# Higher order calculus

In order to reach those truly big numbers like Graham's number the ordinary math isn't enough. The rate of growth is an important part of this process as it lets us get to the bigger numbers faster. Even though the exponential growth is a common term and is indeed pushing our ability of comprehension It's like staying in place compared to where we are aiming for.

So there are the addition, multiplication and exponentiation everybody is familiar with. But it doesn't stop there. After those come the hyperoperations like tetration, pentation and so forth.

Let's take a step back and focus for a moment for the traditional ones still and look how we get from one to the next one. Addition is level 1 operation. We can add numbers together like 3+3. We can keep doing this, but to be more efficient we level up to level 2: the multiplication. So instead of writing out 3+3+3 we can just reduce it to 3*3. The same is true for level 3 or the exponentation: 3*3*3 becomes 3^3.

How about 3^3^3 then? Well we can keep going just like with the traditional operators. The next one is called tetration, so the previous example becomes 3 tetrated to 3. Pentation in turn is the same for tetration.

To express those hyperoperations we also need a new operator: Knuth's up-arrow notation, denoted by an up arrow (↑). Basically it tells how many times we want to repeat the previous operation. Starting with exponents 3↑3 means 3*3*3, 3↑↑3 in turn 3^3^3 e.g. the tetration. To express 3↑↑↑3 in exponential power tower it would have over 7 trillion ^3s. By adding more arrows we get into those higher and higher hyperoperations and also increasingly fast growth rates.

By the way 3↑↑5 is already several orders of magnitude bigger than googolplex and 3↑↑↑↑3 is bigger than the number we got by filling all the Planck's volumes in every universe in the multiverse.