ReExpOp - A NEW MATHEMATICAL OPERATION

Thomas Nguyen

I.   Introduction:

What do you see from the following problem?    

               5 + 5 + 5 + 5 + 5 + 5 = 10 + 5 + 5 + 5 + 5 = 15 + 5 + 5 + 5 = 20 + 5 + 5

The birth of “multiplication” ? Yes, instead of doing several additions, we can do only one multiplication operation:

                                                  5 x 6 = 30

We can access the Time Table or use multiplication button on calculator to get the answer. Similar, let’s take a look at:    

                4 x 4 x 4 x 4 x 4  = 16 x 4  x 4 x 4 = 64 x 4 x 4 =  256 x 4 = 1024

We can short cut the problem as follows:

                                                  4^5  = 1024

Instead of repeating many multiplications, we just do one operation “exponential”.

So what happens if we have this problem:

                3^3^3^3 = ?

Of course, you can do that problem as:

                 (3^3)^3^3 = (27^3)^3 = 19683^3 =  7,625,597,484,987

But, how do you think about this?           3@4 = 7,625,597,484,987        

Where “@” is a new mathematical operation called “ReExpOp” and “7,625,597,484,987”

is came from “ReExpOp Table” below.

We just repeat the pattern: There is one new operation “ReExpOp” to replace for a bunch of other “Exponential” operations.

That is a reason for the birth of "Re-Exp-Op" = "Repeated Exponential Operation", a new mathematical operation. It was created about two years ago.

II. Definition of ReExpOp:

The general form of a "ReExpOp" is:   a@b    (read "a reexpop b")    

                               Where a and b are integers.

                               "a" is called the base and "b" is called the top.

a@n  =  a^a^a^a^.................^a^a           where    n > 0   and  n  is an integer number.                              

             <--------- n  ------------>                                             

a@1  =  a                

a@0  =  a @' 1     (anti-ReExpOp; look at part c)

a@(-n)  =  a @' (n+1)

II. ReExpOp Table: (for numbers from 2 to 9) :

“ReExpOp table” will be stopped when my calculator (not supercomputer) shows “infinity”.

     2@2 = 4                          

     2@3 = 16                           

     2@4 = 256                          

     2@5 = 65,536

     2@6 = 4.3 x 10^9 = 28.7 AU

     2@7 = 1.8 x 10^19  = 1.8 x 10^6 lys   = 1.8 million light-years

     2@8 = 3.4 x 10^38                               = 3.4 x 10^25 lys  =  34 septillions light-years

     2@9 = 1.158 x 10^77                           = 1.158 x 10^64 lys  = 11.58 vigintillions lys

     2@10 = 1.34 x 10^154                         = 1.34 x10^141 light-years                                          

     3@2 = 27

     3@3 = 19683

     3@4 = 7.6 x 10^12   ~  10^13  ~  1 ly   ~ 1 light-year

     3@5 = 4.4 x 10^38                                = 4.4 x 10^25 lys  =  44 septillions light-years

     3@6 = 8.7 x 10^115                              = 8.7 x 10^102 light-years

     4@2 = 256 = 2@4

     4@3 = 4.3 x 10^9 = 2@6 = 28.7 AU

     4@4 = 3.4 x 10^38 = 2@8                    = 3.4 x 10^25 lys  =  34 septillions light-years

     4@5 = 1.34 x 10^154 = 2@10              = 1.34 x 10^141 light-years

     5@2 = 3125

     5@3 = 2.98 x 10^18                              = 0.3 billion light-year

     5@4 = 2.35 x 10^87                              = 2.35 x 10^74 light-years

     6@2 = 46656

     6@3 = 1.0 x 10^28                                = 10^15 lys =  a quadrillion light-years

     6@3 = 1.2 x 10^168                              = 1.2 x 10^155 lys 

     7@2 = 823543

     7@3 = 2.57 x 10^41                               = 2.57 x 10^ 28 lys  =  25.7 octillions lys

      7@4 = 7.4 x 10^289                              = 7.4 x 10^ 276 lys

     8@2 = 16,777,216  ~ 1/10 of AU

     8@3 = 6.3 x 10^57                                  = 6.3 x 10^44 lys  = 0.63 quattuordecillion lys

     9@2 = 387,420,489 ~ 2.58 AU

     9@3 = 1.97 x 10^77                                = 1.97 x 10^64 lys  =  19.7 vigintillions lys

     10@2 = 10^10                                         = 1/1000 of a light-year

     10@3 = 10^100                                       = 10^87 light-years

III. Properties of "ReExpOp":

1. Change to smaller base:  In some special cases of "a", we can change the base to smaller base.

            For example:

            if  a = 4 (special case), then we can change the base into a = 2.

