# How are hyperoperations of rational and irrational numbers calculated?

Let's say I have the following problem, where x represents a real number.

x = 2 * 3.4

I'm sure there are many ways one could solve this problem, but the way I would solve this is by adding 3.4 to 3.4. And I would get the following result:

x = 2 * 3.4 = 3.4 + 3.4 = 6.8

But what if I thought about this problem in a different way? What if I tried to solve it by adding 2 3.4 times? This is where my question begins. How does it make sense to add 2 3.4 times? Also, what I noticed is that this problem can be rewritten (mostly) in terms of addition. For example:

x = 3.4 * 2 = 2 + 2 + 2 + 2 * 0.4

While I can add 2 several times, I'm still left with 2 * 0.4 at the end. In the following example, y represents a real number:

y = 2^3.4 = 2 * 2 * 2 * 2^0.4

With "higher" operations, I noticed that I can rewrite them in terms of "lower" operations. But I still end up with something that is in terms of the original operation. Just like the problem above, I'm left with 2^0.4 at the end.

What makes exponentiation and tetration and the next hyperoperations more difficult is that they're not commutative. So if I were trying to solve the following:

y = 2^3.4

Unlike multiplication, I can't rewrite the problem as:

y = 3.4^2 = 3.4 * 3.4

Because exponentiation is not commutative.

My final example is the following, where z represents a real number:

z = 2 tetrated 0.5 times

Wikipedia states that the answer to this problem is approximately 1.459.

My question is, how are values like 2 tetrated 0.5 times even calculated? What about hyperoperations after tetration such as pentation and hexation?

Secondly, is there any way to explain how a number, such as 5, can be multiplied by itself, let's say, pi times? That would look like the following:

5^pi = ?

The answer turns out to be about 157, but how does it make sense to multiply a number by itself pi times? In other words what is the exact definition of exponentiation for rational and irrational numbers? Surely the following definition doesn't make sense for the above problem:

a^b = a * a * a * a *... *a <- b times

• The definition of multiplication in terms of "adding repeatedly" might be how one would choose to define it for natural numbers. Using the definition for the naturals we can then define multiplication for the integers. Using that definition, we can then use it for the rationals and then finally for the real numbers as a whole. The further down the chain we go however, the original definition loses its meaning. Suffice to say, the definition you cite has its use still in very primitive contexts, but it is not how we define it for the real numbers, hence your confusion. – JMoravitz Oct 7 '17 at 4:32
• Even for the rationals, it no longer has the meaning you are thinking of. Given two rational numbers $\frac{a}{b}$ and $\frac{p}{q}$ we define their multiplication to be $\frac{a\cdot p}{b\cdot q}$ where the multiplication that occurs in the numerator and denominator was the previously defined multiplication on the integers. In your example, $2\cdot 3.4=\frac{2}{1}\cdot\frac{34}{10}=\frac{2\cdot 34}{1\cdot 10}$. Similarly, the other operations you describe lose their meaning of "repeated application" as you leave naturals and go to reals. – JMoravitz Oct 7 '17 at 4:35