Exploring Up-Arrow Notation

One of the greatest things about math is that there are so many ways to do the same thing. We can express the same equation in multiple fashions and still keep the meaning the same. If you stop and think for a moment, you too will realize how awesome this is! How many different ways can you write the word “the” in English? Take a moment to mull this one over; it’s a tough one. Now if you came up with more than one way, you and I need to have a sit down conversation because you are most likely a genius. Anyways I digress, the point is the fact that math allows us so much flexibility in the ways we represent things is beyond amazing.

Take a look at the following exponential expression: , if you are familiar with how exponents work, you’ll recognize that this equals 8. Unfortunately, if you are unfamiliar with this particular area of mathematics, this expression is basically meaningless to you. Luckily for us, math allows us to put this into a different form. So now we can take and represent it instead as . This shows us that 2 raised to the power of 3 is really just 2 multiplied by 2 multiplied by 2. Still not simple enough for you? Well hallelujah, we can express it in another fashion. We can change to the simpler version of . So now we have 2 plus 2 plus 2 plus 2 for the grand total of 8. At the beginning we only had one form to look at this expression but by the end we have 3. There are so many ways to represent things in math that people began to push the limits of that fact. Much like exponential or scientific notation, other mathematicians came up with their own notations (ways of representing expressions in math). The one that I am going to focus on today is Knuth’s Up-Arrow notation.

Here we have ourselves a very fun way of representing math. The closest thing that it can be compared to is exponential notation. Looking at the above example, we can draw a couple of conclusions about the mathematical expressions we used. First, multiplication is just a series of addition operations. What I mean by that is, the expression is really just a short way of writing 2 plus 2 plus 2 plus 2 (a series of addition operations) Can you imagine how awful it would be if every time you had to double something, you had to write out every addition operation it took? It would be pure anarchy! Along the same vein, exponential notation is just a way of expressing a series of multiplication operations. So in this instance, would be the short way of writing . So now enter Knuth’s Up-Arrow notation, this notation gives us a way to represent a series of exponential operations.

The whole idea of multiple exponential operations can be a little daunting so let’s go over an example. Lets take a regular exponential operation that we are used to seeing, like . So this is easy enough to understand, lets convert it into up-arrow notation. As you can guess from the name, up-arrow notation uses an “up-arrow” as its symbol, so 32 turns into 3↑2. And there you have it, up-arrow notation! Just kidding, we have barely scratched the surface that is the awesomeness of up-arrow notation. That was an easy example; so let’s take it one step further. Let’s say that you wanted to write out 3 raised to the power of 3 raised to the power of 3 (333), that wouldn’t be too bad right? What if you wanted to take it out one more step? How about two more steps? Eventually you are going to hit your limit of how small you are actually able to write. But don’t you fret your little mathematician head; up-arrow notation is here to save the day. 3 raised to the power of 3 raised to the power of 3 (333) becomes the nice and simple expression 3↑↑3. Whew that was a whole lot easier and shorter to write out. This could continue until you were blue in the face. For example, if we take 3↑↑4, this doesn’t translate to 3 raised to the power of 3 raised to the power of 4 (334), this actually is equivalent to 3 raised to the power of 3 raised to the power of 3 raised to the power of 3, or 3333. So as you can see, these numbers begin to get bigger very quickly!

Moving on to something even more complex (yay for complexity!!! Wait….), let’s look at when we add a third arrow into the mix. Basically when you add an arrow, you create a series of up arrow operations. So if you have “n” arrows, you can expand it out into a series of (n-1) arrow operators. So looking at the example 3↑↑↑2, let’s expand this out into a series. The problem would then look like 3↑↑↑2 = 3↑↑3 = 3↑3↑3 = 7625597484987. So we had n=3, so when we expanded it out, it became a series of two arrow operations, and then we expanded that out to a series of one arrow operations. So now when we change this to 3↑↑↑3, we can again do this expansion. This time we get 3↑↑↑3 = 3↑↑(3↑↑3) = 3↑↑(3↑3↑3). If we look back at our double arrow example, we know that we will have 3 raised to the power of 3 raised to the power of 3 and so on 7625597484987 times! So as you can see, the triple arrow form grows unbelievably faster than even the double arrow form.

I hope that you are beginning to see the usefulness of this notation. You can take unnecessarily complicated expressions and shorten them to something much easier to read and much easier to write. Up-Arrow notation even goes as far as a quadruple arrow notation, but we will save that for another time. In the future, I hope you’ll use what you’ve learned here to confuse a friend or show off at a classy math party. Until then, just enjoy this fun little notation.



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