            Take a look at "reexpop" table for 4:

            4@2 = 2@4        Why?     Since      4@2 = 4^4 = (2^2) ^(2^2) = 2^2^2^2 =2@4

            4@3 = 2@6

            4@4 = 2@8

            4@5 = 2@10

2. Change to bigger base: Similar, in some special cases of “a“, we can change from small base to bigger base.

            For example:   

            3@5 = 3^3^3^3^3 = (3^3)^3.3.3 = 27^(3.3.3) = 27^27 = 27@2

            Check:     3@5 = 4.434264882430378 x 10^38

                          27@2 = 4.434264882430378 x 10^38

             5@7 = 5^5^5^5^5^5^5 = (5^5)^5.5.5.5.5 = 3125^(5.5.5.5.5)

                      = 3125^3125 = 3125@2

             We have:   4@3 = 2@6     and     27@2 =3@5     and      3125@2 = 5@7

                          (2^2)@3 = 2@6            (3^3)@2 = 3@5               (5^5)@2 = 5@7

Let’s find the pattern:

              4@3 = 4^4^4 = (2^2)^(2^2)^(2^2) = (2^2).2.2.2.2 = 2@6   (2 + 4 = 6)

              27@2 = (3^3)^(3^3) = (3^3).3.3.3 = 3@5                             (2 + 3 = 5)

              3125@2 = (5^5)^(5^5) = (5^5).5.5.5.5.5 = 5@7                   (2 + 5 = 7)

Apply the pattern:    

              256@3 = (4^4)^(4^4)^(4^4)  the pattern is (2 + 8 = 10)  

               Therefore:   256@3= 4@10

So, the special cases, that we are talking above, are the case in which the base "a" can be written in the form:                              a = n^n   

 Examples:    a = 4 = 2^2;  a = 27 = 3^3;  etc.

3. Extension Property:

It is clearly that if we continue to exponential a "reexpop" then the top part will change to higher level.

                                                (a@b)^(a@c)  =  a@(b+c)

For example:        (4@2)^(4^3)  =  (4^4)^(4^4^4)  =  4^4^4^4^4  =  4@5

                       (3@3)^(3@4)  =  (3^3^3)^(3^3^3^3)  =  3^3^3^3^3^3^3  =  3@7

4. Shrinking property: (Anti-ReExpOp)

Most of mathematical operations have their anti-operations.

For examples:   (add, subtract)  (multiply, divide)  (exponential, root)  (derivative, integral) etc.

Similar, "reexpop" has its own anti-operation called "anti-reexpop". It’s represented by the symbol  @'

             (a@b)^(a @' c)  =  a@d                                       if    (b - c) = d >1

             (a@b)^(a @' c)  =  a@1  = a                                if    (b - c) = d =1

             (a@b)^(a @' c)  =  a@0  =  a @' 1                      if     (b - c) = d = 0 

             (a@b)^(a @' c)  =  a@(-d) = a @' (d+1)             if     (b - c) = d < 0   

Where a,b,c, and d are integers

We know    a@3 = a^a^a ;  how about    a @' 3 ?

                                                                 a @' 3  =  a^(1/a)^(1/a)^(1/a)  = a^[1/(a^3)]

Summary:                  a@n  =  a^a^a^......^a^a^a 

                                               <-------- n --------->

                                   a @' n  =  a^[1/(a^n)]

                                   a@0  =  a @' 1

                                   a@(-n)  =  a @' (n+1)

For examples:                               

                                  3 @' 2  =  3^[1/(3^2)]  =  3^(1/9)

                                  5 @' 3  =   5^[1/(5^3)]  = 5^(1/125)

