occupy a Planck length 10 -35 m, then you could get 10 35 x 10 35 x 10 35 digits ( ie 10 105 ) in a cubic meter, there would still not be enough space in the universe to write it down
47. We do know the last 500 digits of his number - the last digit is a 7 Also the lower limit is 13

More than any other question the infinite always has moved so deeply the human soul

https://slideplayer.com/slide/14909234/ created by Richard Waltz 5 years ago

More than any other question the infinite always has moved so deeply the human soul. More than most other ideas the infinite has affected so inspiringly and fruitfully the human mind. But more than any other notion the infinite is in need of elucidation.

  —David Hilbert

But the consideration of the unlimited has a difficulty. For, there result a lot of impossibilities, may we assume that it does not exist or that it does exist.

  —Aristotle

History of the Infinite

• I Naturally infinite II Towards infinity
• III Alogos
• IV Infinitesima
• lV Unlimited
• VI Microscopical
• VII Cosmical
• VIII Eternal
• X Theological
• lX Transcendental
• XI Transfinite
• XII Infinite

Naturally Infinite

Four categories of being

Finite ᅳ

Unlimited ᄋ

Potentially infinite 1,2,3,4,5 ...

Actually infinite {1,2,3,4,5 ...}

The infinite exists only in potential form. There is no actual infinity (except the divine).

There are only finite numbers. The finite would be eliminated and destroyed by the infinite if this existed.

  —Aristotle

Infinitum actu non datur.

  —Medieval Scholastics

John Wallis ( ) Used the symbol ∞ for the first time 1655 in his Arithmetica Infinitorum.

Latin: 100 millions

Greek: Hippopede

We recognize that there is infinity but don't know anything about its nature. It is not even, it is not odd.

By +1 it is not changed.

  —Blaise Pascal

God made the integers. The rest is man-made.

  —Leopold Kronecker ( )

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

• 
  Son of an obviuously good-natured Pisa consular officer nicknamed Bonaccio.
• Called himself Fils Bonacci (Son of Bonaccio).
• Merchant, travelled to Egypt, Syria, Greek, Sicily, Provence, became acquainted with Arabic (= Indian) numerals.
• Later at the court of Friedrich II showed exhibition mathematics:
• x = 1°22'7''42'''33''''4'''''40'''''' =Mathematica gives 1.36880810782...

Fibonacci sequence, first recurrence sequence (outside india)

$a_{n-1}=a_{n}+a_{n-a}$

$a_{1}=a_{2}=1$

Gentle and content and kindly disposed toward every friend of mathematics.

But: "Hand the boy a coin!"

  —Museion of Alexandria

The Elements (στοιχεια), about 1500 printed editions.

All former text books lost without trace - later unknown.

Books I-VI: plane geometry

Books VII-X: arithmetic

Books XI-XIII: spatial geometry

Euclidean form: definition, theorem, proof, final clause.

quod erat demonstrandum.

όπερ εδει δείξαι.

There are more than any given number of primes.

Let P = P1P2P3P be the product of all primes.

(P + 1) divided by one of these primes leaves remainder 1.

Therefore Pis a prime itself or it contains another prime Pn+1

2-3-5-7-11 13 + 1 = 30031 = 59-509

2-3-5-7 11 13-1= 30029

But there are gaps between primes larger than any number

n. n! + 2, n! + 3, n! + 4, ..., n! + n

All terms are divisible. n! is divisible by all natural numbers 2, 3, n. n! + 2 is divisible by 2, n! + 3 is divisible by 3,

Archimedes (287 - 212 BC)

Educational journey to Egypt, Alexandria, Museion Correspondence with Eratosthenes.

• buoyancy force (heureka)
• rule of lever
• block and tackle ("give me a fixed point ...")
• calculating the center of mass
• water screw
• exhaustion
• π
• parabola
• spiral

Machinery of war (block and tackle, catapult, concave mirror). He defended Syracuse over two years nearly alone against the Romans. Defeat by betrayal.

Many people believe, o my King Gelon, the number of sand grains be infinite. Others think that this number is not unlimited but that never such a large number could be named. But I will try to show that among the numbers that I have determined already there are numbers surpassing the number of sand grains not only in a heap of sand as large as our earth but even if the whole universe was filled with sand.

The Sand Reckoner

Less than a myriad (10000) grains of sand are due to a poppy seed.

Less than 40 poppy seeds laid side by side make up more than a finger-breads (2 cm). 1 stadion (185 m) < 1 myriad finger-breadths (200 m).

The diameter of the earth < 100 myriad stadia. The diameter of universe < 1 myriad earth diameters.

64 x 10⁵⁷ grains of sand can be placed in the universe known at his times, 10⁶³ inside the sphere of fixed stars.

"And we can move on."

Archimedes could move on, but without powers:

ai myriakismyriostas periodou myriakismyrioston arithmon myriai myriades

= 10^{8 x 1016}