Category: mathematics

It’s a bit more complicated than this

Forget your times tables – mathematicians have found a new, faster way to multiply two numbers together. The method, which works only for whole numbers, is a landmark result in computer science. “This is big news,” says Joshua Cooper at the University of South Carolina.

To understand the new technique, which was devised by David Harvey at the University of New South Wales, Australia, and Joris van der Hoeven at the Ecole Polytechnique near Paris, France, it helps to think back to the longhand multiplication you learned at school.

We write down two numbers, one on top of the other, and then painstakingly multiply each digit of one by each digit of the other, before adding all the results together. “This is an ancient algorithm,” says Cooper.

If your two numbers each have n digits, this way of multiplying will require roughly n² individual calculations. “The question is, can you do better?” says Cooper.

Lots of logs

Starting in the 1960s, mathematicians began to prove that they could. First Anatoly Karatsuba found an algorithm that could turn out an answer in no more than n^1.58 steps, and in 1971, Arnold Schönhage and Volker Strassen found a way to peg the number of steps to the complicated expression n*(log(n))*log(log(n)) – here “log” is short for logarithm.

These advances had a major impact on computing. Whereas a computer using the longhand multiplication method would take about six months to multiply two billion-digit numbers together, says Harvey, the Schönhage-Strassen algorithm can do it in 26 seconds.

The landmark 1971 paper also suggested a possible improvement, a tantalising prediction that multiplication might one day be possible in no more than n*log(n) steps. Now Harvey and van der Hoeven appear to have proved this is the case. “It finally appears to be possible,” says Cooper. “It passes the smell test.”

“If the result is correct, it’s a major achievement in computational complexity theory,” says Fredrik Johansson at INRIA, the French research institute for digital sciences, in Bordeaux. “The new ideas in this work are likely to inspire further research and could lead to practical improvements down the road.”

Cooper also praises the originality of the research, although stresses the complexity of the mathematics involved. “You think, jeez, I’m just multiplying two integers, how complicated can it get?” says Cooper. “But boy, it gets complicated.”

So, will this make calculating your tax returns any easier? “For human beings working with pencil and paper, absolutely not,” says Harvey. Indeed, their version of the proof only works for numbers with more than 10 to the power of 200 trillion trillion trillion digits. “The word ‘astronomical’ falls comically short in trying to describe this number,” says Harvey.

While future improvements to the algorithm may extend the proof to more humdrum numbers only a few trillion digits long, Cooper thinks its real value lies elsewhere. From a theoretical perspective, he says, this work allows programmers to provide a definitive guarantee of how long a certain algorithm will take. “We are optimistic that our new paper will allow us to achieve further practical speed-ups,” says van der Hoeven.

Harvey thinks this may well be the end of the story, with no future algorithm capable of beating n*log(n). “I would be extremely surprised if this turned out to be wrong,” he says, “but stranger things have happened.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Gilead Amit*

The number 33 has surprising depth

Add three cubed numbers, and what do you get? It is a question that has puzzled mathematicians for centuries.

In 1825, a mathematician known as S. Ryley proved that any fraction could be represented as the sum of three cubes of fractions. In the 1950s, mathematician Louis Mordell asked whether the same could be done for integers, or whole numbers. In other words, are there integers k, x, y and z such that k = x³ + y³ + z³ for each possible value of k?

We still don’t know. “It’s long been clear that there are maths problems that are easy to state, but fiendishly hard to solve,” says Andrew Booker at the University of Bristol, UK – Fermat’s last theorem is a famous example.

Booker has now made another dent in the cube problem by finding a sum for the number 33, previously the lowest unsolved example. He used a computer algorithm to search for a solution:

33 = 8,866,128,975,287,528³ + (-8,778,405,442,862,239)³ + (-2,736,111,468,807,040)³

To cut down calculation time, the program eliminated certain combinations of numbers. “For instance, if x, y and z are all positive and large, then there’s no way that x³ + y³ + z³ is going to be a small number,” says Booker. Even so, it took 15 years of computer-processing time and three weeks of real time to come up with the result.

