Fun! Facts about Factorials!

By Leo Lam

(Test Preparation (ACT/SAT/SSAT), Math, Physics at The Edge Learning Center)

Students often first encounter the idea of factorial when they learn about probability, particularly combinatorics. They often see it again in binomial expansion or binomial distribution. If they go into advanced calculus, they would see it popping up in Taylor series and Maclaurin series. Most students will just remember that factorial is defined as “multiplying all the positive integers up to the particular number.” Today, I would like to share some facts about this exciting(!) operator in mathematics and certain rules to follow when working with it.

Read more on the importance of math in life in Leo’s blog “A link between Math and Life”

You don’t need to yell

The exclamation mark notation was introduced by French mathematician Christian Kramp in 1808. The idea of multiplying consecutive whole numbers has been around for centuries; people had been using this method to find out the distinct way to order multiple objects (i.e. permutation). Having this notation, as Kramp personally suggested, makes writing down things much easier.

You might see it as Pi

The need of multiplying numbers is not unique to the consecutive whole numbers. As a matter of fact, just like the need to add certain terms together (which we use the Sigma notation, Σ, to denote such operation), it is more common to see the product notation Π. A typical Pi notation works just like a Sigma notation; instead of adding, you just multiply each term! Here is an example:

It gets huuuuge!

If you have a single-line display scientific calculator with only 2 digit displayable exponent, you would likely find that typing 70! will give you an error. It doesn’t mean the value cannot be calculated. 70! is bigger than 1 googol, which makes it impossible for such calculator to display. Here is a list of some of the bigger factorial values:

Your calculator/computer doesn’t just multiply

Mathematically speaking, calculating factorial is easy: you just multiply a bunch of numbers together. However, simple as it may seem, most computers don’t find the answer by just multiplying. The sheer value means there are certain limitations. Factorials are always integers because it’s the result of multiplying integers together. Modern computers covert our everyday numbers into binary, using only 0s and 1s, before they do any calculation. Older computers are limited to working with 32 binary digits, or bits, which translate to a maximum of 2,147,483,647 in decimal. For the newer computers with 64-bit architecture, it can store an integer in decimal value up to 9,223,372,036,854,775,807 . Which is a big number. But if you look at the table, you will see that a 32-bit computer can only calculate up to 12!, while 20! is the limit for a 64-bit computer. Beyond these boundaries, most common computers have to resort to approximation to provide an answer.

It only works for whole numbers… or does it?

Most students know that we can’t do 2.1! or (-3)! because factorials are limited to positive whole numbers. However, mathematicians have extended the application of factorial, called the Gamma function, into complex numbers (with the exception of negative integers). This application is often utilized in mathematical analysis and complex analysis.

Above is the graph of the Gamma function plotted for real values.

At this point, you would probably say, “yeah, that’s exciting(!) and all, but it’s not like I will ever need to apply any of that. Show me something I can use now!”

So here it is, certain rules and tricks that you want to remember about factorials.

0! = 1

As weird as it may sound, this is a fact that we must remember. There are several ways to explain why this must be true. In mathematics, the empty product rule (which is very different than the “null factor law.” I have written an application of the null factor law in my previous blog, which you can find here) states that when we are multiplying no product, the result is set as default to the multiplicative identity, which is 1. 0! Literally means we are not multiplying any number, so it falls into the empty product rule. Thus, the result should be 1.

Another way to explain 0! is to look at it like finding ways to order items. We learn from permutation that ! gives you the number of unique ways to order  items in distinct sequence. Now think about what 0! represents; it is kind of like “how many ways are you able to put nothing in order.” And the answer is… 1 way – by ordering nothing. More importantly, this definition fits into the model or permutation and combination.

A third way to look at this is to see how consecutive factorials relate to each other. For example, we know that:

but we can also rewrite it as:

Which relates the two consecutive numbers. If we continue to move our list down, we can see that:

 

nCr and nPr will always give you a whole number

This is a reminder to those who is given a non-calculator question that involves combination and permutation. If you need to calculate these values, the result will always be a non-zero whole number. Keep in mind that both of these functions are finding “number of ways” to perform something, so getting a fractional result would not make any sense.

You can’t “cancel”, “split”, or “combine” factorials

A common mistake that students make is doing something like:

Which is very tempting to do, because they look just like a fraction. However, if we expand the terms, we will see that:

which is different. We can think of the factorial as an operation that needs to precede multiplication and division, which limits how we can simplify terms.

Thus, we also need to remember that:

Simple counter examples will help us check that the operations above are invalid.

However, you can still manipulate factorials

One of the most important techniques regarding factorials is that we can rewrite a factorial as products, which can be rewritten again if we are being careful. For example:

While we are not splitting the 15 into smaller numbers, we are able to “peel” each layer off and create a different factorial. This means the following ways to rewrite a factorial are all valid:

It is also possible to “add layer” by making the factorial bigger, like this:

Thus, here is another way to make a factorial look different:

Now let us take a look at an example and see how we can actually make certain factorials look simpler.

or

The key is to work with the larger factorial and change it to a multiplication between individual term(s) and a smaller factorial.

And now you know more about this unique operator in mathematics. Next time when you encounter the exclamation mark in your math assignment, don’t fret! Remember the rules and tricks and you can shred them with ease.

Hopefully you won’t have this face anymore when you are given a factorial question.

 


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