From 1 + 1 = 2 to Amazon.com

On Quora, someone asked how you could explain to a 5-year-old why 1 + 1 = 2 and not 3, 4, or some other number.  It’s an interesting question, and I’ll repeat my answer, plus a little more insight into why the question is important.

Roughly speaking, this is actually a proof question:  Prove 1 + 1 = 2.  I’ve talked a little bit elsewhere about why mathematicians do proofs, and why proof is important, but a quick justification for why you’d want to prove 1 + 1 = 2 follows.

The reason 1 + 1 = 2 is that mathematically, “+1” means “the thing after.”  So 1 + 1 is “the thing after 1,” which is to say 2.  Likewise, 4 + 1 is “the thing after 4,” namely 5.  The reason this is useful is that once you’ve extended the idea of arithmetic this way, you can develop an entirely new type of arithmetic and ultimately arrive at the basis for internet security.

It’s a long path, but here are the highlights…

Once we recognize “+1” as “the thing after,” we can do arithmetic with anything that has a definite ordering.  Thus Sunday + 1 is “the thing after Sunday,” which we might say is Monday, and write Sunday + 1 = Monday.

We can extend our ideas to +2:  Sunday + 2 = Tuesday, Sunday + 3 = Wednesday, and so on.  However, an interesting thing happens when you go too high:  What’s Sunday + 7? It should be the seventh day after Sunday…which is Sunday again.  So we have Sunday + 7 = Sunday.

Now here’s a peculiar thing:  Sunday + 7 = Sunday.  It seems that “+7” has changed nothing.  In particular, it’s just like adding 0.  In higher math, we have a name for this:  7 is the modulus of the arithmetic system, and the type of arithmetic we produce is called modulo arithmetic.

We can work with other moduli.  The only thing we have to remember is that the modulus is just like 0, so if we add the modulus to a number, we get the same number back.  For example, consider a wacky number system with modulus 12, so we’d write things like 3 + 12 = 3.  Or (since 24 = 12 + 12), we could even write 7 + 24 = 7 and so on.

Now who on Earth would use such a bizarre number system?  The answer is:  you do, every time you check your schedule.  Thus 9 + 4 is the fourth hour after 9:  it’s 1, so we’d add 9 + 4 = 1.  (There are about a million reasons why everyone should learn to tell time on an analog clock…this isn’t one of them, though being able to tell time on an analog clock will make this sort of arithmetic a lot easier)

Let’s change gears a bit, and consider modern internet security.  When you send your credit card number to Amazon.com, you’d really prefer it if no one else could read it.  Thus you encrypt it.  Modern encryption techniques are based around a central idea:  The number you want to keep secret is the solution to a mathematical equation that is very hard to solve.

Unfortunately, in ordinary arithmetic, most equations can be solved easily through the use of what mathematicians call the intermediate value theorem, though the rest of us might call it the Goldilocks Principle.  Consider this:  My secret number solves the equation x^{3} = 729.

Now even if you don’t know how to solve this equation, you can still use the Goldilocks Principle as follows.  Guess any number:  for example, 3.  We observe that 3^{3} = 27, which is too small.  Now guess another number:  for example, 10.  We observe 10^{3} = 1000, which is too big.  Since 3 is too small and 10 is too big, some number between is just right.

So we guess again:  5.  We have 5^{3} = 125, too small.  So now we know 5 is too small and 10 is too big, and again something in between is just right.  If we keep doing this, we’ll be able to find the number that’s just right, and determine the secret number.  Not what you want if the secret number is your credit card number!

So how does modulo arithmetic help?  Modulo arithmetic helps because the intermediate value theorem fails.  Suppose we’re working with a modulus of 12.  Again, suppose my secret number satisfies x^3 = 5.  We try again:  we guess 1, and find 1^{3} = 1, so 1 is too small.  Our next guess is 2, and find 2^{3} = 8, so 2 is too big.

But there’s no number between 1 and 2 that will be just right!

Yet there is a solution:  5^{3} = 5 when we’re working with a modulus of 12.  (That’s because 5^{3} = 125 = 12 \times 10 + 5, and if 12 is like 0, this means 125 = 0 \times 10 + 5 = 5)

The moral of this story is this:  Don’t break into other people’s houses and eat their porridge.  Whups, wrong story.