For some numbers, finding a solution to the equation k = x³ + y³ + z³ is simple, but others involve huge strings of digits. “It’s really easy to find solutions for 29, and we know a solution for 30, but that wasn’t found until 1999, and the numbers were in the millions,” says Booker.

Another example is for the number 3, which has two simple solutions: 1³ + 1³ + 1³  and 4³ + 4³ + (-5) ³ . “But to this day, we still don’t know whether there are more,” he says.

There are certain numbers that we know definitely can’t be the sum of three cubes, including 4, 5, 13, 14 and infinitely many more.

The solution to 74 was only found in 2016, which leaves 42 as the only number less than 100 without a possible solution. There are still 12 unsolved numbers less than 1000.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Donna Lu*

Today I’d like to share an idea. It’s a very simple idea. It’s not fancy and it’s certainly not new. In fact, I’m sure many of you have thought about it already. But if you haven’t—and even if you have!—I hope you’ll take a few minutes to enjoy it with me. Here’s the idea:

So simple! But we can get a lot of mileage out of it.

To start, I’ll be a little more precise: every matrix corresponds to a weighted bipartite graph. By “graph” I mean a collection of vertices (dots) and edges; by “bipartite” I mean that the dots come in two different types/colors; by “weighted” I mean each edge is labeled with a number.

The graph above corresponds to a 3×23×2 matrix MM. You’ll notice I’ve drawn three greengreen dots—one for each row of MM—and two pinkpink dots—one for each column of MM. I’ve also drawn an edge between a green dot and a pink dot if the corresponding entry in MM is non-zero.

For example, there’s an edge between the second green dot and the first pink dot because M21=4M21=4, the entry in the second row, first column of MM, is not zero. Moreover, I’ve labeled that edge by that non-zero number. On the other hand, there is no edge between the first green dot and the second pink dot because M12M12, the entry in the first row, second column of the matrix, is zero.

Allow me to describe the general set-up a little more explicitly.

Any matrix MM is an array of n×mn×m numbers. That’s old news, of course. But such an array can also be viewed as a function M:X×Y→RM:X×Y→R where X={x1,…,xn}X={x1,…,xn} is a set of nn elements and Y={y1,…,ym}Y={y1,…,ym} is a set of mm elements. Indeed, if I want to describe the matrix MM to you, then I need to tell you what each of its ijijth entries are. In other words, for each pair of indices (i,j)(i,j), I need to give you a real number MijMij. But that’s precisely what a function does! A function M:X×Y→RM:X×Y→R associates for every pair (xi,yj)(xi,yj) (if you like, just drop the letters and think of this as (i,j)(i,j)) a real number M(xi,yj)M(xi,yj). So simply write MijMij for M(xi,yj)M(xi,yj).

Et voila. A matrix is a function.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

I am often asked the question, “How do you do it?!” Now while I don’t think my note-taking strategy is particularly special, I am happy to share! I’ll preface the information by stating what you probably already know: I LOVE to write.* I am a very visual learner and often need to go through the physical act of writing things down in order for information to “stick.” So while some people think aloud (or quietly),

I think on paper.

My study habits, then, are built on this fact. Of course not everyone learns in this way, so this post is not intended to be a how-to guide. It’s just a here’s-what-I-do guide.

With that said, below is a step-by-step process I tried to follow during my final years of undergrad and first two years of grad school.**

‍Step 1

Read the appropriate chapter/section in the book before class

I am an “active reader,” so my books have tons of scribbles, underlines, questions, and “aha” moments written on the pages. I like to write while I read because it gives me time to pause and think about the material. For me, reading a mathematical text is not like reading a novel. It often takes me a long time just to understand a single paragraph! Or a single sentence. I also like to mark things that I don’t understand so I’ll know what to look for in the upcoming lecture.

STEP 2

Attend lecture and take notes

This step is pretty self-explanatory, but I will mention this: I write down much more than what is written on the chalkboard (or whiteboard). In fact, a good portion of my in-class notes consists of what the professor has said but hasn’t written.

