Brians Number Big Numbers Take the next 30

Brian’s Number

Big Numbers Take the next 30 secs to write down the largest (real) number that you can think of. • Rules: • It must fit on a 3 x 5 card • Numbers only, no names • Can’t say “one divided by plank’s constant” • Operations are okay • Instead of just 99, 9999, 999 you could write (99, 9999, 999)9 • Go!

Some big numbers • 9, 999, 999, 999, 999, 999 • Impressed? • 1 less than 1055 • No longer as impressive • In fact, there are more possible ways to shuffle a standard deck of card. • 52! = 80, 658, 175, 170, 943, 878, 571, 660, 636, 856, 403, 766, 975, 289, 505, 440, 883, 277, 824, 000, 000 • Roughly 8*1067 or 80 unvigintillion

Mine was 34 • Not 34 • The 3 is in the subscript. • Ever seen this notation before? • (I made it up) • It’s pretty big, but I guess I need to explain the notation first.

Operations • How many (binary) operations do you know of? • Elementary Arithmetic Operations • Addition, Subtraction, Multiplication, Division • But multiplication is just repeated addition, right? • So what’s repeated multiplication? • Exponentiation • What’s repeated exponentiation?

Hyperoperations • Infinite sequence of operations. • Defined iteratively as repeating the prior operation • • • 1 st is addition 2 nd is multiplication 3 rd is exponentiation 4 th is called tetration 5 th is called pentation (0 th is just counting)

Hyperoperations •

Notation •

An example •

My number was 34 • This is not tetration! • It’s the next hyperoperation, called pentation • At least, it’s how I imagine one would write pentation with Rucker’s notation • ab already taken with exponentiation • and ba is already take with Tetration • So where do you “put the b” ? • Why not “bottom left”?

Pentation •

Let’s start smaller •

10 2 is (almost) a nice round number • Doubling game • • What is 22 ? 23 ? 24 ? 210 is slightly more than 1, 000. • Aka 103 • (which is only “round” because of our arbitrary base 10 system, but I digress)

Back to 24 •

Keep simplifying, this is fascinating! •

Enough simplification! •

Click a pen •

Clicking assumptions •

Collective pace •

What about me? • As the only human that isn’t pen clicking, I’ve got to find a way to entertain myself • I’ve always wanted to visit Mt Everest. • But travel is impossible with everyone clicking. • What if I brought Mt Everest to me? • Suppose I genetically modified a species of ants. • Let’s call them Ant-thony’s. • I trained these guy to walk from SOU to Mt Everest with an ant wheelbarrow and bring me back one grain of dirt. • The grain is roughly spherical with a radius of 0. 2 mm

Ant-thony’s pace • The good news: • An ant’s walking speed is 2 inches/second. • 12 hour of walking per day = 86, 400 inches or a little over 2 km per day. • Ant-thony’s are relentless and live only to move the wheelbarrow and procreate.

Ant-thony’s pace • The bad news: • I made the wheelbarrow too heavy. • It’s only once every million years that an ant-thony comes along strong enough to move the wheelbarrow. • And he only moves it 2. 4 cm, then dies.

The Journey Begins • So in the year 1, 002, 017 along comes Ant-thony the Great, who pushes the wheelbarrow 2. 4 cm toward Mt Everest, then dies. • The colony hangs out there until 2, 0002, 017 when Ant-thony the great 2 (ATG 2) pushes the wheelbarrow another 2. 4 cm. • RIP ATG 2. • And so it goes…

Ant-thony’s 12, 000 Km Journey

Dedicated Ants • It takes them a while • 2. 4 cm per million years • They need to walk 12, 000 kilometers • Aka 12, 000 meters • Aka 1, 200, 000 centimeters • So it would take 500, 000 “Anthony the Greats” • So 500, 000, 000 years • But they also need to get back, so double it. • 1015 years!

Welcome home Ant-thonys! • It’s the year 1, 000, 002, 017 when the Anthony’s return • It’s worth noting that in 1015 years our sun has long since burned out. • In fact, all stars currently alive will be gone. • But the ants keep marching! • And the people keep clicking. • We done yet?

Click count • 1015 years with 1018 digits per year • • our computer has written 1033 digits That’s a lot. More digits than has been written by humanity in its history (by a lot). But nowhere near 2*10153 digits.

I’m not satisfied • It’s 1, 000, 002, 017 and the ants have delivered a grain of Everest • Not to appear ungrateful, but… • I wanted Mt Everest, not one grain! • “Thanks, Ant-thonys, but it’s back to work for you guys. ” • Every 1015 years they drop off another grain of Everest to King Brian.