                                  (4@5)^(4 @' 3)  =  4@(5-3)   =  4@2

                                  (2@5)^(2 @' 4)  =   2@(5-4)  =  2@1  =  2

                                  (7@3)^(7 @' 3)  =  7@(3-3)  =  7@0  =  7@' 1   =  7^(1/7)

                                  (9@4)^(9 @' 7)  =   9@(4-7)  =  9@(-3)   =  9 @' 3   =   9^[1/(9^3)] 

5. Multiplication:

                                                        a@b * a@c = a^[a^(b-1) + a^(c-1)]

Example:                 2 @3 * 2@4 =  2^[(2^2) + (2^3)]

                                                      =  2^(4 + 8)  =  2^12

6. Division:

                                                        a @b ¸ a@c =  a^[a^(b-1) - a^(c-1)]

Example:                  3@5 ¸ 3@3 = 3^[3^4 - 3^2]  =  3^(81 - 9)  =  3^72

IV.  Application in Astronomy:

ReExpOp can be used to describe big numbers in Astronomy.

Common Units, which are used to measure distance in Astronomy, are AU (astronomical unit), ly (light year), and pc (parsec).

                     1 AU = Distance between the Earth and the Sun  ~  150,000,000 Km

                     1 AU = 0.4 of  9@2 Km

For examples: Some special distances in our Solar system are:

Mercury can be said to be about 1/3 AU from the Sun ~ 0.1 of  9@2

Venus 0.7 AU ~ 0.3 of  9@2

Earth 1 AU  ~  0.4 of  9@2

Mars 1.5 AU  ~  0.6 of  9@2

Asteroid Belt 2.3 - 3.3 in scale of  9@2

Jupiter 5.2 AU ~ 2 of 9@2

Saturn 9.5 AU  ~  3.7 of  9@2

Uranus 19.6 AU  ~  7.6 of  9@2

Neptune 30 AU  ~  2@6

Pluto  39 AU  ~  1.4 of  2@6

Diameter of our Solar system is about 79 AU  ~  2.8 of  2@6

A light year (" ly ") is  a distance that light can travel in one year.

This unit is usually used for outside of our Solar system.

 Speed of light ~ 300,000 Km/sec

 One year ~ 365 days 

Therefore, one light year = (300,000 km/sec) * (365 days x  24 hours/day x  60mins/hour x 60secs/min)

                    1 ly  =  light year ~  100,000,000,000 ~10^13 km ~ 3@4 Km

                     1 pc  =  parsec  =  3.26 light-years

Let' s express some other common distances in universe with ReExpOp:

1. Diameter of our Sun is about:

                                        1,391,980 Km  ~  1.7 times of 7@2

2. Our galaxy is The Milky Way galaxy. It is about:

                                        150,000 light years across  ~  1/2 of 5@3

                                         1000 light-years thickness  ~  1/40 of 5@3

3. Distance from The Milky Way to nearest galaxy "Andromeda" is:

                                          21 x 10^18 Km  ~  7 times of 5@3

4. Distance from Earth to next nearest star "Proxima Centouri" is:

                                          40 x 10^ 12 Km  ~   4.24 light years  ~  5 times of 3@4

5. The Crab supernova remnant is:

                                          4,000 light years away  ~ 0.01 of 5@3

6. Typical distance between galaxies is about:

                                           20-40 the sizes of a galaxy  ~  10-20 times of 5@3

7. The diameter of the observable universe is at least:

                                           93 billion light-years or  8.8 x 10^26 m  ~  310 times of 5@3

In summary, names of large numbers which used in astronomy such as “vigintillion, quattuordecillion, septillion, etc” now can be replaced by simple "ReExpOp".

For example:

Instead of saying "Distance from our galaxy to the nearest galaxy "Andromeda" is about 2.1 million light-years"; We can say "Distance from our galaxy to the nearest galaxy "Andromeda" is about 2@7."

 Instead of saying "The diameter of the observable universe is at least 93 billion light-years."; We can say "The diameter of the observable universe is at least  310 times of 5@3."

V. Other potential application areas for ReExpOp:

Beside its application in Astronomy, "ReExpOp" can be applied in other areas such as Virus Production, Nuclear Reaction, Radioactive Decay, etc.

San Diego , January 3^rd, 2010.

Thank You very much for your time.

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