‍My arsenal

‍STEP 3

Rewrite lecture notes at home

My in-class notes are often an incomprehensible mess of frantically-scribbled hieroglyphs. So when I go home, I like to rewrite everything in a more organized fashion. This gives the information time to simmer and marinate in my brain. I’m able to ponder each statement at my own pace, fill in any gaps, and/or work through any exercises the professor might have suggested. I’ll also refer back to the textbook as needed.

Sometimes while rewriting these notes, I’ll copy things word-for-word (either from the lecture, the textbook, or both), especially if the material is very new or very dense. Although this can be redundant, it helps me slow down and lets me think about what the ideas really mean. Other times I’ll just rewrite things in my own words in a way that makes sense to me.

A semester’s worth of notes!

As for the content itself, my notes usually follow a “definition then theorem then proof” outline, simply because that’s how material is often presented in the lecture. But sometimes it’s hard to see the forest for the trees (i.e. it’s easy to get lost in the details), so I’ll occasionally write “PAUSE!” or “KEY IDEA!” in the middle of the page. I’ll then take the time to write a mini exposition that summarizes the main idea of the previous pages. I’ve found this to be especially helpful when looking back at my notes after several months (or years) have gone by. I may not have time to read all the details/calculations, so it’s nice to glance at a summary for a quick refresher.

And every now and then, I’ll rewrite my rewritten notes in the form of a SaiBlog post! Many of my earlier posts here at Math3ma were “aha” moments that are now engrained in my brain because I took the time to SaiBlog about them.

STEP 4

Do homework problems

Once upon a time, I used to think the following:

How can I do problems if I haven’t spent a bajillion hours learning the theory first?

But now I believe there’s something to be said for the converse: 

How can I understand the theory if I haven’t done a bajillion examples first?

In other words, taking good notes and understanding theory is one thing, but putting that theory into practice is a completely different beast. As a wise person once said, “The only way to learn math is to DO math.” So although I’ve listed “do homework problems” as the last step, I think it’s really first in terms of priority.

Typically, then, I’ll make a short to-do list (which includes homework assignments along with other study-related duties) each morning. And I’ll give myself a time limit for each task. For example, something like “geometry HW, 3 hours” might appear on my list, whereas “do geometry today” will not. Setting a time gives me a goal to reach for which helps me stay focused. And I may be tricking my brain here, but a specific, three-hour assignment sounds much less daunting than an unspecified, all-day task. (Of course, my lists always contain multiple items that take several hours each, but as the old adage goes, “How do you eat an elephant? One bite at a time.”)

By the way, in my first two years of grad school I often worked with my classmates on homework problems. I didn’t do this in college, but in grad school I’ve found it tricky to digest all the material alone – there’s just too much of it! So typically I’d first attempt exercises on my own, then meet up with a classmate or two to discuss our ideas and solutions and perhaps attend office hours with any questions.

As far as storage goes, I have a huge binder that contains all of my rewritten notes*** from my first and second year classes. (I use sheet protectors to keep them organized according to subject.) On the other hard, I use a paper tray like this one to store my lecture notes while the semester is in progress. Once classes are over, I’ll scan and save them to an external hard drive. I’ve also scanned and saved all my homework assignments.

Well, I think that’s about it! As I mentioned earlier, these steps were only my ideal plan. I often couldn’t apply them to every class — there’s just not enough time! — so I’d only do it for my more difficult courses. And even then, there might not be enough time for steps 1 and 3, and I’d have to start working on homework right after a lecture.

But as my advisor recently told me,”It’s okay to not know everything.” Indeed, I think the main thing is to just do something. Anything. As much as you can. And as time goes on, you realize you really are learning something, even if it doesn’t feel like it at the time.

Alright, friends, I think that’s all I have to share. I hope it was somewhat informative. If you have any questions, don’t hesitate to leave it in a comment below!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Mathematicians are stunned by the discovery that prime numbers are pickier than previously thought. The find suggests number theorists need to be a little more careful when exploring the vast infinity of primes.

Primes, the numbers divisible only by themselves and 1, are the building blocks from which the rest of the number line is constructed, as all other numbers are created by multiplying primes together. That makes deciphering their mysteries key to understanding the fundamentals of arithmetic.

