A student asked this question today: "Why is 0! (zero factorial) equal to one, instead of zero?"

Good question! Let's begin by making sure everyone knows what the "!" (factorial) notation means. n! means "the product of all the integers that lie between n and 1, inclusive."

Thus, 4! = 24, because 4! = 4(3)(2)(1) = 24.

6! = 6(5)(4)(3)(2)(1) = 720

The strange thing, though, is that 0! = 1, and that doesn't really seem to match our definition. After all, the integers between 0 and 1 inclusive are 0 and 1, and when you multiply them together, you get zero, not one!

Okay, so maybe our definition is flawed. We'll come back to that later.

The thing is, though, we don't want 0! to be equal to zero, because it's not useful. You see, we use factorials when we're calculating combinations of things, or when we're expanding a binomial to a power.

If you have something like (x + 1)⁵, the n^th term in the expansion of that is (₅C_n-1)x^6-n. That's the binomial theorem.

The problem is, that theorem doesn't work if we say that 0! = 0. Why? Because ₅C₀ = ^5!/_(0!·5!), and if 0! is zero, then we have a division by zero problem! On the other hand, if we say that 0! = 1, then this works out perfectly to ₅C₀ = 1.

And really, if you think about it, that makes sense: If you have five objects, in how many different ways can you choose none of them? Uh, one!

We can see that 0! = 1 makes sense using patterns, too. Consider this:

^7!/_6! = 7
^6!/_5! = 6
^5!/_4! = 5
^4!/_3! = 4
^3!/_2! = 3
^2!/_1! = 2

Now to continue this pattern, what do we need next? 

^1!/_0! = 1

Solve this equation for 0!, and you get: 0! = 1.

Okay, so it makes sense with the combination notation to say 0! = 1, and we can even see from patterns that it must equal 1. So the real problem is our definition. So maybe we should reword our definition a little bit.

I like this way of saying it: For all non-negative n, n! is the product of 1 with all the positive integers less than or equal to n.

Does that work? Sure! It keeps everything else the same, but since there no positive integers less than zero, we're left with 1.

That's one way of getting around it. Another way is to just say For all non-negative n, n! is the product of all the positive integers less than or equal to n. This way of defining it forces us to use the "empty product" definition, which says that multiplying together zero factors gives a result of one.

Or you can define it recursively by saying : 0! = 1, and for all integer n > 0, n! = n(n - 1)!.

Or, you can simply do this: For all positive n, n! is the product of 1 with all the positive integers less than or equal to n, and 0! = 1.

This last one defines n! when n>0, and then gives a special definition for 0!.

No matter how you choose to define it, the real point is that mathematicians chose to define it to be one instead of zero simply because it was of practical use to do so!

Eleventh grade Max from England has a question for us, which references our Vortex Based Math pages:

"Hello prof! Lately I've been finding myself doodling little stars in the margins of my workbooks, seeing what patterns I can make with x number of points in a circle, a rule for arranging lines, making notation for different rules and so on. So far it's just been harmless messing around with patterns, but I saw your post on vortex maths and I'm concerned at the similarities I'm seeing with my own work. I don't think of myself as particularly 'mystic' but I do appreciate the patterns in mathematics, a la ViHart on YouTube and I'd rather not end up like Marko Rodin. My question is, should I keep on doodling and doing pseudo-mathematical patterns or move on and find something else? If so, what related things can I doodle that might bring me a bit closer to actual maths? Thanks for your time"

Well Max, I've got to tell you, this is one of the best questions I've seen in awhile. I like the way you're thinking. Before I answer your question, though I'm going to tell you a little story that your comment about doodling reminded me of.

When I was a kid, maybe in third or fourth grade, while all my classmates were learning addition and subtraction and the like, I was (I confess) a bit bored with math class.

So I started doodling. I doodled a tiny five-pointed star in the middle of my note paper. Then I drew a pentagon around it. Then I drew another star outside of that, and then another pentagon, and then another star, until my entire page was filled with stars within stars within stars.

And then I realized there was someone standing over my shoulder watching what I was doing.

It wasn't my teacher. Oh no, it was worse than that. It was the District Mathematics Curriculum Coordinator. And when you've been caught doodling by someone with such an impressive sounding title, it's a bit terrifying.

But Mr. Tame just asked me quietly, "What have you learned?"

I was a bit flustered (to say the least) but I had enough presence of mind to stumble out a response about how my first star wasn't perfectly drawn, and the more stars I drew, the more those errors got magnified.

He nodded thoughtfully and said, "That's a pretty important lesson," and then moved to look over the next student's shoulder.

I breathed a big sigh of relief and turned my attention back to addition and subtraction.

