Some Basic Facts About Irrational Numbers

Thu Apr 30, 2009 8:31 am, by The Captain

Every high school student learns how to use reductio ad absurdum to prove the irrationality of $\sqrt {2}$ , but that is not the case for $\pi$ or $e$ . We often take the irrationality of these numbers for granted at school. Is there an easy way to prove this argument? Perhaps I. Niven has the credit for the most well-known proof of this fact by a clever application of elementary calculus. Following Niven, Parks has shown the following in " $\pi$ , and $e$ , and other irrational numbers, AMM, 93 (1986)722--723":

Theorem.
Let $c$ be a positive real number and let $f(x)$ be a continuous function on $[0, c]$ , positive on $(0, c)$ . Suppose there are (antiderivatives) $f_{1}(x), f_{2}(x), \dots$ with $f_{1}'(x) = f(x)$ and with $f_{k}'(x) = f_{k - 1}(x)$ for all $k\ge 2$ , and such that $f_{k}(0), f_{k}(c)$ are integers for all $k\ge 1$ . Then $c$ is irrational..

Sketch of Proof

It follows immediately from the above theorem that
If $0 < |r|\le \pi$ and $\cos(r), \sin(r)$ are rational numbers, then $r$ is irrational.
If $r$ is a positive rational number and $r\ne 1$ , then $\ln(r)$ is irrational.
(In fact the above numbers are transcendental according to Lindemann's theorem.)

Hence, $\pi, e$ are irrational numbers.

An alternative proof for the irrationality of e

Compare the first corollary of the above theorem with this famous result:
If $\cos(r), \sin(r)$ are rational numbers distinct from $\pm 1$ , then $r$ is not a rational multiple of $\pi$ .

Question. Can you establish the validity of our last statement with the aid of Kronecker's theorem which reads "if $x$ is an irrational number, then the sequence $a_{n > 0} = nx - [nx]$ is dense in $[0, 1]$ "?

Remark. Here is a good characterization for the curious reader. If $\gcd(a,b) = 1$ and $b\ge 4$ , then $\cos(\pi \frac {a}{b})$ is irrational.
(For a proof see, for example, Combinatorial Geometry in the Plane (translation of Kombinatorische Geometrie in der Eben))

Pure vs. Applied

Fri Feb 27, 2009 5:58 am, by The Captain

[ Currently: Enjoying Prof. Stewart's Cabinet ]

Relations between pure and applied mathematicians are based on trust and understanding. Pure mathematicians do not trust applied mathematicians, and applied mathematicians do not understand pure mathematicians.
Mr. Green

Infinite Unitary Rings

Thu Feb 19, 2009 10:03 am, by The Captain

Let $R$ be an infinite unitary ring and consider these equations over $R$ :

$a_1x_1 = b_1, a_2x_2 = b_2,\dots, a_kx_k = b_k$ where $a_i\neq0$ for $0\leq i\leq k$ .

Prove or disprove that there exists a subset $S$ of $R$ such that $|S| = k + 2$ and each of the above equations has at most one solution in $S$ .

The above problem is proposed by me, and in my mind it is not that easy to solve. Good news for those looking for new and non-trivial algebra problems Wink

How to Construct $(1-ab )^{-1}$

Tue Nov 25, 2008 7:38 am, by The Captain

Now, I want to quote another famous problem whose matrix version I saw for the first time when I was in high school. I quote it from T. Y. Lam's Exercises in Classical Ring Theory, a true masterpiece in its category. Be sure that you don't miss the comment at the end of the solution, since, in my mind, that is more important than the proof itself.

Problem.[Ex. 1.6] Let $a, b$ be elements in a ring $R$ . If $1 - ba$ is left-invertible (resp. invertible), show that $1 - ab$ is left-invertible (resp. invertible), and construct a left inverse (resp. inverse) for it explicitly.
Solution. The left ideal $R(1 - ab)$ contains $Rb(1 - ab) = R(1 - ba)b = Rb,$

so it also contains $(1 - ab) + ab = 1$ . This shows that $1 - ab$ is left-invertible. This proof lends itself easily to an explicit construction: if $u(1 - ba) = 1$ , then

$b = u(1 - ba)b = ub(1 - ab),$ so

$1 = 1 - ab + ab = 1 - ab + aub(1 - ab) = (1 + aub)(1 - ab).$

Hence, $(1 - ab)^{ - 1} = 1 + a(1 - ba)^{ - 1}b,$

where $x^{ - 1}$ denotes a "left inverse" of $x$ . The case when $1 - ba$ is invertible follows by combining the "left-invertible" and "right-invertible" cases.