Although whether a number is prime or not is pre-determined, mathematicians don’t have a way to predict which numbers are prime, and so tend to treat them as if they occur randomly. Now Kannan Soundararajan and Robert Lemke Oliver of Stanford University in California have discovered that isn’t quite right.

“It was very weird,” says Soundararajan. “It’s like some painting you are very familiar with, and then suddenly you realise there is a figure in the painting you’ve never seen before.”

Surprising order

So just what has got mathematicians spooked? Apart from 2 and 5, all prime numbers end in 1, 3, 7 or 9 – they have to, else they would be divisible by 2 or 5 – and each of the four endings is equally likely. But while searching through the primes, the pair noticed that primes ending in 1 were less likely to be followed by another prime ending in 1. That shouldn’t happen if the primes were truly random – consecutive primes shouldn’t care about their neighbour’s digits.

“In ignorance, we thought things would be roughly equal,” says Andrew Granville of the University of Montreal, Canada. “One certainly believed that in a question like this we had a very strong understanding of what was going on.”

The pair found that in the first hundred million primes, a prime ending in 1 is followed by another ending in 1 just 18.5 per cent of the time. If the primes were distributed randomly, you’d expect to see two 1s next to each other 25 per cent of the time. Primes ending in 3 and 7 take up the slack, each following a 1 in 30 per cent of primes, while a 9 follows a 1 in around 22 per cent of occurrences.

Similar patterns showed up for the other combinations of endings, all deviating from the expected random values. The pair also found them in other bases, where numbers are counted in units other than 10s. That means the patterns aren’t a result of our base-10 numbering system, but something inherent to the primes themselves. The patterns become more in line with randomness as you count higher – the pair have checked up to a few trillion – but still persists.

“I was very surprised,” says James Maynard of the University of Oxford, UK, who on hearing of the work immediately performed his own calculations to check the pattern was there. “I somehow needed to see it for myself to really believe it.”

Stretching to infinity

Thankfully, Soundararajan and Lemke Oliver think they have an explanation. Much of the modern research into primes is underpinned G H Hardy and John Littlewood, two mathematicians who worked together at the University of Cambridge in the early 20^th century. They came up with a way to estimate how often pairs, triples and larger grouping of primes will appear, known as the k-tuple conjecture.

Just as Einstein’s theory of relativity is an advance on Newton’s theory of gravity, the Hardy-Littlewood conjecture is essentially a more complicated version of the assumption that primes are random – and this latest find demonstrates how the two assumptions differ. “Mathematicians go around assuming primes are random, and 99 per cent of the time this is correct, but you need to remember the 1 per cent of the time it isn’t,” says Maynard.

The pair used Hardy and Littlewood’s work to show that the groupings given by the conjecture are responsible for introducing this last-digit pattern, as they place restrictions on where the last digit of each prime can fall. What’s more, as the primes stretch to infinity, they do eventually shake off the pattern and give the random distribution mathematicians are used to expecting.

“Our initial thought was if there was an explanation to be found, we have to find it using the k-tuple conjecture,” says Soundararajan. “We felt that we would be able to understand it, but it was a real puzzle to figure out.”

The k-tuple conjecture is yet to be proven, but mathematicians strongly suspect it is correct because it is so useful in predicting the behaviour of the primes. “It is the most accurate conjecture we have, it passes every single test with flying colours,” says Maynard. “If anything I view this result as even more confirmation of the k-tuple conjecture.”

Although the new result won’t have any immediate applications to long-standing problems about primes like the twin-prime conjecture or the Riemann hypothesis, it has given the field a bit of a shake-up. “It gives us more of an understanding, every little bit helps,” says Granville. “If what you take for granted is wrong, that makes you rethink some other things you know.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

Yesterday I received a disheartening 44/50 on a homework assignment. Okay okay, I know. 88% isn’t bad, but I had turned in my solutions with so much confidence that admittedly, my heart dropped a little (okay, a lot!) when I received the grade. But I quickly had to remind myself, Hey! Grades don’t matter.

