From the time they can crawl, babies are ready to manipulate objects. Among the toys babies commonly play with should be included various measuring devices. They should see pictures of people measuring things and comparing measurements. They should fumble around with puzzle pieces. They should hear the story of Hippasus being assassinated by his fellow Pythagoreans for his immeasurable worth. That way, they will fully understand the human condition.
Once they can stand without Mommy or Daddy's help, children are ready to tackle Zeno's Paradox. To be properly prepared to get away from the messes they make, they must first learn how to get at least halfway between trouble and safety. With each successive half, the consequences of their trouble diminish, but never vanish. This realization is important for moral development. Most importantly, without knowing why motion is possible, toddlers will never be able to steal cookies from the jar, a tragedy for posterity. Until they resolve it, they will feel forever stuck and their developing sense of agency will be stunted.
When children are ready to spend half their day among strangers without crying from fear of abandonment, due to Zeno's argument against their parents ever being able to make the distance to pick them up from school, they are ready to square circles made of Plato. They may also begin the process of linearizing the sphere by constructing all five Platonic solids out of papier-mâché. As their parents will undoubtedly attest, by the end of kindergarten the children have such well-developed methods of exhaustion even Eudoxus would have wept.
By the end of fifth grade, children should know the difference between captivity and opportunity. That is to say, most of them will have embraced one or the other. Developing their ability to distinguish between the rational and the irrational will help children to decide whether they want to be students or reluctantly occupy an assigned seat. Along the way, they should also gain comfort with the fact that certain letters are literal numbers, like π, φ, and e, or must be treated like numbers, such as, well, any letter from at least two different alphabets. To really drive the point home, it should be discussed that some letters may stand for specified, yet unspecified numbers, like a, b, and c in ax + b = c, while x stands for an unknown number that will be known once the specified, yet unspecified numbers are specified. Additionally, to prepare them for middle school analysis, students should become well-versed in the sometimes difference, and sometimes equivalence, between natural numbers and whole numbers. No matter who you ask, it's much ado about nothing. As well, familiarity with the relations between the integers, the rationals, and the irrationals is necessary for the study of the real numbers, which makes it sound like the all those other numbers were fake and we were just wasting everyone's time until they grew up. You see, you needed the fake numbers to help you understand the real numbers which, because they lack imagination, aren't that complex after all. Alas, there is not much analysis going on in elementary school.
By the end of eighth grade, children have abandoned analysis. You see, puberty is just about upon them. They are drunk from a cocktail so potent (and stinky) it will put hair in places they don't want hair. Their constant inebriation is just preparation for doing things which will make their toes curl. Of course, this won't happen until they realize the anti-climactic effects of doing things which will make their toenails curl. Wait. What are we talking about? Isn't this a math blog. Maybe we should see an analyst. Then we will learn that everything is about sex. Therefore, math is sexy. Try convincing a middle school child of that argument!
After surviving freshman year, teenagers will begin to analyze their lot in life. They will sort themselves into different groups as they struggle to sort themselves out. They will begin to question authority. Even the authority of math... and math teachers. The one subject in life whose established authority is infallible, and they dare question its value, its validity. Scandalous! If you are crazy enough to try to sell adolescents the idea that math is everywhere, is beautiful, is the ticket to salaries above the national average... well. Where's the evidence? Their cups are filled to the brim with everything but math. Then we go and pour some math in it? That's less rational than the square root of 2! On the other empty hand, math teachers' salaries aren't necessarily above the national average, and many of them see analysts to cope with their lots. What's attractive about that? Let's analyze high school math, grade by grade.
Yes, we're off on a rant... good feelings gone.
Just about all freshman in American public high schools must take algebra. It doesn't matter that few of them are prepared for the rigor-mortis... I mean, the "rigor" of high school algebra. The standard American high school algebra textbook is neither algebraic nor textbook. Discuss. We tell ninth graders that everybody does it. When was the last time you talked with someone about someone else who gets paid to solve quadratic equations all day? Or calculate diagonals of rectangles seventeen hours, seven days a week? Or determine when a property of the real numbers is or isn't being used? That's what we thought. For the sake of argument, let's stipulate that there are such jobs. It still doesn't come close to what mathematicians actually do. There is no beauty in handing out mathematical shortcuts like candy. FOIL? Don't make us laugh. The quadratic formula song?? Don't make us cry. Long division??? I'm getting verklempt! And what is the point of expecting them to memorize the nine properties of the real numbers? Are algebra students expected to justify that assertion by constructing the real numbers? Are they expected to study those properties in the context of groups, rings, and other fields? Are they expected to prove that the probability of a square root being rational is zero? Why not? Otherwise, there is absolutely no mathematical point in requiring that such facts be recited. It seems there is not much mathematics being done in algebra classes. And not much algebra; it's really just pre-pre-pre-pre-calculus. It's more akin to what goes on in Hogwarts Academy. Just memorize a bunch of spells and use them at the right times. Who can convert 4/7 into decimal without a calculator? Billy? You solved it?? When did you... Billy, return my magic wand this instant, young man!