I appreciated Mr. Tame's willingness to let me get away with doodling. Because I did figure out something important about error and percent error (which I certainly could not have put into words at the time, but as Mr. Tame said, it was an important lesson).

So don't be afraid to doodle. Don't be afraid to explore. Don't be afraid of patterns. But don't stop just because you've found your pattern. Now ask the question, "Why?"  That's where Marko and his buddies fell apart. They thought the pattern was the end of the quest. But the pattern was just the beginning. From there they should have gone on to ask questions like, "What is the cause of these patterns?"

I love that you referenced ViHart. Have you watched her video about the golden ratio and Fibonacci numbers? If you haven't you should. There are some amazing, astounding patterns related to the Fibonacci sequence and the golden ratio that show up in nature. But ViHart doesn't just say, "Have you noticed that the leaves of a plant spiral in patterns related to Fibonacci numbers?" She says, "These are astounding patterns, and let's see if we can figure out why a plant's leaves tend to grow in this way!"

That's the difference between a Rodin and a ViHart. One is satisfied to find a pattern, the other wants to know why the pattern exists. Some of the most interesting mathematics and science stems from people discovering patterns and then trying to figure out why they happen.

A wise old king (Solomon) once said, "It is the glory of God to conceal a thing: but the honour of kings is to search out a matter." The universe is filled with extraordinary and beautiful patterns. They're like a buried treasure hidden just below the surface. But don't just find them. Dig them up. Search them out. Understand them.

And if you can't answer the question "Why?" Don't get discouraged. Go find something else to explore, and maybe someday down the road your explorations will connect some dots, and you'll have an answer to the question that you couldn't answer today!

Thanks for writing, Max!

Professor Puzzler

Michael from Los Angeles asks, "Let's say the first man, Adam did NOT eat the forbidden fruit. Then he would have procreated with Eve in the Garden of Eden and produced a family who would have procreated with each other to produce a bigger family who would have procreated with each other to have an even bigger family and on and on. How long would take for the population of this 'family' to reach, say, 5 million persons? Keep in mind these persons not only never die, they never age past 20 years physically. There is no sickness and they are ALL fertile at age 16. There is no pain assoiciated with child birth and every child is born perfectly healthy. Let's say they are equally male and female and begin to procreate at age 16 and, to make the math simpler, each couple only has one child per year. How long would it take for their number to equal 5 million?"

Well, Michael, once in awhile I get a question that interests me enough that I don't just answer it, I do additional exploration to amuse myself. And you're in luck, because I'm sharing my additional exploration with you.

You see, today while I was on my lunch break between teaching Algebra One and Algebra Two, I created a "population explosion simulator," which allows you to enter a variety of parameters, such as initial population, death age, number of years between child births, etc.

The default parameters for the simulator are the parameters which satisfy your question.

Population Explosion Simulator!

Enjoy!

Kobe from Mephis asks, "What conversions are used to convert miles per hour to miles per minute?"

Well, Kobe, unit conversion is a very important topic to understand, so rather than just giving you a quick answer to your question, I'm going to show you how to go about figuring it for yourself, okay?

Let's suppose you have the quantity 60 ^mi/_hr, and you want to convert it to ^mi/_min.

In order to do this conversion, you're going to have to multiply 60 ^mi/_hr by one or more conversion factors. What is a conversion factor? Well, I like to tell my students that a conversion factor is just one. After all, if we're going to multiply 60 ^mi/_hr by something without changing its value, that "something" has to be the number 1. It might not look like the number one, but it has to be equal to 1.

So let me ask you a question: How many minutes are there in an hour? Hopefully you said "60." 

So 1 hr = 60 min, right? But if 1 hr = 60 min, then ^{1 hr}/_{60 min} must be equal to one, because the numerator and the denominator are both equal to the same thing. This is a conversion factor. Of course, ^{60 min}/_{1 hr} is also a conversion factor, right? We're going to use one or the other of those two conversion factors; we just have to figure out which one. So we write out an equation:

60 ^mi/_hr× _____ = ? ^mi/_min.

The big question is, what is the conversion factor I'm going to put in that blank space? Well, I'm going to choose the conversion factor that causes the unit "hours" to disappear from the equation. After all, I don't want to have "hours" in my answer, right?

So I look first at where the "hours" unit is in my equation. It's in the denominator of the units. That means I want to use the conversion factor that has "hours" in the numerator. Why? Because if there's "hours" in the denominator, and "hours" in the numerator, they'll cancel each other out, which is what I want!

So we do this:

60 ^mi/_hr× ^{1 hr}/_{60 min} = ? ^mi/_min.

