Mathematics

In a previous post I mentioned the question of why is mathematics possible. Among the interesting comments to the post, here is a comment by Tim Gowers: “Maybe the following would be a way of rephrasing your question. We know that undecidability results don’t show that mathematics is impossible, since we are interested in a tiny fraction of mathematical statements, and in practice only in a tiny fraction of possible proofs (roughly speaking, the comprehensible ones). But why is it that these two classes match up so well? Why is it that nice mathematical statements so often have proofs that are of the kind that we are able to discover? I think part of the answer is that the evolution of mathematics follows what we can prove: it is quite easy to come up with simple statements that look hopeless (is zeta(3) a normal number? etc. etc.), so the match-up just mentioned is far from perfect. I think we develop an instinct for what kinds of statements are likely to be amenable to our techniques, and from time to time we are surprised by a statement like the Poincare conjecture that turns out to be very very hard. I wonder whether the true answer might be something like this: there is a ‘random variable’ associated with ‘natural’ mathematical statements, which takes as its value the length of the shortest ‘humanly discoverable’ proof. We know that this variable can take very large values, or even be infinite, but on average it is fairly small (if only because a lot of statements you can write down have simple counterexamples). We get interested in the statements for which the random variable takes a large but, it seems to us, not too large, value. And empirically such statements exist. Of course, all this is wild speculation. I don’t know how to formulate a precise question, but it would be fascinating if one could somehow do rigorous justice to the intuition that most natural statements have short proofs but a few are more difficult and a few are much more difficult. It would be something like a ‘quantitative statistical version of Godel’s theorem’.”

I love using rectangles as a model for multiplication. In this video, Mike & son offer a pithy demonstration of WHY a negative number times a negative number has to come out positive: Get all our new math tips and games: Subscribe in a reader, or get updates by Email.

Look for KEY concepts when solving word problems in Algebra. These examples will explain how to solve Algebra word problems.

Writing equations in standard form is easy with these examples!

Disclaimer: I have never taught Algebra 2/Trigonometry in high school. Yes, I am familiar with the material. No, I am not familiar with the acutal curriculum. On any test, you have to vary the level of difficulty of the questions. You can't make the students sweat every single question. It's the rollercoaster approach: let them catch their breath on the next climb before the next drive drop. That said, my first look at last Friday's (June 14, 2013) Algebra 2/Trigonometry Regents shows some questions which my Algebra students might have been able to answer. However, as I look at this, I can't help wondering: if it's on this test, why am I cramming so much material into the Algebra 1 curriuculum? Some questions my Algebra 1 students should've been able to answer (whether or not I covered the material as is): 7. What is the graph of the solution set of |2x - 1| > 5?I didn't cover this, but I covered inequalities, and simple checking of the answers would yield the correct choice. (My students could get this only as a multiple-choice.) 8. What is the range of the function below? (graph omitted) I covered domain and range, so they should have gotten this one easily. Which ordered pair is in the solution set ... ?It doesn't matter the actual equations. The fastest solution here is to plug in x and y and check. Using the simple equation, 3y - x = 0, choices (1) and (3) can be eliminated immediately. Choice (2) is obviously wrong for the first equation even if you accidentally reversed the co-ordinates, which I'm expecting is what the test makers were expecting when they wrote it. The answer, by quick elimination before I even checked it, is (4). 13. Sue invests $500 ... Compound interest is now in Algebra, rather than just simple interest. However, they do make you solve for time in these problems. However, because its multiple choice and the fact that the formula is given and the avaiability of calculators, it doesn't take long to plug in all four answers. 17. Which problem involves evaluation 6P4?Okay, fun time. Do I or don't I explain 6P4 on the blackboard? Do I cover permutations and how do I cover it? Yes, I cover permutations, but there's a time issue. I may explain how it's done without writing "6P4", and certainly without writing a complicated factorial formula for it. However, this year's Algebra regents had a factorial problem totally devoid of any statistics, so maybe I should be teaching that. In any case, each of the four choices should be answerable in Algebra - How many different four-digit ID numbers can be formed using 1, 2, 3, 4, 5, and 6 without repetition? - How many different subcommittees of four can be chosen from a committee having six members? - How many different outfits can be made using six shirts and four pairs of pants? - How many different ways can one boy and one girl be selected from a group of four boys and six girls? The answers, respectively, are 6P4, 6C4, 6 * 4, and 4 * 6. (The last two are basically the same question.) 18. Which equation is represented by the graph below?I have one Geometry class per day. This was covered. There's nothing special about this problem that those students couldn't have done it. 24. Which expression is equivalent to ... ?A fraction with variables and negative exponents. Nothing tricky about it. Laws of exponents. So I hope everyone caught their breath on those and saved their your energy for the rest of the exam. Some questions were trickier than others, but some (especially the multiple choice) weren't too hard to work out. One last question, which is beyond my regular Algebra class: 30. Find the number of possible ten-letter arrangements of STATISTICS.There are 10! arrangements of the any 10 letters, but there are duplicates you have to divide by 3! for the repeated S's, by 3! for the repeated T's, and by 2! for the repeated 2!. So the answer is 10! / (3! * 3! * 2!), which is 50,400. Frankly, I'm surprised at this question, because that's pretty much t

