Big numbers

From FMNWiki
Jump to: navigation, search

Truly large numbers are sometimes referred to in the world of physics and mathematics. These numbers are so big that they defy naming conventions, and are often hard to wrap one's head around. But they are important, so one has to be able to write them down. This page covers such notation, attempts to explain the enormity of the numbers, and indicate their importance in the various fields for which they were developed.

Contents

Hyper

"Hyper" notation is probably the simplest to explain. Hyper operations are an infinite set of operations, generally denoted "hyper0", "hyper1", etc. The first five can be explained without the need for additional notation; hyper5 and beyond essentially require the "Up Arrow Notation" which is discussed later.

operation arguments result description
hyper0 colspan="2" a a + 1 increment
hyper1 a b a + b addition
hyper2 a b ab multiplication
hyper3 a b ab exponentiation
hyper4 a b {\underbrace {a^{a^{a^{\cdots^a}}}}_b}={\underbrace {a \uparrow a \uparrow a \uparrow \cdots \uparrow a}_b} tetration--a "tower of powers"
hyper5 a b {\underbrace {a \uparrow\uparrow a \uparrow\uparrow a \uparrow\uparrow \cdots \uparrow\uparrow a}_b} pentation
hyper6 a b {\underbrace {a \uparrow^3 a \uparrow^3 a \uparrow^3 \cdots \uparrow^3 a}_b} hexation

Knuth's Up Arrows

Donald Knuth devised a method for representing large numbers in the following way. In the same way that multiplication can be seen as iterated addition and exponentiation can be seen as iterated multiplication, there were no simple ways to define iterated exponentiation.

a\times b={\underbrace {a+a+a+\cdots+a}_b}

a^b={\underbrace {a\times a\times a\times\cdots\times a}_b}

a\uparrow b={\underbrace {a^{a^{a^{\cdots^a}}}}_b}

Thus,

a\uparrow\uparrow b={\underbrace {a\uparrow a\uparrow a\uparrow \cdots\uparrow a}_b}

And

a\uparrow^3 b={\underbrace {a\uparrow\uparrow a\uparrow\uparrow a\uparrow\uparrow \cdots\uparrow\uparrow a}_b}

Etc.

Example

Numbers here grow very rapidly. For instance, 3\uparrow3=3^3=27. Then Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert): 3 \uparrow\uparrow 3 = 3 \uparrow 3 \uparrow 3 = 3^3^3 = 7625597484987 . But 3\uparrow\uparrow\uparrow3=3\uparrow\uparrow3\uparrow\uparrow3={\underbrace {3\uparrow3\uparrow3\uparrow\cdots\uparrow3}_{7625597484987}}, a number far too large to be written.

Conway's Chained Arrows

John Conway also invented a way of representing large numbers, occasionally in a more concise fashion than Knuth's arrows allow.

a\to b=a^b

a\to b\to 1=a\to b

a\to 1\to c+1=a

a\to(b+1)\to(c+1)=a\to(a\to b\to(c+1))\to c

Examples

Example 1

1\to X This will eventually, no matter what X happens to be, reduce to 1y = 1

Example 2

2\to2\to4

This becomes 2\to(2\to1\to4)\to3 which reduces to Failed to parse (lexing error): 2\to\2\to3 .

Continue this step and you end up with 2\to2\to1=2\to2=2^2=4. So any chain 2\to2\to X results in 4.

Example 3

2\to3\to2\to2

2\to3\to(2\to3\to1\to2)\to1

2\to3\to(2\to3)=2\to3\to(2^3)=2\to3\to8

2\to(2\to2\to8)\to7

2\to4\to7

2\to(2\to3\to7)\to6

2\to(2\to(2\to(2\to(2\to(2\to(2\to4)\to1)\to2)\to3)\to4)\to5)\to6

2\to(2\to(2\to(2\to(2\to(65536)\to2)\to3)\to4)\to5)\to6

Result...really, really, really big. This one can be expressed using Knuth's up arrow notation like so: 2\uparrow^8 3

Ackermann Function

Wilhelm Ackermann created the recursive function that bears his name. It is defined as follows:

\begin{cases} A(m,n,0)=m+n \\ A(m,0,1)=0 \\ A(m,0,2)=1 \\ A(m,0,p)=m \text{ for } p>2 \\ A(m,n,p)=A(m,A(m,n-1,p),p-1) \text{ for } n>0 \text{ and } p>0 \end{cases}

This is actually extremely similar to the hyper operator. A(m,n,0) = m + n (hyper1), A(m,n,1)=m\times n (hyper2), A(m,n,2) = mn (hyper3). A(m,n,p)\neq m\uparrow^{p-1}n for p > 2, however--the slight difference is that A(m,n,p)=m\uparrow^{p-1}n+1 in this case.

A simplified version of the Ackermann function, technically called the Pèter-Ackermann function, is defined as follows:

A(m,n)= \begin{cases} n+1 & \text{if } m=0 \\ A(m-1,1) & \text{if } m>0 \text{ and } n=0 \\ A(m-1,A(m,n-1)) & \text{if } m>0 \text{ and } n>0 \end{cases}

Example

Because it is defined recursively, A(3,4):

  • A(3,4) = A(2,A(3,3))
    • A(3,3) = A(2,A(3,2))
      • A(3,2) = A(2,A(3,1))
        • A(3,1) = A(2,A(3,0))

etc.

It takes 10307 steps to arrive at the answer in this fashion, but A(3,4) = 125.

Thing is, massive numbers like 2^{2^{2^{16}}}-3 can be written as simply A(4,3).

Steinhaus-Moser Notation

Steinhaus-Moser notation is interesting, in that it uses shapes to define numbers. n in a triangle denotes nn. Then n in a square can be re-written as n inside n nested triangles. n in a pentagon can be re-written as n inside n nested squares. Obviously, the numbers grow exceptionally rapidly here, as well.

Examples

Triangle

The number 2 in a triangle is 22 = 4.

Square

The number 2 in a square is equivalent to the number 2 inside 2 nested triangles. Which is thus equivalent to 22 = 4 inside one triangle, or 44 = 256.

Pentagon

The number 2 in a pentagon is equivalent to the number 2 inside 2 nested squares. As we evaluate from the inside out, and we already know that the number 2 inside a square is 256, we now have the number 256 inside a square. Which is equivalent to the number 256 inside 256 nested triangles. Get it now? This number here is almost too massive to comprehend, but there are numbers defined using this notation:

Mega

Mega is the number 2 in a pentagon. 10\uparrow\uparrow257<\text{ mega }<10\uparrow\uparrow258.

Megiston

Megiston is the number 10 in a pentagon.

Moser's Number

Moser's number is the number 2 in a polygon with mega sides. Try that one on for size.

Retrieved from "http://findingmyname.com/wiki/index.php/Big_numbers"
Personal tools