Author Archives: jwanlass8

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.

Thinking in different bases

Very few people ever stop to think about why numbers are the way they are. Have you ever stopped to consider why you live in a society that uses a ten based (or decimal) number system? This is a system where you start at 1 and go all the way up to 9 before you reset with one followed by a zero (or 10). Well if you haven’t taken the time to ponder this deeply complex situation that you find yourself in, fear not, for I will explain why. For the most part, human beings find themselves in the possession of 10 fingers. So the leading hypothesis for why we use a base 10 system is because it was convenient since we could count to 10 on our hands. For that single and simple reason, society decided that its number system would have a base of 10. There you have it, one of the great mysteries of life has been cleared up for you.

Now that you have an open space where a mystery used to rest, allow me to fill that spot with a new mystery. Why don’t we use a differently based number system? No really, think about it! Throughout history, multiple societies chose not to use a decimal system. For example, Babylonians used a base 60 number system and the Mayans used a base 20 system. Likewise, there are quite a few key things in our lives today that don’t really rely on the conventional base 10 way of doing things.

binaryFor example, we are surrounded by computers, which utilize Binary, a base 2 number system. At every position there is either a 1 or a 0, so the number ten in binary looks like 1010.

Another example of something that isn’t really 10 based is a clock. Take a second to look at an old school clock with an hour and minute hand (gasp! Not digital!) . You’ll quickly notice that it goes from 1 to 12 instead of 1 to 10. Weeks are broken down into 7 days and minutes as well as hours are in chunks of 60. So as you can see, the case could be made to switch to a different based number system. Let’s take a look at another option that society could use in place of the current base 10 system.

The system that we’ll look at is one of my personal favorites (thanks to my Computer Science bias), the hexadecimal system. As you can probably guess from the name, the hexadecimal system adds six to the base 10 system, leaving us with a grand total of base 16! The system uses 0 to 9 to represent numbers 0 to 9 and then uses A to F to represent the numbers 10 to 15. Hexadecimal is a positional numeral system just like the decimal system. Just to give you a better idea of what hexadecimal is, lets learn how we can represent a hexadecimal number in decimal.

Let’s take the random hexadecimal number 3FB1. Since it is a positional system, meaning the position of the symbol is part of its value, we can just take the symbol and multiply the symbol’s value by its base to the power of its position (position numbering goes right to left and starts with 0 rather than 1). So we would take (3 x 163) + (15 x 162) + (11 x 161) + (1 x 160). Simplifying this further we get (3 x 4096) + (15 x 256) + (11 x 16) + (1 x 1). At the end of this we are left with 16,305. So right off the bat we can begin to see the some of the potential benefits that would come with using a base 16 system. Firstly, we notice that it takes fewer symbols to represent numbers. Where in decimal we had to use 5 symbols (6 if you count the comma) to represent 16305, in hexadecimal we only had to use 4. We can also note that because of this space bonus, we could potentially represent higher numbers in the same number of hexadecimal characters. Even though this all sounds great, there are some disadvantages that would come with using the hexadecimal system. For one, performing mathematical operations on base 16 numbers can get complicated quickly (a base 16 multiplication table has 256 instead of 100 elements). Try performing long division on two hexadecimal numbers! Also, I personally believe that it would be trickier to set up equations with variables due to the fact that the characters “A”, “B” and “C” could no longer be used (there goes the quadratic formula song). On a similar note, there are many people who think that we would be better off on a duodecimal system (base 12), but that is a conversation for another time.

So next time when you are counting on your fingers, take some time to think about the effects of you simply having 10 fingers!