It's been almost a week, and the tests are graded, so I don't suppose anyone still cares about this, but I'll go ahead anyway. First off, if you weren't familiar with the word bivariate, you could have broken it down into bi-, meaning "two", and -variate, which looks like "variable", right? So bivariate: two variables. Which of the tables is measuring two variables and will give a scatter plot, as opposed to a bar graph. The answers, unfortunately, doesn't matter because the questions was thrown out. A "lack of specificity" was the reason. Likewise, if you took the test in Chinese, two answers were accepted to question number 1 because of a translation error. That happens a lot. As for the Factorial question, a.k.a. "the question with the exclamation point", I was able to guess that answer without doing any work for one simple reason: the last step was to subtract 10, but only one answer was 10 less than another. That answer was the correct one. (I checked my guess afterward, of course.) For the record: 6! + 5!(3!)/(4!) - 10 can be done with the scientific calculator, if you know where to look, but it isn't necessary. 6! = 720, 3! = 6, 5!/4! = 5, so 720 + 5 * 6 - 10 = 720 + 30 - 10 = 740 I wanted to review some of the open-ended questions. Question 31. An inequality with a negative multiplier. The trick was to remember to reverse the direction of the inequality symbol. That is, -5(x - 7) , when divided by -5 becomes (x - 7) > -3. The final answer is x > 4. Question 32. A volume question on the Algebra test. Silly. If they at least gave the Volume and asked to find, say, the height, you could argue it was an Algebra task, but, as is, it's a middle-school problem. The formula for volume of a cylinder was in the back of the booklet: V = (pi)r^2*h. The trick here is that the gave the diameter instead of the radius, so you had to divide 13 by 2 to get 6.5. If you didn't, your answer was four times larger than it should've been, but you most likely got 1 out of 2 points. The final answer is 1,014*pi. Note: The question said "in terms of [pi]", so if you multiplied by 3.14 or used the pi key on your calculator (i.e., you did extra work!), you lost a point for not answering the question that they asked. Question 33. A distance question with big numbers, with a conversion added on. Two questions on the test involved converting between hours and days and weeks. This was one of them. The distance from Earth to Mars is 136,000,000 miles. A spaceship travels at 31,000 miles per hour. Determine, to the nearest day, how long it will take the spaceship to reach Mars. Divided 136,000,000 by 31,000 to get the number of hours (4387.096774...) and then divide by 24 to get the number of days (182.795698...). The final answer is 183 days. Question 34. The Counting Principle. How many options are on the menu? They've given this question many times before, but this is the largest number of items that they've ever used. The Principle remains the same. There are five main courses, three vegetables, five desserts, and three beverages. To find the number of possible means, multiple the four of them: 5 * 3 * 5 * 3 = 225. How many have chicken tenders? That's 1 * 3 * 5 * 3 = 45, which is one-fifth of the total. How many have pizza (1), corn or carrots (2), a dessert (5) and a beverage (3): 1 * 2 * 5 * 3 = 30 If you showed your work, you likely got one point for each correct answer. Question 35. Trigonometry. Find the angle of elevation. You have a right triangle with a height of 350 feet and a base of 1000 feet, and you want to find the angle on the ground. You have the opposite (350) and the adjacent (1000), but you don't know the hypotenuse, so that means that you need to use tangent to solve the problem. So tan(x) = (350)/(1000) and, therefore x = tan-1(350/1000), which is approximate 19.29. The final answers is 19 degreesPartial credit likely for using sine or cosine, or if you answer is expressed in radians or if rounded incorrectly. Qu

