Readers, after a post on wealth inequality, I decided to write about something that truly captivates me- mathematical inequalities! All kidding aside, I have always had a passion for mathematics. Knowing this, various friends let me know that if I wrote a blog post about math that I’d never go back. They also told me that I would be completely incoherent. I’ll try to avoid both but there are no guarantees. To those of you who aren’t big fans of math, don’t fret, this post will not be littered with foreign mathematical concepts (but there is a lot of math) and [hopefully] will be interesting!
I guess I’ll start with a bit of background; math was my passion ever since I could read numbers. There were times when it definitely got the best of me: for example, I tried to play around with Newton polynomials in a notebook while biking up hills when I was little (did I mention that I’m terribly clumsy?). For the most part though, it was wonderful because it could accompany me anywhere and I saw it everywhere. My dad had taught me the operations (addition, subtraction, multiplication, division) by grouping coffee beans and now I saw it in food and in particular, breakfast. I actually remember switching from eating waffles to bagels so I could make this (http://georgehart.com/bagel/bagel.html) on the weekends. Really though, I did enjoy my childhood! A friend had once said that I had learned knot theory before learning to properly tie my shoes, which is more a testament to how late I actually learned how to correctly tie my shoes. All anecdotes aside, I’ve always loved the subject because of how most interesting problems have a crux move that changes your perspective completely. Moreover, the elegance of particular solutions based on simple concepts is awe-inspiring and powerful. That being said, I was deterred by math for much of high school because I realized that for math contests, which I thoroughly enjoyed, many people simply memorized formulas from a math book for glory. Still, I maintained an appreciation for math and one of my favorite topics was inequalities, which I thought was particularly elegant due to both its simplicity and power. I figured I would show a glimpse of its elegance (I apologize in advance for the lack of depth to math whizzes and if I use concepts that seem a bit bizarre to those of you that aren’t math lovers). Here’s a word document version of the post in the case that the formatting and haziness of math notation on wordpress makes you cringe: On Personality
The result here seems simple but not that relevant at first glance but it is a delightful first step into inequalities. It is one case of a famous inequality known as the Arithmetic Geometric Mean Inequality (or AM-GM ). I’ve omitted the proof for this post because of its many cases, but I can provide pieces of it in the comments section if you’d like.
This inequality (AM-GM) states that as long as
This is definitely a cool result, but I’m sure you’re wondering how it could be applied. Here’s an example of another inequality that can be proven using our newfound inequality.
for positive real numbers a, b and c (Nessbit’s Inequality)
How do we begin?
Well, we know from AM-GM inequality that for any positive real (x,y,z)
(cross multiplication and AM-GM)
Let’s write the inequality and similar components as variables:
Initially, maybe this setup seems really arbitrary and just assigning variables rather than being interesting. But in this setup there lies the crux move.
What we can gather from these variable declarations is that:
By adding the last two equations, we get
Which is what we set to prove in the first place!
Ok, so maybe that wasn’t super applicable, but here’s an elegant proof of Cauchy-Schwartz inequality (http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality) using AM-GM inequality, which was the basis in some sense for the Heisenberg Uncertainty Principle and an inequality you’ve probably seen in a lot of Linear Algebra textbooks (oh, the horror!). If you’re not interested in a proof sort of question with a bit of summation notation, I’d recommend looking at the next example!
We know that based on the definition of AM-GM:
and we also know that
Let’s say that:
Would you not agree that the result is pretty amazing?
I remember that when I learned AM-GM, I didn’t realize its power until I saw it used to solve International Math Olympiad problem (which all of you can solve now too!).
Here it is (1964 IMO, Problem #2):
It certainly looks complicated at the start, but yet again, just assigning variables and the use of AM-GM makes this mathematical hodgepodge of letters and numbers more elegant.
Then the expression becomes:
After the simplification of the messy expansion, we get:
What does this simplified expression remind you of? AM-GM inequality of course! I really appreciate inequalities because you have to creatively use simple expressions to solve much more complex ones; memorization is virtually useless (credit to Art of Problem Solving for posting the solution and not making me write it out on the computer).
I guess I’d like to end on a somber note. The American education system, particularly in elementary school, middle school and high school, seems to be bent towards the regurgitation of formulas, plugging and chugging and repetition to teach mathematics. I definitely don’t have a particular way to resolve this issue, but I’d like to hope that our system will eventually embrace creativity. The question of whether it is possible to teach creativity is certainly a fascinating debate topic that I would love to look more into in the future. I definitely have no background on it, but if you have any insight on it, feel free to let me know. Additionally, if there are any interesting inequality ideas you’d like to point out or something you would like me to explain further than I did in this post, just leave a comment! I apologize to those of you in advance who found the post too long and didn’t see any elegance in my examples; I’ll write about something less absurd soon!
In case you’re interested, I have posted another problem that is slightly harder than the ones preceding it; not in the number of steps but rather in how daunting it looks. It was on the shortlist for the International Mathematics Olympiad and looks like a problem a lot of people would not even choose to attempt (feel free to skip it if you’re not feeling mathy).
IMO 2010 SL, A3
The problem looks pretty difficult, right?
Then by rearranging the problem statement, we get that
By AM-GM Inequality, we see that
(Ah, the Haziness!)
Whee! We have successfully solved a very complicated looking summation problem using basic inequalities!