The six points were deducted from two problems. (Okay, fine. It was three. But in the third I simply made an air-brained mistake.) In the first, apparently my answer wasn’t explicit enough. How stingy! I thought. Doesn’t our professor know that this is a standard example from the book? I could solve it in my sleep! But after the prof went over his solution in class, I realized that in all my smugness I never actually understood the nuances of the problem. Oops. You bet I’ll be reviewing his solution again. Lesson learned.

In the second, I had written down my solution in the days before and had checked with a classmate and (yes) the internet to see if I was correct. Unfortunately, the odds were against me two-to-one as both sources agreed with each other but not with me. But I just couldn’t see how I could possibly be wrong! Confident that my errors were truths, I submitted my solution anyway, hoping there would be no consequences. But alas, points were taken off.

Honestly though, is a lower grade such a bad thing? I think not. In both cases, I learned exactly where my understanding of the material went awry. And that’s great! It means that my comprehension of the math is clearer now than it was before (and that the chances of passing my third qualifying exam have just increased. Woo!) And that’s precisely why I’m (still, heh…) in school.

So yes, contrary to what the comic above says, grades do exist in grad school, but – and this is what I think the comic is hinting at – they don’t matter. Your thesis committee members aren’t going to say, “Look, your defense was great, but we can’t grant you your PhD. Remember that one homework/midterm/final grade from three years ago?” (They may not use the word “great” either, but that’s another matter.) Of course, we students should still work hard and put in maximum effort! But the emphasis should not be on how well we perform, but rather how much we learn. Focus on the latter and the former will take care of itself. This is true in both graduate school and college, but the lack of emphasis on grades in grad school really brings it home. And personally, I’m very grateful for it because my brain is freed up to focus on other things like, I don’t know, learning math!

So to all my future imperfect homework scores out there: bring it on.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

A maths problem previously tackled with the help of a computer, which produced a proof the size of Wikipedia, has now been cut down to size by a human. Although it is unlikely to have practical applications, the result highlights the differences between two modern approaches to mathematics: crowdsourcing and computers.

Terence Tao of the University of California, Los Angeles, has published a proof of the Erdős discrepancy problem, a puzzle about the properties of an infinite, random sequence of +1s and -1s. In the 1930s, Hungarian mathematician Paul Erdős wondered whether such a sequence would always contain patterns and structure within the randomness.

One way to measure this is by calculating a value known as the discrepancy. This involves adding up all the +1s and -1s within every possible sub-sequence. You might think the pluses and minuses would cancel out to make zero, but Erdős said that as your sub-sequences got longer, this sum would have to go up, revealing an unavoidable structure. In fact, he said the discrepancy would be infinite, meaning you would have to add forever, so mathematicians started by looking at smaller cases in the hopes of finding clues to attack the problem in a different way.

Last year, Alexei Lisitsa and Boris Konev of the University of Liverpool, UK used a computer to prove that the discrepancy will always be larger than two. The resulting proof was a 13 gigabyte file – around the size of the entire text of Wikipedia – that no human could ever hope to check.

Helping hands

Tao has used more traditional mathematics to prove that Erdős was right, and the discrepancy is infinite no matter the sequence you choose. He did it by combining recent results in number theory with some earlier, crowdsourced work.

In 2010, a group of mathematicians, including Tao, decided to work on the problem as the fifth Polymath project, an initiative that allows professionals and amateurs alike to contribute ideas through SaiBlogs and wikis as part of mathematical super-brain. They made some progress, but ultimately had to give up.

“We had figured out an interesting reduction of the Erdős discrepancy problem to a seemingly simpler problem involving a special type of sequence called a completely multiplicative function,” says Tao.

Then, in January this year, a new development in the study of these functions made Tao look again at the Erdős discrepancy problem, after a commenter on his SaiBlog pointed out a possible link to the Polymath project and another problem called the Elliot conjecture.

Not just conjecture

“At first I thought the similarity was only superficial, but after thinking about it more carefully, and revisiting some of the previous partial results from Polymath5, I realised there was a link: if one could prove the Elliott conjecture completely, then one could also resolve the Erdős discrepancy problem,” says Tao.