Quick Everest Aside •

Volumes • Ant-thony’s grain of dirt was spherical with radius 0. 2 mm • So Volume is 0. 034 mm 3 • That means everest is made up of 333, 561, 741, 000, 000 / 0. 034 = 1022 grains of dirt. • Need 1022 trips, each lasting 1015 years for the ant-thony’s to transport Everest to SOU! • That’s 1037 years! Basically everything is gone. Here’s what wikipedia says: “…this means that after 1037 years, one-half of all baryonic matter will have been converted into gamma ray photons and leptons through proton decay. ” Whatever.

Done! • In the year 10, 000, 000, 000, 002, 017 Mt Everest arrives at SOU! • The sun has long since burnt out. • In fact, all stars have long since burnt out. • But I have my very own Everest. • How’s the clicking going? • 1037 years, 1018 clicks per year, so 1055 clicks • Long way to go.

Now what? • With nothing better to do, I decide to ask the ants to return Mt Everest to Nepal. • In the year 20, 000, 000, 000, 002, 017 they finish! • The grateful Nepalese government awards me a $1 T bill • Inflation would be rough, the bill would be worthless with any (constant) inflation. • (Try to calculate the inflation rate to make a $1 T bill worth one millionth of a penny)

Stackin’ Dollars • I take my worthless $1 T bill, place it on the ground • Currency is. 1 mm thick, by the way • I ask the ants to entertain themselves by bringing Everest back and forth. • Every 2*1037 years I get another $1 T bill. • I place each $1 T bill on top of the previous one. • When I’ve earned 100 million of these bills, my stack is 10, 000 mm high, aka 10, 0000 m. • My dollar stack is taller than Mt Everest itself.

Stack • 100 Million is 100, 000 aka 108 bills, each taking 2*1037 to earn • So it’s been 2*1045 years • Or 2*1063 clicks. • Long way to go.

Proxima Centuri • The closest star to us (other than our sun) • • 4. 25 light years away Roughly 5*1013 km Or 5*1019 mm Or 5*1020 bills • Our stack reaches Proxima Centuri after (5*1020) * (2*1037) years • Or, at least, where it used to be. It’s gone now. • 1058 years, or 1076 clicks.

This is getting ridiculous • 1076 is nowhere near 2*10153… so repeat. • Each time we reach Proxima Centuri, I pick an atom on earth and name it. • When I finish with earth I tackle another planet in our solar system • When I’m done with our solar system, I pick another star in our galaxy. • There are 1068 atoms in our Galaxy • When I’m done naming every atom in the galaxy, the click count is up to 10144 • What just happened? Can anyone recap the entire sequence of events?

Almost done? • 144 is close to 153 • 10144 is not close to 10153 • In fact, it’s 0. 0000001% of the way there • And our number is twice as long as a number with 10153 digits • TL; DR: 24 is really big

My number was not 24 • It was 34. • So what? • The towers height is the number that the computer didn’t even finish writing! • Let that sink in. It’s pretty amazing.

This deserves another slide To be clear: • 34 is so absurdly large that we couldn’t even fathom writing it in any standard notation. • Best we can do without tetration is write it as an exponent tower. • But the height of the tower is that crazy number from before. • The computer couldn't finish writing that crazy number. • The height of our tower is not the length of the number, but the size. • When the tower had height 4, it was that absurd number. • Height 5 would be insane. • Height 6? Forget about it. • But my number has height

Mind blown? • If you can even begin to wrap your head around pentation you’re doing better than I am. • Remember that we used small numbers. • It would be like saying you understand multiplication because you can almost imagine how big 2 * 4 is. • But just suppose you understand pentation. • Remember pentation is just the 5 th operation in our infinite hyperoperation sequence.

Graham’s Number • • The 6 th is hexation. It’s nuts. Incomprehensible. But what about the 100 th operation in the sequence ? Or what about the 34 th operation in that sequence? Graham is not impressed.

Defining Graham’s number •

g 1 is NOT Graham’s number g 2 is defined as below. Let the ridiculousness of this sink in… • g 2 doesn’t use the 2 arrows (tetration), or 3 arrows (pentation) or even 4 (hexation) • It uses g 1 arrows. • g 1 Arrows! • g 1 ARROWS!!!!

g 2 is not Graham’s number! Prepare yourself…

A couple last thoughts… • What about ggraham’s number? • Graham’s number of layers. • LAYERS!!! • waitbutwhy. com has an entertaining article on Graham’s number.

Last thought… • TREE(n) is a function • TREE(1) = 1, TREE(2) = 3 • TREE(3) > g 64 • Actually, TREE(3) > g 187196 • TREE(4)? • TREE(3)) • Before you ask, I cannot explain the tree function. Wikipedia will try if you want.

The End • Questions? • (Slides available at http: //webpages. sou. edu/~stonelakb/math)