Riddle me this: Reading is a pre-requisite for college admission. Reading a tape measure is not. Nuff said? No. Writing a decent essay is a pre-requisite for graduating college with a bachelor's degree. Reading a tape measure is not. But wait, it gets worse. Getting original research past a panel of experts in your field is a pre-requisite for obtaining a PhD. Reading a tape measure is not. Now, I'm not saying a PhD should at least be able to read a tape measure. I'm not even saying a college graduate should at least be able to read a tape measure. I'm certainly not saying no college should accept you unless you can at least read a tape measure. What I AM saying is, YOU SHOULDN'T BE ALLOWED TO GRADUATE HIGH SCHOOL UNTIL YOU CAN READ A FUCKING TAPE MEASURE.
Most colleges require math majors to take no less than six courses of analysis: (1st) single variable differential calculus, (2nd) single variable integral calculus, (3rd) multivariable and vector calculus, (4th) ordinary differential equations, (5th) probability, (6th) real analysis. It wasn't until I had left teaching in public high schools that I realized something really stinks in real suburbia. The concept of the real number is troubling. I mean, every high school algebra student is told that the square root of 2 is irrational, but the rest of the story might as well be a state secret. Yet, we are never asked to deal with the irrationality of irrational numbers. We are only expected to use rational approximations. Thus, we studied the properties of real numbers without ever having to extract them from numbers that were actually real.
Mastery is what those we consider masters still seek.
I don't know, but I'll find out and get back to you. In the mean time, let's analyze the situation. Listen up, kids.
In 1492, Columbus sailed the ocean blue. But he couldn't have unless, as it happened long before that, a religious sect allegedly murdered one of their own to conceal the discovery of an immeasurable length. Remember what the Joker asked Batman in The Dark Knight? What happens when an immovable object meets an unstoppable force? We'll never know because that crazy cult didn't want us to know. Actually, we know. And the account of the cover up is apocryphal.
All this business about analysis began with Pythagoras. He led a monastery in which he taught, among other things, number theory and geometry. We owe musical scales and the invention of many musical instruments to him. You see, Pythagoras enforced a peculiar edict about numbers. He held that all numbers were commensurable. Then comes Hippasus. This guy proved that the diagonal of a unit square was not commensurable. And their whole world fell apart. So, they killed him... as the story goes.
The Babylonians were aware of the difficulty of trying to get exact answers for certain calculations. In the case of square roots, it turned out that most of them cannot be known. Compare this to something like 3 / 7. The Babylonians knew that calculating the sexagesimal value of 3 / 7 ceased being fun at some point. We know now that we can know 3 / 7 in a way that we can't know the square root of 2; 3 / 7 has an infinite decimal expansion with period 6 while √2 requires an infinite decimal expansion with no period. In other words, we at least know what 3 / 7 will be doing at the end of time, but we have no idea what √2 will be doing. Except, we do. Thanks to a recent idea called continued fractions, we have discovered a pattern in the behavior of all square roots. For more information, look no further.
Since the sides of a square are uniquely determined by its area, we may well-define the concept of a square root.
Informal Definition: The square root of n is the side length of a square with area n.
What does this conclusion mean? If √2 is not rational, then what is it? Where does it belong on the number line? Let's start with a guess. We know that 1 < √2 < 2. 1½ is a reasonable place to start. How close is it? (1½)^2 = 2¼. Solving (3/2)x = 2 gives us 4/3. (4/3)^2 is 1 7/9. Thus, we have improved our bounds: 4/3 < √2 < 3/2. Taking the mean of 3/2 and 4/3 gives us an improved estimate for √2: 17/12. How close is it? (17/12)^2 = 2 1/144, which is extremely close, but we are trying to understand what is going on with √2. The error is found by solving (17/12)x = 2 which gives us x = 24/17. (24/17)^2 = 1 287/289. We are closing in! 1 < 4/3 < 24/17 < √2 < 17/12 < 3/2 < 2. Is there an end to this? If not, it would suggest that the magnitude √2 confounds all of our measurement algorithms.
If you kept improving the bounds, then you would eventually tire of the tedium for it seems to never end. So, now we have expanded the universe of numbers by one object: ℚ ∪ {√2}. Can we do the usual arithmetic in this expanded universe of numbers? If we can't encapsulate √2 in any meaningful way, then we will need a new arithmetic. Without a finite rational expression equivalent to √2, the best we can do is approximate or write things like a ⊛ √2 or √2 ⊛ b, where a and b are rational and ⊛ represents +, −, ×, or ÷. This is not practical. However, further investigation leads to some extremely important mathematical discoveries, like The Calculus.
The diagram above shows that the square built from the diagonal has twice the content of the square upon whose diagonal it was built. Thus, it shows us how to construct a square containing exactly two square units, and so the side of that square will be the square root of 2. This observation is the foundation of the Pythagorean Theorem.
One way to understand √2 is as a Dedekind cut. We redefine √2 as the set of all rational numbers q such that q2 is less than 2. We may do this for other troublesome roots like √3 ∛50. In fact, we can reformulate the number line as the union of the rationals and the totality of cuts. By doing this, we get a new number system, but it has a strange arithmetic.