^{b}can be denoted a ↑ b, the fourth one, tetration, sometimes denoted as

^{b}a, can be denoted a ↑↑ b. In fact for all integers x > 2, the xth hyperation is denoted as a then x - 2 up arrows then b or a ↑

^{x - 2}b.

The hyperations after exponentation acting on pairs of numbers where the first number is a positive integer and the second one is a nonnegative integer are defined as follows: a ↑

^{x}b = 1 and a ↑

^{x + 1}(b + 1) = a ↑

^{x}(a ↑

^{x + 1}b). For any real number a > 0, just like for positive integers, we also define tetration to the base a and integral exponent to satisfy the 2 formulae:

- a ↑↑ 0 = 1
- a ↑↑ (b + 1) = a ↑ (a ↑↑ b)

- a ↑↑ 0 = 1
- For all real numbers b, if a ↑↑ b exists, then a ↑↑ (b + 1) = a ↑ (a ↑↑ b)
- a ↑↑ b approaches the upper limit exponentially.
- There exists no other function satisfying the previous properties that has a number not in the domain of this function in its domain.

- a ↑↑↑ b is only defined for a ≥ 1
- When a = 1, a ↑↑↑ b is defined for b > -1 and is equal to 1
- There exists a real number d which is the largest real number such that d ↑↑↑ b has an upper limit for integer exponents b. When 1 < a < d, a ↑↑↑ b is defined on R such that a ↑↑↑ (b + 1) = a ↑↑ (a ↑↑↑ b). and a ↑↑↑ b approaches a constant exponentally as b approaches ∞.
- When a ≥ d, a ↑↑↑ b is defined to be the analytic continuation of a ↑↑↑ b defined for 1 < a < d with respect to a.

^{x - 2}b is not defined for a < 1, when a = 1, it's defined for b > -1 and is equal to 1. There exists a largest real number r such that for integers b, a ↑x - 2 b has an upper limit. When 1 < a < r and x > 3, a ↑

^{x - 2}b is defined such that a ↑

^{x - 2}(b + 1) = a ↑

^{x - 3}(a ↑

^{x - 2}b) if a ↑

^{x - 2}b exists and a ↑

^{x - 2}b approaches a constant exponentially. When a ≥ r, a ↑

^{x - 2}b is defined to be the analytic continuation with respect to a of a ↑

^{x - 2}b defined for 1 < a < r. It turns out that for any x-ation where x ≥ 3, for any a > 1, a ↑

^{x - 2}b is defined for all real numbers b with a lower limit on a ↑

^{x - 2}b if x is odd and there exists a real number above which b must be in order for a ↑

^{x - 2}b to be defined with no lower limit on a ↑

^{x - 2}b if x is even.