The moral of this story is this:  Modulo arithmetic allows us to set up equations that are very hard to solve.

Of course, if you’re thinking ahead, you’ll realize that Amazon has to be able to solve the equation.  But that’s a post for another day.

The Art of Estimation

So unless you’ve been living under a rock,  you know many people have concerns about Donald Trump being President.  This is from one of the marches.  One quick political statement, and then on to the math:  I prefer to think of them as marches, not as protests:  Trump is the President of the United States, and while I don’t like the fact, protesting it doesn’t do anything useful.  Rather, it’s a reminder…in this case, that women make up 51% of the population of the U.S., so the powers-that-be should remember this.

Back to the math.  During this event, the question came up:  How many people showed up? You could, of course, try to count the number of people, but this is difficult, since people are moving (this is the problem the Census Bureau faces…you can read a little bit about the reason why it’s important in my article from Math Horizons, and read quite a bit more about it in the first few chapters of my book, Constitutional Calculus).

If an exact number is important, there are a number of statistical techniques that can be used.  But often, a rough estimate is enough.  There are several ways to make such an estimate.

The first is what you’d call an order of magnitude estimate.  To get a sense of what this means, pick any whole number:  for example, you might take n = 2.  The powers of n form a sequence of numbers:  2^{0} = 1, 2^{1} = 2, 2^{2} = 4, 2^{3} = 8, 2^{4} = 16, and so on.  An order of magnitude estimate occurs when we identify one of these as close to our number.

For example, in the picture, we might try to find an order of magnitude estimate for the number of people holding signs:  Is it closest to 1, 2, 4, 8, 16, 32, 64, 128 (or more)?  Don’t try to count the number of people, as this defeats the purpose of making the estimate!  (I’m thinking 16).  More commonly, we take powers of 10:  1, 10, 100, 1000, 10000, and so on.  Thus, the number of people holding signs is around 10.

The nice thing about order of magnitude estimates is that they’re easy to make if you have any experience with the so-called real world.  Thus, here’s an order of magnitude estimate for the number of marchers:  10,000.  I get this number because I know what a crowd of 100 looks like (a crowded auditorium) and have some sense of what 100,000 people look like (a stadiumful of people), and it seems that the number of marchers is somewhere between the two.

The order of magnitude estimate is fast and easy to obtain…but you get what you pay for:  it’s not very precise.  To gain more precision, you’ll need to make do a little work and deal with some facts.  (As I’ve noted elsewhere, one of the reasons mathematics is important to a free society is that it forces you to ask yourself what you really believe and why you believe it)  In this case, we have the following data points:

  • The picture is from the Walkway Over the Hudson, a former railway bridge converted to a linear park.  The bridge portion is a little over a mile in length.
  • The marchers occupied (in the sense of took up space, not in the sense of Wall Street) the whole length of the bridges.

The art of estimation begins by using approximations to the actual values.  This takes a little experience and a little knowledge.  In this case, the knowledge is that the bridge is a little over a mile in length, and 1 mile is 5280 feet.  We’ll estimate the bridge’s length as 6000 feet.

To estimate the number of marchers, we need to know how many people there are per foot of bridge.  An average person can squeeze into a one-foot space, which means that if packed to capacity, one linear foot of the bridge can have a lot of people across it:  imagine drawing two lines across the bridge, one foot apart.  You could probably fit a good dozen people in there.

Now you can see from the picture, the bridge isn’t that packed.  However, we can use this to provide an upper bound on the number of marchers:  the fully packed bridge could hold 12 \times 6000 = 72,000 persons.

Why is this useful?  Suppose someone reports on the march, and says that “Over a million people attended.”  Given that a fully packed bridge would only hold around 72,000, we might suspect this number is a gross over-estimate.  Good mathematics helps us identify questionable claims.

At the same time, consider another report:  “Only a few hundred people attended the march.”  The bridge is 6000 feet long.  If there were only a few hundred people on the bridge, each person would have to be some distance from the other.  Note that you could divide 6000 by however many hundred the report claimed, but you could also do an estimate:  If each person stood 10 feet apart from the next, then the bridge would hold 600 people.  But the picture clearly shows people standing closer than 10 feet apart, so the number of marches had to be in excess of 600.

