Math Backwards: Part II

This post contains a little more technical stuff, with less explanation, than many of the posts on this blog. If there is something you don’t understand, feel free to ask about it below. Or skim it – you won’t miss out on the main idea.

In the last post, we looked at this mystery number:

We left off with a question: what is the value of this sum, written as a “regular” number?

To keep things from getting too messy, I’m going to make up a new name for the mystery number. I’m going to call it “M”: that’s a nice short name, and it reminds me of the word “mystery.”

The letter M here is an example of notation. I am choosing to represent the mystery number as M, instead of as a big fraction. This type of neatness is generally useful. The more abstract an idea, the more important it is to have a good way to put it on paper.

Unfamiliar notation can make written math look intimidating, at least until you learn what each symbol stands for. But good notation makes things clearer. An eloquent discussion of notation can be found in a classic book on problem solving, George Pólya’s How To Solve It.

Now let’s talk about M.

I have two ways to approach it, though there are certainly more out there that I didn’t think of. My first method involves some algebra.

Method Number One

Take a look at this number again.

Notice that if we peel the outermost “1+1/” off this number, we end up with another copy of M, albeit in a slightly smaller font. Because of the fractal nature of M, the bottom half of the fraction is the same as the entire number. That gives us two different ways to write the mystery number: as M, or as 1 plus 1 over M. We can write that as an equation:

 1+frac{1}{M} = M.

By writing M in two different ways, we learned something about it. This equation means that if you take 1 plus 1 over the mystery number, you get the mystery number again.

Now, if we can find a “regular” number that satisfies this equation, we know it must be the mystery number.

Well, if 1+frac{1}{M} and M are equal, that means they both represent the same number. They are two different ways of writing the same thing. So, if I subtract 1 from each of those terms, I’ll get a new number, but both sides will still be equal. That’s the idea behind a lot of algebraic manipulation: when two terms are equal, they represent the same number, so if you do the same thing to both of them, they will still be equal.

 1+frac{1}{M}-1= M-1.

Now, 1+frac{1}{M}-1 is the same as frac{1}{M}. So,

 frac{1}{M} = M-1.

Since these two terms are equal, we can multiply each by M and again get two equal terms. So,

 M times frac{1}{M} = M times (M-1).

Multiply this out, and the M’s on the left cancel. On the right side, we multiply everything in the parenthesis by M. So,


We still don’t know which number makes this true, but the equation is starting to look workable. Now, subtract 1 from both sides again.

 0 = M^2-M-1 .

This is a special type of equation called a quadratic equation, and it has a 0 on one side. Using something called the quadratic formula, which I will not explain in depth right now (though it does have good reasons behind it if you want to learn about it!), we find

 M=frac{1+sqrt{5}}{2} .

Hmm, that’s interesting.

Before we talk about why that’s interesting, let’s take a different perspective on this number.

Method Number Two

Let’s look at a number that’s similar to the one we’re interested in: 1+frac{1}{1+frac{1}{1}}. It almost looks like a mini-version of M, but it’s not quite the same, because it stops repeating itself pretty quickly. There is no “…” in this number.

First, let’s look at the fraction in the second term of the denominator: frac{1}{1}. This is equal to 1.

Next, the denominator as a whole: 1+frac{1}{1} is equal to frac{2}{1}, or 2.

Finally, let’s look at the bigger fraction: 1+frac{1}{1+frac{1}{1}} is equal to frac{3}{2}.

What if we added another layer, another “1+1/”, to the outside of this fraction? We’d have 1+frac{1}{1+frac{1}{1+frac{1}{1}}}. We know what most of this fraction looks like already. The whole thing is equal to 1+frac{1}{1+frac{3}{2}}, which is the same as frac{5}{3}.

So far, by working outwards, we’ve got a little sequence: there’s frac{1}{1}, then frac{2}{1}, then frac{3}{2}, then frac{5}{3}. If you add another layer to the outside of the fraction, you get frac{8}{5}. Add another, and you get frac{13}{8}.

You might notice a pattern here. These fractions involve a sequence of positive integers. Each fraction is made up of some integer in the sequence, divided by the previous integer in the sequence. And the integers are: 1, 1, 2, 3, 5, 8, 13…

It’s the start of the Fibonacci sequence.

That’s weird.