Comment. The formula for $(1 - ab)^{ - 1}$ above occurs often in linear algebra books (for $n\times n$ matrices). Kaplansky taught me a way in which you can always rediscover this formula, "even if you are thrown up on a desert island with all your books and papers lost." Using the formal expression for inverting $1 - x$ in a power series, one writes (on sand):

$(1 - ab)^{ - 1} = 1 + ab + abab + ababab + \dots = \\ 1 + a(1 + ba + baba + \dots)b = 1 + a(1 - ba)^{ - 1}b.$

Once you hit on this correct formula, a direct verification (for $1$ -sided or $2$ -sided inverses) is a breeze. $\dagger$

For an analogue of this exercise for rings possibly without an identity, see Exercise $4.2$ [in the aforementioned book].

$\dagger$ This trick was also mentioned in an article of P. R. Halmos in Math. Intelligencer $3 (1981), 147 - 153.$ Halmos attributed the trick to N. Jacobson.

Balancing Vectors

Fri Sep 26, 2008 1:39 am, by The Captain

Here is a famous problem to which I want to quote an elegant solution from The Probabilistic Method by Alon and Spencer.

Problem. Let $v_1,\dots, v_n\in \mathbb{R}^m$ , all $|v_i| = 1$ . Then there exist $\epsilon_1,\dots, \epsilon_n = \pm1$ so that
$|\epsilon_1v_1 + \dots + \epsilon_nv_n|\leq\sqrt {n},$
and also there exist $\epsilon_1,\dots, \epsilon_n = \pm1$ so that
$|\epsilon_1v_1 + \dots + \epsilon_nv_n|\geq\sqrt {n}.$

Solution.Let $\epsilon_1,\dots, \epsilon_n = \pm1$ be selected uniformly and independently from $\{1, - 1\}$ . Set $X = |\epsilon_1v_1 + \dots + \epsilon_nv_n|^2$ .
Then $X = \sum_{j = 1}^{n}\sum_{i = 1}^{n}v_i.v_j\epsilon_i\epsilon_j$ .

Thus $E[X] = \sum_{j = 1}^{n}\sum_{i = 1}^{n}v_i.v_jE[\epsilon_i\epsilon_j]$ .
When $i\neq j$ , $E[\epsilon_i\epsilon_j] = E[\epsilon_i]E[\epsilon_j] = 0$ . When $i = j$ , $\epsilon_i^2 = 1$ so $E[\epsilon_i^2] = 1$ . Thus
$E[X] = \sum_{i = 1}^nv_iv_i = n$ .

Hence there exist specific $\epsilon_1,\dots, \epsilon_n = \pm1$ with $X\leq n$ and with $X\geq n$ . Taking square roots gives the theorem.

Remark. The original problem in the book is in $\mathbb{R}^n$ . But there is no relation between this $n$ and the number of vectors which is also $n$ . Thus, I've replaced the first one with $m$ .

A Challenging Linear Algebra Problem

Sat Sep 20, 2008 1:41 am, by The Captain

[ Currently: Reading about Coverings of Vector Spaces ]

Let $V$ be an $n$ dimensional vector space, and let $S_{1},\dots, S_{m}$ be $(n - 1)$ dimensional subspaces of $V$ . Give a necessary and sufficient condition that implies $V\neq \bigcup_{i = 1}^m S_i$ .

P.S I've found a simple sufficient condition in terms of $S_i$ , and a necessary and sufficient condition in terms of matrix algebra. I'd be glad to have your ideas about this problem.

Some Applications of Eigenvalues and Eigenvectors

Wed May 28, 2008 7:40 am, by The Captain

[ Currently: Selecting some parts of the note of ceee on eigenpairs ]

$\dagger$ Have you ever seen the video of the collapse of the Tacoma Narrows Bridge? The Tacoma Bridge was built in 1940. From the beginning, the bridge would form small waves like the surface of a body of water. This accidental behavior of the bridge brought many people who wanted to drive over this moving bridge. Most people thought that the bridge was safe despite the movement. However, about four months later, the oscillations (waves) became bigger. At one point, one edge of the road was 28 feet higher than the other edge. Finally, this bridge crashed into the water below. One explanation for the crash is that the oscillations of the bridge were caused by the frequency of the wind being too close to the natural frequency of the bridge. The natural frequency of the bridge is the eigenvalue of smallest magnitude of a system that models the bridge. This is why eigenvalues are very important to engineers when they analyze structures. (Differential Equations and Their Applications, $1983$ , pp. $171 - 173$ ).

$\dagger$ Car designers analyze eigenvalues in order to damp out the noise so that the occupants have a quiet ride. Eigenvalue analysis is also used in the design of car stereo systems so that the sounds are directed correctly for the listening pleasure of the passengers and driver. When you see a car that vibrates because of the loud booming music, think of eigenvalues. Eigenvalue analysis can indicate what needs to be changed to reduce the vibration of the car due to the music.