Now we multiply across. Notice that our conversion factor doesn't just get rid of the "hours" unit - it introduces the "minutes" unit into the denominator - which is great - that's the unit we want in the denominator. When we multiply across, the "hours" cancel, and since we have a 60 in the numerator and a 60 in the denominator, they cancel as well, leaving us with:

60 ^mi/_hr× ^{1 hr}/_{60 min} = 1 ^mi/_min.

And that's it, you're done! We can get more complicated than this, because sometimes you'll run into conversion problems which require you to convert two units. For example, converting miles per hour into feet per second! For this one, you'd need to convert miles to feet and you'd need to convert miles to seconds. So you'll have two unit conversion factors in order to solve the problem.

Thanks for asking!

Professor Puzzler

Fifth grader Mario asks, "Is there a limit to the biggest or smallest Number?"

Good question, Mario! Sorry for the long delay in answering; we've been swamped with mail lately! The answer is: no, there's no limit. Numbers keep going on forever in all directions. You might not know the names of the numbers, but that doesn't mean they don't exist!

Big Numbers

What's the biggest number you can name? A million? A trillion? A quadrillion? A googol? (Not a Google, that's is a search engine!) A googol looks like this:

10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

It's a pretty big number! (And it's also called ten duotrigintillion, but I don't even know how to pronounce that, so I'm going to stick to saying "googol.") A googol is a one followed by one hundred zeroes!

But is that the largest number? It isn't! After all, I could add one to it, couldn't I? Then I would have:

10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001

Better than that, I could multiply it by ten:

100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

Or by one hundred:

1 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

Wait a minute! I could even multiply it by itself!

100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

I have no idea what to call that number (except I could call it "a googol times a googol"). But even though I don't have a fancy name for it, that doesn't mean it's not a real number, right? In fact, could I keep multiplying it by itself? Sure! I could even multiply it by itself a googol times, which would have so many zeroes that I couldn't possibly display it on this page.

No matter how big a number is, I can make it even bigger by tacking more zeroes on the end of it! So even though I don't know the names of all those numbers, there is no limit to how big a number can be!

Negative Numbers

When you said "how small" you might have meant a couple different things - you might have meant negative numbers, or you might have meant numbers close to zero. I'll address the question of how close you can get to zero in a second.

But first, in case you were talking about negative numbers, here's something you should know: For every positive number, there is a corresponding negative number. Take any positive number and stick a negative sign in front of it, and you've got the corresponding negative number. 7 and -7. 100 and -100. 1 googol and -1 googol. 

You get the idea, right? If there is no limit to how big the positive numbers can be, that means there's no limit to how big the negative numbers can be, either!

Close to Zero

But maybe, when you said "smallest number" you were talking about how close to zero you can get. And the answer is still "No, there is no limit."

For this, let's grab a calculator (if you have one handy) and check out a couple things. Calculate the following for me: 

• 1/2 = ?
• 1/3 = ?
• 1/4 = ?
• 1/5 = ?

These are called reciprocals. When you divide one by a number, you get that number's reciprocal. Many calculators have a reciprocal button; on my calculator it looks like this:

x^-1

You should get something that looks like this:

• 1/2 = 0.5
• 1/3 = 0.33333333
• 1/4 = 0.25
• 1/5 = 0.2

Notice that the bigger the number, the closer its reciprocal is to zero. Let's try a few more numbers to make sure. This time we'll pick numbers that are powers of ten (a one, followed by some zeroes).

• 1/10 = 0.1
• 1/100 = 0.01
• 1/1000 = 0.001
• 1/10000 = 0.0001

Oh, I'm seeing a definite pattern here! The number of zeroes after the decimal is one less than the number of zeroes in the number we took the reciprocal of. What happens if we divide 1 by a googol?  Well, you don't want to try to enter a googol in your calculator, but we can use the pattern to figure out the answer. (That just shows that you're actually smarter than a calculator!)

1/googol = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 1

Wow! That number is tiny! But is it the tiniest? No! Because there are numbers even bigger than a googol that we could take the reciprocal of. And the bigger the number, the closer its reciprocal is to zero! And since there is no limit to how big numbers can be, there's no limit to how close to zero we can get, either.

Another way of looking at it: No matter how close to zero a number is, we can always make it smaller by sticking another zero after the decimal point.

Now, Mario, when you get into high school or college, you'll probably take a calculus class, and you'll start learning more about these kinds of ideas. You'll start hearing teachers talk about "limits" and you'll think, "But wait! Long ago, Professor Puzzler told me there were no limits!" Don't worry. Keep listening. You'll realize that the way your teacher is using the term "limit" is different from the way you used it back when you were in fifth grade. And you'll also realize that we just scratched the surface of these ideas. When it comes to understanding math...

...the sky is the limit!