So how about an actual estimate?  Here’s what we did (several of us discussed the question):  Along 6000 feet of bridge, there was always more than one person per foot.  This gave us a minimum of 6000 people.  However, we didn’t think there were as many as twelve persons per foot of bridge, so we did the easy thing and doubled the amount for our upper estimate.   Thus we concluded that there were between 5000 and 10,000 marchers.

(Why not 6000 and 12,000?  We could, but given the nature of the numbers, we rounded to multiples of 5000…this too is part of the art of estimation)

I’ve written a follow-up that gives the official and unofficial estimates.

One final note:  It’s important to emphasize that thinking mathematically is a habit.  In this particular case, it’s the habit of asking questions like “How many people do you think are here?”  By answering that question, we are armed with a useful tool (an estimate of the number of marchers), which we can then use to confront the world (disparate claims over the number of marchers).

This number sense will be increasingly important in the years to come.

Neither Borrower Nor Lender Be…

Despite its name, the Sapir-Whorf hypothesis is not something found on Star Trek.  Instead, it’s based on the following idea:  How you speak influences how you think.  I stress this concept in most of the courses I teach, but nowhere is it more important than in elementary mathematics.

Consider the following subtraction:  42 – 27.  If you were taught the standard algorithm, you might have been taught that since you can’t subtract 7 from 2, you have to borrow 1 from the 4, making it 3; the 1 then makes the 2 into a 12, which you can subtract from.  For generations, schoolchildren have been taught to subtract by borrowing.

And for generations, schoolchildren have been taught something that is blatantly false!

If I borrow your car, you expect I’m going to return it to you.  You’d be very unhappy if I kept it, or sold it for scrap.  Yet this is exactly what happens to the 1 “borrowed” from the 4:  it is never returned.

A better term is trade or unpack.  Let’s shift gears a moment, and suppose you need 8 eggs.  You go to the refrigerator, and find to your dismay that you only have 4 eggs, and 1 dozen.  You can’t get 8 eggs, because you only have 4 eggs.  Fortunately, you’re clever and realize “Hey, I can open up the dozen, and I’ll have 12 eggs.”  Indeed:  from the dozen, you’ll take 8 eggs, leaving 4 behind.  (The refrigerator will still have the other 4 eggs, which weren’t touched, so all together you’ll leave behind 4 + 4 = 8 eggs, possibly in two cartons…a frustratingly common situation in our house)  You can watch this video to see this exact process.

What about 42 – 27?  Here you have 4 tens 2, and want to subtract 2 tens 7.  So you unpack one of the tens, giving you 3 tens 12, from which you can remove 2 tens 7, leaving 1 ten 5:  15.

What about 4 hours, minus 2 hours 27 minutes?  Again we unpack:  4 hours is the same as 3 hours 60 minutes, from which we subtract 2 hours 27 minutes, leaving 1 hour 33 minutes.

What about 2 miles minus 153 feet 8 inches?  I dunno…that’s the “English” system of units.  You can tell how useful the system is, since (by GDP) two of the three most powerful countries to use it are Myanmar and Liberia, which most people would be hard-pressed to find on a map.

Fortunately, I collect useless trivia, so I know 1 mile = 5280 feet, and 1 foot = 12 inches.  So unpack 1 of the 2 miles into 5280 feet; then unpack one of the feet into 12 inches.  This means 2 miles becomes 1 mile 5279 feet 12 inches, from which we can remove 153 feet 8 inches, leaving 1 mile 5126 feet 8 inches.

 

Why Study Higher Mathematics?

Just about everyone will tell you that mathematics is important and should be taught in school.  All the arguments are over the type of mathematics that should be taught.  Algebra?  Statistics?  Probability?  Real world problems (“John has 137 erasers to distribute among 17 students…”)?  Math world problems (“How many ways can Ellen arrange six different books on a shelf…”)?