$\dagger$ Eigenvalues can also be used to test for cracks or deformities in a solid. Can you imagine if every inch of every beam used in construction had to be tested? The problem is not as time consuming when eigenvalues are used. When a beam is struck, its natural frequencies (eigenvalues) can be heard. If the beam "rings," then it is not flawed. A dull sound will result from a flawed beam because the flaw causes the eigenvalues to change. Sensitive machines can be used to "see" and "hear" eigenvalues more precisely.

$\dagger$ Oil companies frequently use eigenvalue analysis to explore land for oil. Oil, dirt, and other substances all give rise to linear systems which have different eigenvalues, so eigenvalue analysis can give a good indication of where oil reserves are located. Oil companies place probes around a site to pick up the waves that result from a huge truck used to vibrate the ground. The waves are changed as they pass through the different substances in the ground. The analysis of these waves directs the oil companies to possible drilling sites.

$\dagger$ Eigenvalues are not only used to explain natural occurrences, but also to discover new and better designs for the future. Some of the results are quite surprising. If you were asked to build the strongest column that you could to support the weight of a roof using only a specified amount of material, what shape would that column take? Most of us would build a cylinder like most other columns that we have seen. However, Steve Cox of Rice University and Michael Overton of New York University proved, based on the work of J. Keller and I. Tadjbakhsh, that the column would be stronger if it was largest at the top, middle, and bottom. At the points $\frac 14$ of the way from either end, the column could be smaller because the column would not naturally buckle there anyway.
Does that surprise you? This new design was discovered through the study of the eigenvalues of the system involving the column and the weight from above. Note that this column would not be the strongest design if any significant pressure came from the side, but when a column supports a roof, the vast majority of the pressure comes directly from above.

$\dagger$ Ready to parlay your knowledge of linear algebra into fame and fortune? Read "The $\$$ 25,000,000,000 Eigenvector: The Linear Algebra Behind Google". (the seventh link below)

Some Cool Links!

Sat Mar 15, 2008 10:19 am, by The Captain

I shall not be liable under any circumstances for the contents of these pages!

$0.$ AoPS/Mathlinks!

$1.$ Earliest Known Uses of Some of the Words of Mathematics

$2.$ Images of Mathematicians on Postage Stamps

$3.$ Vignettes on Teaching

$4.$ Women Mathematicians

$5.$ Some Proofs in Abstract Algebra

$6.$ Math Movies

$7.$ The $\$$ 25,000,000,000 Eigenvector: The Linear Algebra Behind Google

$8.$ Declaration of Linear Independence

$9.$ Linear Algebra Close to Earth

$10.$ Matrix Market

$11.$ Math Jokes

Fallacy in Mathematics

Thu Feb 28, 2008 3:08 am, by The Captain

$\spadesuit = \text{Elementary}, \clubsuit = \text{Intermediate}$
$\spadesuit1.$ We will show $0 = 1$ , using the integration by parts formula, ie $\int u dv = uv - \int vdu$ . Taking

$u = \frac 1x$ and $v = x$ , the above formula gives $\int \frac 1x dx = 1 + \int \frac 1x dx$ . Whence $1 = 0$

as required!!!

$\dagger$ Siamak Jafari, Danesh va mardom journal #78-79
(Consult Spivak's calculus, in the case you doubt about the right answer.)

The second fallacy is from an everlasting book in algebra, namely Methods in Algebra, by the great Parviz Shahriari.

$\spadesuit2.$ We want to find all $R$ s such that the circle given by $x^2 + y^2 = R^2$ is tangent to the parabola $y = x^2 + 2$ . First we should note that the problem has a solution $R = 2$ as a simple geometric interpretation shows.

Now we give an analytic solution! Omission of the variable $x$ from two given equations yields the following quadratic equation: $y^2 + y - (R^2 + 2) = 0$ . We want those two curves to be tangent, so we set the discriminant of this equation to zero and seek for its repeated roots: $\Delta = 4R^2 + 9 = 0$ But this last equation has no real roots, contradicting our geometric interpretation!!

What is wrong with our reasoning? Did you find out 'a' gap immediately?

Another Problem I'm Thinking On These Days (A Tough One)

Wed Jan 16, 2008 7:08 am, by The Captain

Let $S$ be a set of invertible matrices, ie $S\subset GL_n(\mathbb F)$ , with the property

that the sum of any two distinct elements of $S$ is non-invertible. What I want to prove

consists of two steps; Prove that $S$ is finite and give an upper bound for its

cardinality.

PS. We discussed this problem on the forum, and as some participants pointed out, $\mathbb F$ should be of a characteristic different from two.

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