Maybe this pattern will stop after a little while – there have certainly been stranger coincidences in mathematics. Or maybe the pattern continues forever. But there’s no way to directly check – no matter how many fractions we evaluate, there is always room to add another “1+1/” on the outside of the fraction.

But I think the Fibonacci pattern continues forever.

Actually, I don’t just think it. I know it. And I can prove it.

Here is my proof.

To make things easier to write, I’m going to call the first Fibonacci number F_1, the second Fibonacci number F_2, and so on. I’m also going to start with the fact that for some number of layers, the fraction is equal to a Fibonacci number divided by the Fibonacci number before it. I’m going to call that number of layers n. So when there are n layers, the fraction is equal to frac{F_n}{F_{n-1}}.

We just calculated this for a bunch of values of n, and we know it’s true for all of those. But for right now, we’re just assuming that n is some specific number of layers for which the fraction is frac{F_n}{F_{n-1}}. We’re not specifying which number n actually is.

Now for some more algebra. By adding another layer to frac{F_n}{F_{n-1}}, we get

 1+frac{1}{frac{F_n}{F_{n-1}}} .

This is equal to


Which is equal to

 frac{F_n+F_{n-1}}{F_{n}} .

So far so good. Now there’s one important thing we know about Fibonacci numbers: each is the sum of the previous two Fibonacci numbers. In other words, F_{n-1}+F_n=F_{n+1}. That tells us that the fraction above is equal to

 frac{F_{n+1}}{F_{n}} .

Let’s review what we just did here. We showed that if a fraction with n layers is equal to frac{F_n}{F_{n-1}}, then a fraction with n+1 layers is equal to F_{n-1}+F_n=F_{n+1}. That’s the same formula as frac{F_n}{F_{n-1}} for n layers, but with n+1 in place of n.

We already know that when n=2, this works: when there are two layers, the fraction is equal to frac{F_2}{F_{1}}, or frac{1}{1}. Now, since the fraction works for n, it works for n+1=3 as well. And since it works when n=3, it must work for n=4. And since it works for n=4, it must work for n=5… and so on. This proves that for any positive integer n geq 2, this fraction with n layers is equal to frac{F_n}{F_{n-1}}. The pattern continues forever.

As n gets bigger and bigger, the value of frac{F_n}{F_{n-1}} gets closer to a particular number. That number is known as the Golden Ratio.

The number M is what you would get if you added infinitely many layers to this fraction. You’d get the number that frac{F_n}{F_{n-1}} approaches as n gets bigger and bigger.

In other words, M is the Golden Ratio. As we showed with Method One, M is frac{1+sqrt{5}}{2}.

I don’t know about you, but I find this connection pretty amazing.

La espiral de Fibonacci

Well, there you have it. We took a strange-looking math idea, played with it, and ended up finding connections to some totally different ideas. Hopefully it was a different experience from that of the drills and rules you might associate with math.

There are a lot of follow-up questions I could ask on this topic.

But I’ll let you ask the questions yourself.


Math Backwards: Part I

This post includes my thought process as I look at a particular piece of math. That part of the post contains a little more technical stuff, with less explanation, than many of the posts on this blog. If there is something you don’t understand, feel free to ask about it below. Or skim it – you won’t miss out on the main idea.

Flowers and Fibonacci

I think most people learn math backwards. For most of my life, I had the impression that math is a neat package of rules. The teacher would say something like “The square root of 2 is irrational,” “a^2+b^2=c^2,” or “Dividing by zero is undefined.” Then I would memorize the rule and apply it to a set of problems. This, unfortunately, is what math class means to most people: you are told a rule, and then use it to get the right answer to a question.

Following rules of math is a bit like running laps around a basketball court. It is related to what you might do in a basketball game. It can even make you a better player. But it is certainly not the same as a game of basketball. Following mathematical rules is a helpful tool in doing real math, but it is not the same thing.

There is certainly value in knowing how to use a rule. But how do we know about the rules in the first place?

I’ll let you in on a little secret. Mathematicians make up their own questions. They ask about what they’re interested in or what they think will be useful. They find their own answers.

That’s how math gets developed.

That’s not to say that there is no structure to math – to come up with a valid mathematical idea, you need to stick to the principles of logic. There are true mathematical statements, and false ones. Some are applicable to physical situations, and some are not. Some are interesting, some are beautiful, some are useful, and some are not. There is a whole scene of math research and development that is thriving nowadays – and it has been around since at least the Ancient Greeks.