The problem is that you can’t answer the question of “What should we teach in school?” until we answer the question of “Why should we have an educated population?”  There are three reasons for getting an education:

  1. To help yourself.
  2. To help your community.
  3. To help your government.

These goals aren’t mutually exclusive:  If you help yourself, for example by getting a better job, then you help the government, because you pay more in taxes (and, because you have a better job, you are more likely to want to keep things the way they are…so you support the government).  Likewise, if you help your community, you’re helping yourself, because you’re making your own living environment better.  At the same time, helping the community might not be helping the government, and vice versa.

This is where things get interesting.  If school mathematics focuses on “real world problems” and “useful mathematics,” then it’s set up to help individuals and the government.  However, there’s no guarantee that this will help the community:  wealthy persons exist, even (and perhaps especially) under repressive regimes.

Here’s why higher mathematics is important.  First, a quick definition (my own):  Higher mathematics is any mathematics that focuses on proofs.  Every branch of mathematics can be taught and learned at a higher level:  3 + 2 = 5 is arithmetic and very basic; higher mathematics occurs when you ask (and answer) why 3 + 2 is 5.

I’ve talked elsewhere about why mathematicians do proofs. Despite my silver-tongued eloquence, not everyone is convinced:  students continue to ask “Why should we have to prove things that everyone knows?”  But consider this:  Throughout history…

  • Everyone knew that men were smarter than women.
  • Everyone knew the sun went around the Earth.
  • Everyone knew that slavery was acceptable.

Progress occurred when people began asking “Well sure, everyone knows these things…but are they really true?”

And this is one of the things that higher mathematics trains you to do:  Proof causes you to ask questions about what “everybody knows.”  Take that “3 + 2 = 5”.  What do we really mean by that?  There’s a few different answers, but one of them is “3 + 2 is the second number after 3.”  (You can go in a few different directions after this point, but that’s another story)

We can go further:  proof requires us to construct a logical argument, with each step based on the step before it.  We can’t make bold leaps, like “Since this happened one time, it must happen all the time.”  Instead, we have to establish a chain of causality, with each step carefully constructed.

We can go further still:  proof forces you to constantly question your own beliefs.  Many people accept 5 \times 3 = 3 \times 5 without question.  But if you’ve been taught to do proofs, there’s a little voice inside your head saying “Why do you believe that?”  If the answer is “Because someone told me it was true”, then you’re uncomfortable…and try to find evidence.

Now for the punchline:  You can slant history to make sure your country is always right, and other countries are always wrong.  Philosophy can be wrangled to serve the state:  China did this for two thousand years.  Even science can be forced into doctrinal correctness:  physics and chemistry did very well under the Soviets.  The arts can be browbeat into submission; literature can be stifled; engineering can be co-opted.

But questioning belief and demanding evidence is an integral part of higher mathematics.  You cannot remove these components from higher mathematics without destroying higher mathematics.

And if the state chooses not to teach higher mathematics?  Then the mathematics needed to solve new problems will not be developed.  Individuals and society will suffer…and the government will be replaced.    Thus I claim:

The development and continued existence of a free society depends on the teaching of higher mathematics.

Two Is the Oddest Number

One of the biggest problems facing anyone in a creative field is:  How do I create something new?  For obvious reasons, you can’t be taught how to be original.  But there are some ways you can make it easier for your creativity to emerge.  Here’s one strategy:  Two is the oddest number.

What does that mean?  One of my thesis advisors (and the original source of the saying) explained it like this.  To a mathematician, there are only three numbers:  zero, one, or infinity.  Either something doesn’t exist at all (zero); it’s unique (one); or it happens infinitely often (infinity).

For example, suppose you have a line and a point.  In Euclidean geometry, there is a unique straight line through the point that is parallel to the given line.  In spherical geometry, there are zero straight lines through the point that are parallel.  In hyperbolic geometry, there are an infinite number of straight lines through the point that are parallel.  Thus zero, one, or infinity.

Or consider primes.  There is a unique even prime:  2.  All other primes are odd.

So how does this help you create something new?  One way is to try and find a second example of something.  Here’s an example, that requires some background in permutation groups.  Here’s the short version, though if you really want to delve into the topic, take a look at my (in progress) videos on abstract algebra.