A brilliant opinion piece by ALICE CRARY and W. STEPHEN WILSON in the Opinion Pages of the New Your Times. A highlighted message: Mastering and using algorithms involves a special and important kind of thinking. Read the whole paper. It also contains a great quote from  John Dewey: the goal of education “is to enable individuals to continue their education.”   [With thanks to muriel]

So that now we have a clearer view of sensing modalities [1], here are some examples that showed up on our collective radar screen in the past month: The first one is akin to functional compressive sensing of the type: Hx = N(Ax). There, we are not interested in image reconstruction but rather the reconstruction of a specific item in a series of measurements (the moving objects). Compressive Object Tracking using Entangled Photons Omar S. Magaña-Loaiza, Gregory A. Howland, Mehul Malik, John C. Howell, Robert W. Boyd We present a compressive sensing protocol that tracks a moving object by removing static components from a scene. The implementation is carried out on a ghost imaging scheme to minimize both the number of photons and the number of measurements required to form a quantum image of the tracked object. This procedure tracks an object at low light levels with fewer than 3% of the measurements required for a raster scan, permitting us to more effectively use the information content in each photon. Next, we have the typical indirect imaging of the coded aperture of X-ray observatories ( x = L(Ax) ). Let us note the reference to potential use of compressive sensing. It is just a question of time before they come to our side of the Force. Simultaneous analysis of large INTEGRAL/SPI datasets: optimizing the computation of the solution and its variance using sparse matrix algorithms L. Bouchet, P. Amestoy, A. Buttari, F.-H. Rouet, M. Chauvin Nowadays, analyzing and reducing the ever larger astronomical datasets is becoming a crucial challenge, especially for long cumulated observation times. The INTEGRAL/SPI X-gamma-ray spectrometer is an instrument for which it is essential to process many exposures at the same time in order to increase the low signal-to-noise ratio of the weakest sources. In this context, the conventional methods for data reduction are inefficient and sometimes not feasible at all. Processing several years of data simultaneously requires computing not only the solution of a large system of equations, but also the associated uncertainties. We aim at reducing the computation time and the memory usage. Since the SPI transfer function is sparse, we have used some popular methods for the solution of large sparse linear systems; we briefly review these methods. We use the Multifrontal Massively Parallel Solver (MUMPS) to compute the solution of the system of equations. We also need to compute the variance of the solution, which amounts to computing selected entries of the inverse of the sparse matrix corresponding to our linear system. This can be achieved through one of the latest features of the MUMPS software that has been partly motivated by this work. In this paper we provide a brief presentation of this feature and evaluate its effectiveness on astrophysical problems requiring the processing of large datasets simultaneously, such as the study of the entire emission of the Galaxy. We used these algorithms to solve the large sparse systems arising from SPI data processing and to obtain both their solutions and the associated variances. In conclusion, thanks to these newly developed tools, processing large datasets arising from SPI is now feasible with both a reasonable execution time and a low memory usage. Next we have some interesting phase retrieval imaging of the type: x = N2(N1(x)) Single-pixel digital "ghost" holography Pere Clemente, Vicente Duran, Enrique Tajahuerce, Victor Torres-Company, Jesus Lancis Since its discovery, the "ghost" diffraction phenomenon has emerged as a non-conventional technique for optical imaging with very promising advantages. However, extracting intensity and phase information of a structured and realistic object remains a challenge. Here, we show that a "ghost" hologram can be recorded with a single-pixel configuration by adapting concepts from standard digital holography. The presented h