In school, we usually are usually given a rule, given a problem, and asked to find an answer. When a person studies math on their own, they ask their own questions. They look at an idea from their own perspective. They make their own mini-discoveries about it.

That’s what I’d call doing math forwards. To fully appreciate math, you need to mess around with some ideas on your own.

Take, for instance, the irrationality of the square root of 2. You may remember it as an “obvious” fact, or something that you accepted because your teacher said so. While this is a true fact, it is not at all an obvious one. At the time of its discovery, it was scandalous – perhaps even cause for murder. There’s a reason sqrt{2} is irrational. But people had to discover that fact. They discovered it by looking at the number and asking their own questions about it.

A bad math class lacks the drama, the debate, and the flashes of insight that make the subject interesting. I know many people who love math. But I would be hard-pressed to find someone who loves following a rule that he doesn’t understand. The way people approach math on their own is very different from the way it is presented on a seventh-grader’s homework.

In this post and the next, I will take you through my thought process as I look at a particular piece of math. It is not going to represent the way all mathematicians think, or even all of the ways I look at math. It is just a sample.

This thought process is a bit messy in places. There are unanswered questions. And that’s fine, because I am not trying to solve anything.

I’m just playing with an idea.


Let’s look at this mathematical object.

Here’s the first thing I would ask: What kind of object is it?

Math deals with all types of objects, like numbers, shapes, vectors, functions, and sets. The list is practically endless. Each type of object has its own special properties.

We make this kind of categorization all the time in everyday life, without even thinking about it. It makes no sense fry a baseball or play catch with pancake batter. In the same way, it makes no sense to multiply two circles or rotate the number 78. Before I can do anything interesting with a mathematical object, I need to know what kind of object it is.

In this case, I’d say the object we’re dealing with is a number. Now I know which steps are possible, and which steps are not. For instance, I might add 5 to it, but I won’t draw a line through it.

It may have been easy for you to see that this is a number. Truthfully, in most situations, I do not spell out this kind of thought at all. It usually happens unconsciously. But I find that this can be an important first step to articulate. When I am faced with more complicated objects, this question can be extremely helpful.

An important thing to notice about this number: This is a fraction, which has a fraction inside it, which in turn has a fraction inside it, and so on. It starts with   . The “…” is replaced by another   , which all together makes the fraction. Next it becomes  , and so on. No matter how many layers are explicitly written, it represents the same number. The “…” tells the reader that the pattern repeats forever.

As another example, 2+2+2+2+2+… means that you start with 2, then 2+2, then 2+2+2, and so on. In this case, the “…” means that the pattern of adding 2 is repeated forever.

Next thought: Is there any obvious idea I might overlook?

Have you ever struggled with the volume controls on your TV, only to realize that it was on mute? That is the kind of thing I want to avoid by asking this question. In this case, I think it makes sense to check whether our mystery number,, is equal either to 0 or to infinity. I want to rule out those possibilities, because neither are particularly interesting.

I know to check this because of experience. I have seen a lot of sums with a “…” that equal infinity or equal 0. For example, 2+2+2+2+2+… gets bigger than any particular number: it is infinite. And    gets smaller and smaller. It is equal to 0.

These examples really deserve more explanation in the context of infinite series. But that’s a separate discussion. Back to our number.

Well,  is 1, plus a positive fraction (I don’t know what the fraction is equal to yet). So  can’t be equal to 0.

I just ruled out something — that’s good news in my book!

Infinity seems more difficult to deal with, but it turns out that I can use the same trick. We just found out that   is greater than 1, which tells me that frac{1}{1+frac{1}{...}} is less than 1.* That tells me that we are adding 1 to a number less than 1. So that sum must be less than 2. If our mystery number is less than 2, it certainly can’t be infinity.

*If that is confusing, think of it this way. 2/3 < 1 because 2 < 3: if you split 2 cakes among 3 people, everyone gets less than a whole cake. A positive fraction is

Now we know something about our mystery number. Not only is it not 0 and not infinite, it is somewhere between 1 and 2. That’s way more than I knew just a few minutes ago.

So far, I have been asking questions without a particular goal in mind. No one gave me a question and told me to answer it. My purpose was simply to get information, because the mystery number looked kind of weird and interesting.