Suppose I have a set of distinct symbols a, b, c, d, \ldots.  A permutation occurs when I rearrange the symbols.  It’s best to think of the permutation as what happens when you do a “replace all” in a document.  A compact way to represent these permutations is cycle notation, where an expression like (abc) indicates you’re going to replace all as with bs, all bs with cs, and all cs with as.  Because this cycle has three elements, it’s called a 3-cycle.

We can also have 2-cycles:  (ab).  These occur so often that we have a special name for them:  they are transpositions.  

We can juxtapose two cycles (of any length) and form a composition.  For example, (ac)(ad).  For somewhat technical reasons, we read these from right to left.  Thus first we replace all as with $latex $d$s, and all ds with latex $a$s; then we replace all as with cs and cs with as.  The net effect is that all as have been replaced with ds; all ds with cs (because the first cycle replaced them with a a, and the second cycle replaced the a with a c); and all cs with as.  So (ac)(ad) is the same as (adc).

In a similar manner, we can write (ac)(bc) as (acb).

At this point, the mathematician says “Hmmm…two times we’ve managed to replace a composition of two transpositions with a 3-cycle.  But two is an odd number…maybe we can always replace a composition of two transpositions with a 3-cycle?”

So now we try it on (ab)(cd).  And we find…the first component of a rather important theorem in the study of permutations.

Flip Your Class, Part Two

Earlier, I described a new approach to teaching that is very old:  the flipped or inverted classroom.  Here I’ll start describing some of the nitty-gritty details, and some of the pitfalls.

The most important thing to remember:  A video lecture is NOT an ordinary lecture that has been videotaped.

A lot of folks have videotaped their lectures and put them online.  These videotaped lectures have their role, but in general, they don’t work for flipped courses for two important reasons:

  • A lecture is 50+ minutes with a captive audience.
  • A human visual field covers 120+ degrees.

What this means is that you can talk for a long time on a topic, and you can cover the board with lots of information.

In contrast, a person watching a video on the internet has an attention span of about 5 minutes.  (Be honest with yourself:  How often do you sit through the entirety of a 15 minute YouTube video, and how often do you fast forward until you get to the interesting bits?)  This means if you videotape a 50 minute lecture, most people will watch it for five minutes…and it is nearly impossible for someone to fast forward through the lecture to find the relevant parts.

There’s another problem:  that video also covers maybe 60 degrees of the person’s visual field.  This means you have considerably less space to present information.  When PowerPoint first came out, most users forgot about that, and loaded slides with hundreds of words and loads of information.  These days…most users still don’t understand that if a slide contains more than about 100 words, it will be unreadable.

What does this mean for math teachers?  I’ve adopted the ten minute rule:  If the material takes longer than 10 minutes to present, break it down into 9-minute segments, and include 1 minute at the beginning to review the material of the previous 10-minute segment.

So here’s the first step in flipping a course:  breaking the entire curriculum into 10-minute segments.  If you’re a gonzo mathematician like myself, you’ll do this all at once, and even for courses you have never taught like abstract algebra.  But if you’re a normal, sane person who wants to stay that way, you’ll take a more sensible approach.  I’ll talk about that in a future post.

Welcome to the 21st Century!

Once upon a time, a teacher could assign homework questions to students, and the students could be relied upon to do the work on their own.

Then writing was invented.

A little later, homework solutions manuals became popular (and Cliff notes, for those in the humanities).  Still later, you have the internet where, if you can’t find the answer to a question, the answer doesn’t exist.

This can be a source of dismay to some teachers:  How can they ask a question and prevent their students from simply looking up the answer?  There’s an easy answer to this:  You can’t.

You have to accept the reality that students will look up answers…and adjust your expectations accordingly.  For example, here’s a question that you could look up online:  How many 3-digit numbers increase their value when the digits are reversed?

As worded, the answer “37” is correct (or not…I’m not going to give it away).  All that’s necessary is for a student to enter the question into Google or post it onto a message board, and someone will happily supply the answer.  This makes it a bad question in the 21st century.

What would make this same question a better question?  One added instruction:  “Explain your reasoning process.”  One of three things will happen:

  • A student will work it out, and be able to explain their reasoning process.
  • A student will go online and get the answer.  If they’re honest, their explanation is “I had no clue, so I asked someone to tell me.”  Otherwise, they might try to come up with a reasoning process…that is completely nonsensical.
  • A student will go online and get the answer and the respondent will explain how they got it.  In order to explain the reasoning process, the student would have to sift through the explanation.  This means that, even though they didn’t originate the thought process, answering the question still has learning value.

