Tetration

Tetration is an operation used in googology.

Why Tetration?

Googology is the study of large finite numbers. Mathematicians sometimes need new operations to write these numbers down. Some numbers, you can write with exponentiation. For example, a million is written by $10^6,$ and a googol is written with a $10^{100}.$ What about googolplex, $10^{10^{100}}?$ It can still be written with exponentiation. Let's say we invented a new number that had the value of \[10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}.\] Whoa. I don't think you would want to write ten 10's every time!

Fundamentals

We all know that multiplication is repeated addition. We learned that in elementary school. The definition of $ab$ is $a+a+a+\cdots+a+a$ with $b$ $a$'s.

We also know that exponentiation is repeated multiplication.

Have you ever wondered what was repeated exponentiation?

Yep, you guessed it. Tetration is repeated exponentiation.

Explanation

In order to write \[10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}\] with tetration, you would write $10 \uparrow \uparrow 10,$ because there are ten 10's. Similarly, $3 \uparrow \uparrow 2$ would be $3^3 = 27,$ (there are 2 three's) and $3 \uparrow \uparrow 3$ would equal $3^{27},$ which is a really big number (around 7 trillion)!

What's Next?

Now, are you wondering what repeated tetration is? Click on the link to find out!

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