But now I have a question I really want to answer. I’m curious about this.

What “regular,” “non-…” number is our mystery number equal to? From the discussion so far, we know it is equal to something between 1 and 2. But what? 1.5? 1.000007? sqrt{3}?

I worked on this for a while, and got an answer to this question in two completely different ways. The mystery number, it turns out, has something to do with this flower.

SunFlower: the Fibonacci sequence, Golden Section

I’ll talk about both techniques in the next post.

Count to 1023 On Your Fingers

If you have “31” potatoes, how many do you have?

Really. It’s not a trick question. So give it a try — what do you think? How many potatoes do you have when you have “31”?

Assortment of Organic Fingerling, Russian Banana, & Aztec Potatoes, Scottsdale Old Town Farmers' Market

When I wrote 31, you likely assumed that I meant “three groups of ten, plus one more.” You understood that 31 means that quantity. It was like seeing the written string of letters “pie” and thinking of a round, fruit-filled pastry dessert.

But you may also know that the letters “pie” don’t always refer to a dessert. If I were writing in Spanish, “pie” would refer to someone’s foot.

The string of letters P-I-E has two meanings. Each makes sense in its own context.

The same thing is true of 31. In most situations, 31 means “three groups of ten, plus one more.” That’s because nowadays, we use what is called the base ten system.

We all relate to base ten as “normal.” But it is important to understand what is going on when we go from 9 to 10.

To understand what base ten really does, imagine that I’m a potato farmer, with, say, 1436 potatoes to ship to my friends outside of Idaho. I have special crates to put my potatoes in. Each potato crate holds exactly ten potatoes: no more, no fewer.

I use my potatoes to fill as many crates as possible. Any single potatoes left over, after all the crates are full, are put on the side. Those single potatoes are represented by the rightmost digit (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9). So, with 1436 potatoes, and crates that hold ten each, I will have 6 potatoes left over.

You probably know how this pattern continues. The second-to-rightmost digit works like the rightmost digit, but on a bigger scale. I pack all the ten-potato crates into big sacks. Each sack holds ten crates – that’s ten crates, not ten potatoes. I fill as many sacks as possible, and then count how many crates are left over. 1436 potatoes leaves me with 3 crates not in sacks along with the 6 potatoes not in crates.


And of course, the third digit to the right works the same way. Not only am I shipping my friends some of my mountain-grown potatoes, I actually have an order from a distributor on the East Coast. I load the sacks into trucks, and each truck holds ten sacks. The third-to-rightmost digit represents the number of sacks left when I have filled as many trucks as possible. If there are 1436 potatoes, that means that once the trucks are filled, there are 4 sacks, 3 crates and 6 potatoes left.

You might have noticed a pattern here. Each crate holds ten potatoes. Each sack holds ten groups of ten potatoes: that’s ten times ten potatoes, or one hundred potatoes. Each truck holds ten times ten times ten potatoes: one thousand potatoes. Every step up in size – that is, every digit further to the left – holds ten times as many potatoes as the last one. If, say, there were a barge that held ten trucks, it would hold ten times ten times ten times ten potatoes: ten thousand potatoes.

This is why we can also write 1436 in a way similar to something you may remember from grade school: “1 group of one thousand potatoes, 4 groups of one hundred potatoes, 3 groups of ten potatoes, and 6 single potatoes.” It is the same as writing, “1 group of (ten times ten times ten) potatoes, 4 groups of (ten times ten) potatoes, 3 groups of ten potatoes, and 6 single potatoes.” Likely none of this is revolutionary: you probably realized it already, on some level.

But with this understanding now explicit, other base systems suddenly become interesting. To jump to another base, there is only one thing you need to do. Change “ten” to another number.

DSC01426 - Ross Farm

Imagine you go to a new Base Sixteen potato farm, and are shocked to see larger crates! Base sixteen represents how potatoes would be sorted at that farm. Instead of crates that hold ten potatoes, sacks that hold ten crates, and so on, the crates hold sixteen potatoes, the sacks hold sixteen crates, and so on. To the people who store those potatoes, 1436 means “1 group of (sixteen times sixteen times sixteen) potatoes, 4 groups of (sixteen times sixteen) potatoes, 3 groups of sixteen potatoes, and 6 single potatoes.” That’s a lot more potatoes than the “1436” from my farm.

