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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
= 25 + 5 = 30
The birth of “multiplication” ? Yes, instead of doing several additions, we can do only
one operation:
5 x 6 = 30
We can access the Time Table or 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”
comes from “ReExpOp Table” below.
We just repeat the pattern: There is one new operation to replace for a bunch of other
operation.
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.
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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
<--------- n ------------->
where n > 0 and n is an integer number.
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
2@8 = 3.4 x 10^38
2@9 = 1.158 x 10^77
2@10 = 1.34 x 10^154
= 1.8 million light-years
= 3.4 x 10^25 lys = 34 septillions light-years
= 1.158 x 10^64 lys = 11.58 vigintillions lys
= 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
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5@2 = 3125
5@3 = 2.98 x 10^18
5@4 = 2.35 x 10^87
6@2 = 46656
6@3 = 1.0 x 10^28
6@3 = 1.2 x 10^168
= 0.3 billion light-year
= 2.35 x 10^74 light-years
= 10^15 lys = a quadrillion light-years
= 1.2 x 10^155 lys
7@2 = 823543
7@3 = 2.57 x 10^41
7@4 = 7.4 x 10^289
= 2.57 x 10^ 28 lys = 25.7 octillions lys
= 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
10@3 = 10^100
= 1/1000 of a light-year
= 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
4@3 = 2@6
4@4 = 2@8
4@5 = 2@10
Why?
Since
4@2 = 4^4 = (2^2) ^(2^2) = 2^2^2^2 =2@4
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
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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
if
(b - c) = d < 0
(a@b)^(a @' c) = a@(-d) = a @' (d+1)
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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
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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
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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 lightyears."; 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 3rd, 2010.
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Send your opinion to “nguyentn10@netzero.net”