“I have always felt that that project, despite not solving the problem, was a distinct success,” writes University of Cambridge mathematician Tim Gowers, who started the Polymath project and hopes that others will be encouraged to participate in future. “We now know that Polymath5 has accelerated the solution of a famous open problem.”

Lisitsa praises Tao for doing what his algorithm couldn’t. “It is a typical example of high-class human mathematics,” he says. But mathematicians are increasingly turning to machines for help, a trend that seems likely to continue. “Computers are not needed for this problem to be solved, but I believe they may be useful in other problems,” Lisitsa says.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

Felker prefaces the quote by saying,

Giving up on math means you don’t believe that careful study can change the way you think.

He further notes that writing, like math, “is also not something that anyone is ‘good’ at without a lot of practice, but it would be completely unacceptable to think that your composition skills could not improve.”

Friends, this is so true! Being ‘good’ at math boils down to hard work and perseverance, not whether or not you have the ‘math gene.’ “But,” you might protest, “I’m so much slower than my classmates are!” or “My educational background isn’t as solid as other students’!” or “I got a late start in mathematics!”* That’s okay! A strong work ethic and a love and enthusiasm for learning math can shore up all deficiencies you might think you have. Now don’t get me wrong. I’m not claiming it’ll be a walk in the park. To be honest, some days it feels like a walk through an unfamiliar alley at nighttime during a thunderstorm with no umbrella. But, you see, that’s okay too. It may take some time and the road may be occasionally bumpy, but it can be done!

This brings me to another point that Felker makes: If you enjoy math but find it to be a struggle, do not be discouraged! The field of math is HUGE and its subfields come in many different flavors. So for instance, if you want to be a math major but find your calculus classes to be a challenge, do not give up! This is not an indication that you’ll do poorly in more advanced math courses. In fact, upper level math classes have a completely (I repeat, completely!) different flavor than calculus. Likewise, in graduate school you may struggle with one course, say algebraic topology, but find another, such as logic, to be a breeze. Case in point: I loathed real analysis as an undergraduate** and always thought it was pretty masochistic. But real analysis in graduate school was nothing like undergraduate real analysis (which was more like advanced calculus), and now – dare I say it? – I sort of enjoy the subject. (Gasp!)

All this to say that although Felker’s article is aimed at folks who may be afraid to take college-level math, I think it applies to math majors and graduate students too. I highly recommend you read it if you ever need a good ‘pick-me-up.’ And on those days when you feel like the math struggle is harder than usual, just remember:

Even the most accomplished mathematicians had to learn HOW to learn this stuff!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Example 5

A sequence of functions {fn:R→R}{fn:R→R} which converges to 0 pointwise but does not converge to 0 in L1L1.

This works because: The sequence tends to 0 pointwise since for a fixed x∈Rx∈R, you can always find N∈NN∈N so that fn(x)=0fn(x)=0 for all nn bigger than NN. (Just choose N>xN>x!)

The details: Let x∈Rx∈R and fix ϵ>0ϵ>0 and choose N∈NN∈N so that N>xN>x. Then whenever n>Nn>N, we have |fn(x)−0|=0<ϵ|fn(x)−0|=0<ϵ.

Of course, fn↛0fn↛0 in L1L1 since∫R|fn|=∫(n,n+1)fn=1⋅λ((n,n+1))=1.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Suffixes are important!

Did you know that the words

“maximal” and “maximum” generally do NOT mean the same thing

in mathematics? It wasn’t until I had to think about Zorn’s Lemma in the context of maximal ideals that I actually thought about this, but more on that in a moment. Let’s start by comparing the definitions:

Do you see the difference? An element is a maximum if it is larger than every single element in the set, whereas an element is maximal if it is not smaller than any other element in the set (where “smaller” is determined by the partial order ≤≤). Yes, it’s true that the* maximum also satisfies this property, i.e. every maximum element is also maximal. But the converse is not true: if an element is maximal, it may not be the maximum! Why? The key is that these definitions are made on a partially ordered set. Basically, partially ordered just means it makes sense to use the words “bigger” or “smaller” – we have a way to compare elements. In a totally ordered set ALL elements are comparable with each other. But in a partially ordered set SOME, but not necessarily all, elements can be compared. This means it’s possible to have an element that is maximal yet fails to be the maximum because it cannot be compared with some elements. It’s not too hard to see that when a set is totally ordered, “maximal = maximum.”**

How about an example? Here’s one I like from this scholarly site which also gives an example of a miminal/minimum element (whose definitions are dual to those above).