In base sixteen, the set of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is not enough. Eleven, for example, can’t be written in base sixteen as a combination of those digits. Usually, people use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F to represent zero through fifteen in base sixteen. So, 6A means “6 groups of sixteen, plus eleven more.” Base sixteen is also called hexadecimal. It is used a lot in computer science, including as a way to represent colors in HTML. (I’m a fan of #8904B1 and #00FFBF.) This very website, like many others, uses hexadecimal notation to specify the font color and background color.

Screen Shot 2015-02-12 at 11.59.14 AM

You may be wondering how all of this helps you count to one thousand and twenty-three on your fingers.

Believe it or not, it IS possible to count that high. All you need is the notion of base and a little creativity.

I’ll show you how in the next post.

Turtles All the Way Down

The point of this post is not to teach or explain anything directly. Instead of walking through the steps of an idea, I am going to tell you three stories. The stories are meant to get you thinking. In my opinion, all three of them point to a single topic.

By design, there are no straightforward answers given here… not even the name of the topic. (Hint: It’s not “turtles.”) 

The Little Old Lady

A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: “What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.” The scientist gave a superior smile before replying, “What is the tortoise standing on?” “You’re very clever, young man, very clever,” said the old lady. “But it’s turtles all the way down!”

Steven Hawking, A Brief History of Time

The old lady in this story has a strange scenario in mind. The Earth, she says, is sitting on a turtle. That turtle is standing on a second turtle. The second turtle is standing on a third. And so on: turtles all the way down.

Of course, giant space turtles are impossible in a practical sense. No one really takes that part of the argument seriously. But the old lady’s reasoning holds a deeper logical flaw.You probably got a sense of it when you read her words. There is something not quite right, though what exactly is wrong may not be immediately obvious.

The issue lurks somewhere the old lady’s final statement. Even if there are turtles holding up the Earth, they cannot go “all the way down.” Her scenario provokes discomfiting questions: How far is down? How many turtles are there? How can there be no bottom turtle?



The Footrace 

Hawking’s tale is reminiscent of another famous story: the footrace described in one of Zeno’s Paradoxes.

A race has been called between a tortoise and the Greek mythological figure Achilles. Achilles is known for being swift, and the tortoise is slow. Because of this, the tortoise gets a head start.

One imagines the turtle ten yards ahead of the starting point. The race begins. Achilles quickly makes it to the ten-yard line. But by that time, the tortoise has moved further forwards, perhaps to the twelve-yard line.

Achilles can reach the twelve-yard line, but by then, the tortoise has moved still further. Though Achilles gets closer and closer to the tortoise, he can never pass him: whenever he reaches the tortoise’s most recent location, the tortoise will have already moved ahead.

Of course, this is not how things play out in real life. If this paradox governed real motion, all sorts of commonplace things would be impossible. But they are possible, and they happen all the time. A slow runner can lose a race to someone faster. A car in the left lane of a highway can pass a car on the right.

Zeno’s fictional race conflicts with our experience. And yet, when presented this way, his logic is not easy to refute.

finish line


The Endless Notebook

“What the Tortoise Said to Achilles” is another paradoxical piece. This one is written by Alice in Wonderland author Lewis Carroll. It is a story about the same characters mentioned in Zeno’s Paradox, and it can be read here.

The Tortoise strikes up a conversation with Achilles, as the two approach the finish line of their race. They start with an innocuously simple question (which I cannot explain better than Lewis Carroll did!) But soon the Tortoise’s dizzying list of inquiries fills up Achilles’s notebook. His ideas never seem to stop.

And in fact they never will. Just as it was with the old lady and Zeno, there is something wrong with the Tortoise’s logic, but it is hard to put one’s finger on exactly what.



There’s an unsettling catch in each of these three arguments. In Hawking’s story, the turtles go “all the way down.” In the footrace of Zeno’s paradox, Achilles gets “closer and closer” to his opponent. And in Carroll’s story, the Tortoise always writes down yet another hypothetical step.

These scenarios are obviously not naturally possible. And yet they hold an appeal. They seem to make sense. And each seems to make sense because it rests on a subtle logical assumption. Somewhere deep in each argument lies a premise that seems true, but isn’t. But what?

Do each of these stories have something in common, besides turtles and tortoises? Are two alike, and one different? And what is the real problem with each argument?

Feel free to comment below.


via NASA Visible Earth