What about the student who simply cuts-and-pastes someone else’s reasoning process? One way to limit this is to have the student explain it to you, and ask them to elaborate on each step.  If they didn’t properly assimilate the reasoning process, they won’t be able to explain it to you.

Of course, this relies on having the time to teach.  This is the greatest threat to effective teaching:  overloading the curriculum so material can’t be covered deeply; and overcrowding the classroom so that students can’t get the individual attention they need.

Fortunately, technology takes with one hand, but gives with the other.  One way to make it easier for teachers to devote more time to developing student understanding is the inverted or flipped classroom.

Flip Your Class

Over the years, I’ve done quite a few educational videos for math courses, from Math for Elementary Education, through Calculus and Linear Algebra, and Cryptography.  Originally, I’d intended them to be resources for my students, but unknowingly I had started down the path towards the flipped (or inverted) classroom.

What’s a flipped class?  It’s a brand new thing in education…that’s very, very old.  Back in the Good Old Days (which never existed, but that’s another story), students would seek out a teacher.  The teacher would say “Read these books, and we’ll discuss them next time.”  This had several advantages:

  • The student got into the habit of learning on his or her own,
  • The student got more personal interaction with the teacher.

So what are most math classes like nowadays?  The teacher spends an hour or so lecturing on a topic, then sends the students away to do homework.  The lecture is impersonal:  as a student, you’re one of many; as a teacher, you give the same lecture whether there’s one person or a thousand.

But let’s follow the student after the lecture.  Every student has experienced the following situation:  leaving a lecture, thinking that the lecture was brilliant, since everything made sense.  They go home, ready and raring to tackle the homework, and then…discover that they missed something critical, and can’t do half or more of the problems.  They can try to see the teacher during office hours, or ask questions during class, but even if they can, by that time a week has passed by and the class is on the next topic.

The flipped class alters the timing:  Students get the lecture at home, usually through some web video; then they go to class to work problems.  This works best when students take advantage of several key features of online videos, namely:

  • They can be played anytime, anywhere:  students can learn at their own pace.
  • They can be paused and rewound:  students can learn at their own pace.
  • They can be replayed:  students can review at their own pace.

When the students go to class, they work problems.  This works best when students take advantage of several key features of the classroom, namely:

  • The instructor is there to offer guidance and clarification,
  • Other students are there to discuss potential solutions and to cross-check work.

You’ll notice that the flipped classroom model is exactly the “Read these books and we’ll discuss them in class” model of education.  If you enjoy the teaching process, a flipped class is one of the best ways to experience it.

But be aware…there are pitfalls along the way.  More about this later…

Prime and Composite Numbers

If you’re a mathematician or a student of mathematics, you know what prime and composite numbers are.  But if you’re someone who teaches mathematics, you’ll find the standard definition of prime and composite numbers are problematic.  In the following, we’ll take apart the standard definitions, and come up with a better one.

Here’s a common definition:  A prime number is a number that can only be divided by 1 and itself; all other numbers are composite.

At this point, we have to inject some fine print:

  • Actually, any number can be divided by any (non-zero) number:  5 can be divided by 3, since 5 \div 3 = \frac{5}{3}, so we have to specify that we are talking about whole number divisors.
  • Oh, and 1 isn’t prime, even though its only whole number divisors are 1 and itself.
  • Incidentally, 1 isn’t composite either.

Thus, we get the following standard definition:  Any number larger than 1 is prime if it has no whole divisors besides 1 and itself; the number is composite otherwise.

Now there’s nothing really wrong with the standard definition…if you’re a mathematician.  But in terms of doing anything with a prime number, it’s a good example of how our choice of definitions can make mathematics easier or harder.

There are two problem with the standard definition of prime.  First, to most non-mathematicians, divisors implies division, so to find if a number is prime, we see if any number divides it.  Thus if I want to find whether 5 is prime, I’ll check 5 \div 2, 5 \div 3, 5 \div 4, and since none of them work, I’ll conclude 5 is prime.  But this means that in order to verify whether a number is prime or composite, we must divide…and division is the hardest of the elementary operations.   As a result, this definition discourages verification!