Example

Consider the set

where the partial order is set inclusion, ⊆⊆. Then

• {d,o}{d,o} is minimalbecause {d,o}⊉x{d,o}⊉x for every x∈Xx∈
• e. there isn’t a single element in XX that is “smaller” than {d,o}{d,o}
• {g,o,a,d}{g,o,a,d} is maximalbecause {g,o,a,d}⊈x{g,o,a,d}⊈x for every x∈Xx∈X
• e. there isn’t a single element in XX that is “larger” than {g,o,a,d}{g,o,a,d}
• {o,a,f}{o,a,f} is both minimal and maximal because
• {o,a,f}⊉x{o,a,f}⊉x for every x∈Xx∈X
• {o,a,f}⊈x{o,a,f}⊈x for every x∈Xx∈X
• {d,o,g}{d,o,g} is neither minimal nor maximal because
• there is an x∈Xx∈X such that x⊆{d,o,g}x⊆{d,o,g}, namely x={d,o}x={d,o}
• there is an x∈Xx∈X such that {d,o,g}⊆x{d,o,g}⊆x, namely x={g,o,a,d}x={g,o,a,d}
• XX has NEITHER a maximum or a minimum because
• there is no M∈XM∈X such that x⊆Mx⊆M for everyx∈Xx∈X
• there is no m∈Xm∈X such that m⊆xm⊆x for everyx∈Xx∈X

Let’s now relate our discussion above to ring theory. One defines an ideal MM in a ring RR to be a maximal ideal if M≠RM≠R and the only ideal that contains MM is either MM or RR itself, i.e. if I⊴RI⊴R is an  ideal such that M⊆I⊆RM⊆I⊆R, then we must have either I=MI=M or I=RI=R.

Not surprisingly, this coincides with the definition of maximality above. We simply let XX be the set of all proper ideals in the ring RR endowed with the partial order of inclusion ⊆⊆. The only difference is that in this context, because we’re in a ring, we have the second option I=RI=R.

I think a good way to see maximal ideals in action is in the proof of this result:

As a final remark, the notions of “a maximal element” and “an upper bound” come together in Zorn’s Lemma which is needed to prove that every proper ideal in a ring is contained in a maximal ideal. I should mention that an upper bound BB on a partially ordered set (a.k.a. a “poset”) has the same definition as the maximum EXCEPT that BB is not required to be inside the set. More precisely, we define an upper bound on a subset YY of XX to be an element B∈XB∈X such that y≤By≤B for every y∈Yy∈Y.

So here’s the deal with Zorn’s Lemma: It’s not too hard to prove that every finite poset has a maximal element. But what if we don’t know if the given poset is finite? Or what happens if it’s infinite? How can we tell if it has a maximal element? Zorn’s Lemma answers that question:

‍As I mentioned above, it’s this result which is needed to prove that every proper ideal is contained in a maximal ideal***. It actually implies a weaker statement, called Krull’s Theorem (1929), which says that every non-zero ring with unity contains a maximal ideal.

‍Footnotes

*One can easily show that if a set has a maximum it must be unique, hence THE maximum.

** Here’s the proof: Let (X,≤)(X,≤) be a totally ordered set and let m∈Xm∈X be a maximal element. It suffices to show mm is the maximum. Since XX has a total order, either m≤xm≤x or x≤mx≤m for every x∈Xx∈X. If the latter, then mm is the maximum. If the former, then m=xm=x by definition of maximal. In either case, we have x≤mx≤m for all x∈Xx∈X. Hence mm is the maximum.

*** Note this is NOT the same as saying that every maximal ideal contains all the proper ideals in a ring! Remember, maximal ≠≠ maximum!!

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*Credit for article given to Tai-Danae Bradley*