Tag Archives: tetration

Tetration

As mentioned in the last post, 2↑↑2=4, but what does the ↑↑ symbol mean? ↑↑ is a symbol (there are many notation systems, like ^^) for the operation known as tetration. Anyone with a basic knowledge of mathematics will know that multiplication is just repeated addition, and that a*b is the same as adding a to itself b times. Exponentiation is similar, a^b is just a multiplied by itself b times. Tetration, now, is an extension of these. a↑↑b is a to the power of itself b times, so 3↑↑3=3^3^3. As you can probably tell, tetration makes big numbers quickly, and isn’t supported by many calculating devises, or at least not for very long. Here’s some tetration involving 2:

2^^1=2

2^^2=4

2^^3=16

2^^4=65536

2^^5=2^65536≈2*10^19728

And some involving 3:

3^^1=3

3^^2=27

3^^3=7625597484987

3^^4=3^7625597484987≈10^10^10^1.01

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Why Two is My Favorite Number

Many math geeks will pick a favorite number of theirs because of a certain property of said number. Some people choose e, the base of the natural logarithm, some choose π, the ratio between a circles circumference and its diameter, some clever few choose i, the square root of -1, and some people choose a natural number, that is, a positive integer, that has some significance to them, be it 17, 300, 132, or any other number. I choose two (2) as my favorite number and here’s why:

  • 2 defines what numbers are even, and what numbers are odd.
  • 2 is the first prime number, the smallest prime number and only even prime number. 2 and 3 are the only two consecutive primes.
  • The decimal expansion of any simple fraction where the denominator is 2 will always terminate in a 5 or a 0, depending on whether the numerator is even or odd.
  • 2 is the base of binary, the number system with the smallest base in which numbers can be written (relatively) easily.
  • 2+2=2×2=2^2=2↑↑2=…=4. x↑↑y is called tetration, I’ll go over it later.
  • 2 is its own factorial.
  • 2 is part of a bunch of other special prime categories, like Fibonacci primes, Lucas primes, factorial primes.
  • 2 is a highly composite number, meaning it has more positive whole number factors than any number less than it.
  • There are two characters for two in Chinese, 二 and 两.
  • 2 is the atomic number of helium, the first noble gas.
  • Mersenne primes are found using powers of two.
  • The square root of 2 (about 1.414) was the first irrational number to be discovered.
  • 2 is the fewest number of dimensions needed for polygons to exist in geometry.
  • And finally, a superficial one, my birthday is in February, the second month in the Gregorian calendar.

 

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