There’s a second problem:  it defines a prime number by a property it doesn’t have. Determining whether something has a property is relatively easy; determining whether something doesn’t have a property is harder, and tends to produce answers like “It just doesn’t have that property.”

What do I mean by that?  I’ve asked questions like “Prove 5 is prime” on assignments, and more often than not, I get answers like “5 has no divisors, so it’s prime.”  While this is true, it misses the point of a prove question:  A politician can claim something is true without giving any supporting evidence, but a sensible person is supposed to do better.   Mere claims are meaningless:  you’ve got to present the supporting evidence.

So here’s an alternative definition:  A number is composite if it can be expressed as a product of smaller numbers.  A number greater than 1 is prime if it cannot be expressed as a product of smaller numbers.

Note several features of this definition:

  • We don’t have to specify we’re dealing with whole number products.  While it’s true that 5 can be expressed as a product of other numbers, none of these pairs involve smaller numbers.
  • This also makes clear from the beginning that 1 is not composite, because it can’t be written as a product of smaller numbers.
  • We avoid division and focus on the easier operation of multiplication.

What about “Prove 5 is prime?”  It’s still possible to respond with a simple assertion:  “No product of smaller numbers gives 5.”  However, because the focus of the definition is on the product, and not the property, it’s more likely to cause a respondent to recognize they must provide evidence.

More importantly, students sometime have difficulties trying to prove “5 has no divisors.” I suspect it’s because they see “No divisors” and the immediate response is “But the divisors could be anything at all, so where can I start?”  In contrast, “No product of smaller numbers gives 5” automatically limits the scope of the problem:  products of smaller numbers.

I’ve started using the above definition for prime and composite numbers in all the classes I teach.  I hope you’ll start to use it as well.

Why Do Proofs

Proof is an essential part of mathematics, but it’s sometimes not clear why we prove things in mathematics.  By way of example, let’s consider a simple problem:  Multiplying two even numbers.

If you were a scientist, you would collect data. For example:  4 \times 6 = 242 \times 8 = 1610 \times 10 = 100, and so on.  After a few hundred pieces of evidence, you would form an hypothesis:  The product of two even numbers is even.  You might then perform some more experiments to test the hypothesis:  82 \times 14 = 114814 \times 32 = 44811 \times 4 = 44 (then sheepishly realize that 11 isn’t even, so we’ll ignore that), 24 \times 4 = 95 (better recheck that…yup, it’s supposed to be 96), and so on.

But if you’re a mathematician, you’d next try to prove that the product of two even numbers is an even number.  The first thing to realize is this:  No one ever tries to prove anything they don’t already believe to be true.  If you asked me to prove the product of two even numbers is an odd number, I’d say “But that’s not true…why should I waste time trying to prove it?”  What this means is that we don’t prove things because we doubt their truth:  proof isn’t a way of obtaining truth, because in some sense, we already have it.

So why do we prove things?  Let’s consider what we’d have to do.

  • First, what do we mean by “the product of two even numbers?”  We need to clarify what we mean by an even number:  “I know it when I see it” isn’t good enough.  We come to the conclusion:  An even number is any number that can be written as the product of 2 and some other whole number m.
  • So now we multiply two even numbers:  (2m) \times (2n) = 4mn.
  • But we need to write this as a product of 2 and some other number, so we have 4mn = 2 (2mn).

There’s our proof.

But stop and smell the roses.  Halfway through, we found the product of two even numbers was 4mn.  This means that the product of two even numbers is in fact a multiple of 4.  If we go back to our data, we’ll see that it’s there…but it might not have been obvious at the time.

So what has this effort of proof given us?

  • We had to dig down deep and elaborate on what we man by “even number.”  In general, proof requires us to define our terms in a meaningful and useful way.
  • We had to use a little bit of algebra.  In general, proof requires us to review mathematics we’ve already learned.  (In fact, since most terms in mathematics have a standard definition, the first step also requires a review of the mathematics we’ve learned…or the mathematics we should have learned)
  • We discovered something that we might have missed the first time around.