I began to understand that pure mathematics was more than a collection of random tools mainly fashioned for use in the Cambridge treatment of natural philosophy.Andrew ForsythThe 169th day of the year; 169 is the smallest square which is prime when rotated 180o (691) What is the next one?EVENTS1558 Robert Recorde’s will was admitted to probate, after he died in prison. He introduced the equals sign in The Whetstone of Witte (1557) with the words: “And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lenghte, thus: because noe .2. thynges, can be moare equalle.” “Gemowe” (think Gemini )is an old French work meaning “twin.”. *VFR When they are asked what they would use if this was not available, it seems difficult for students to imagine a different symbol. Image from Wikipedia.1584 Jacob Christmann appointed professor of Hebrew at Heidelberg. In 1595 he defended the view that the circle could only be approximately squared. *VFR1864 Lewis Carroll ?nally decided to write up Alice’s Adventures in Wonderland. [Stuart Dodgson Collingwook, The Life and Letters of Lewis Carroll (1898), p. 96] 1908 , Alan Archibald Campbell Swinton took the first x-ray images in Britain in January 1896 and by a year later the medical professions were bringing him surgical cases for analysis. But "on this day he predicted exactly how another magic box would work, in a letter to Nature. He called it ‘Distant Electric Vision’, but we know it now as television." *Keith Moore, http://blogs.royalsociety.org 1928, aviator Amelia Earhart became the first woman to fly across the Atlantic Ocean. She had accepted the invitation of the American pilots Wilmer Stultz (1900-29) and Louis Gordon to join them on the transatlantic flight. The crossing from Newfoundland to Wales took about 21 hours. Amelia Earhart went on to establish herself as a respected role model, tirelessly demonstrating that young women were as capable as men in succeeding in their chosen vocations. In 1935 she crossed the Atlantic solo in record time: 13 hr 30 min. *TIS1983 Sally Ride, astrophysicist, becomes the ?rst American woman in space. The Soviets were ahead by twenty years and two days.*VFR BIRTHS1799 William Lassell (18 June 1799 – 5 October 1880) was a wealthy amateur English astronomer. He set up an observatory at Starfield, near Liverpool. England, He built his own 24" diameter telescope, and devised steam-driven equipment for grinding an polishing the speculum metal mirror. This telescope was the first of its size to be mounted "equitorially" to allow easy tracking of the stars. He discovered Triton, a moon of Neptune, and Ariel and Umbriel, satellites of Uranus. Later, Lassell built a 48" diameter telescope with th same design and took it to Malta for observations with clearer skies.*TIS1818 Pietro Angelo Secchi (18 Jun 1818, 26 Feb 1878 at age 59) Italian Jesuit priest and astrophysicist, who made the first survey of the spectra of over 4000 stars and suggested that stars be classified according to their spectral type. He studied the planets, especially Jupiter, which he discovered was composed of gasses. Secchi studied the dark lines which join the two hemispheres of Mars; he called them canals as if they where the works of living beings. (These studies were later continued by Schiaparelli.) Beyond astronomy, his interests ranged from archaeology to geodesy, from geophysics to meteorology. He also invented a meteorograph, an automated device for recording barometric pressure, temperature, wind direction and velocity, and rainfall.*TIS 1858 Andrew Russell Forsyth (18 June 1858, Glasgow – 2 June 1942, South Kensington) studied at Liverpool College and was tutored by Richard Pendlebury before entering Trinity College, Cambridge, graduating senior wrangler in 1881. He was elected a fellow of Trinity and then appointed to the chair of mathematics at the University of Liverpo

A Palindrome is a word, phrase or sequence which reads the same in both directions. Derived from the Greek palíndromos, meaning running back again, a palindrome reads the same forward and backward, with general allowances for adjustments to punctuation and word dividers. I think most people would agree that palindromes are fun --even folks who dislike math. After tinkering with palindrome numbers, words, phrases and dates, I became inspired to delve further into this topic. Danica McKellar, actress and mathematician, had some puzzles she created. For example: What did the mathematician say when she was offered cake? "I prefer pi" So I spent many hours creating some of my own puzzles. See if you can solve them! Stay tuned to my e-newsletter for the solutions. If you create any of your own puzzles and solutions, please share them on my Facebook page. In the meantime, enjoy!