All currency coins or notes are issued in these 4 denominations: {1, 2, 5, 10} but not {3, 4, 6, 7, 8, 9}.

Reason: These 4 numbers {1, 2, 5, 10} just need at most  2 operations (+ or -) to generate any number from 1 to 10.

(1 = 1)
(2 = 2)
3 = 1 + 2
4 = 2 +2
(5 = 5)
6 = 1 + 5
7 = 2 + 5
8 = 10 – 2 = 1 + 2 + 5
9 = 10 – 1 = 2 + 2 + 5
(10 = 10)

As I’ve already described, I’m worried about the oncoming MOOC revolution and its effect on math research. To say it plainly, I think there will be major cuts in professional math jobs starting very soon, and I’ve even started to discourage young people from their plans to become math professors.

I’d like to start up a conversation – with the public, but starting in the mathematical community – about mathematics research funding and why it’s important.

I’d like to argue for math research as a public good which deserves to be publicly funded. But although I’m sure that we need to make that case, the more I think about it the less sure I am how to make that case. I’d like your help.

So remember, we’re making the case that continuing math research is a good idea for our society, and we should put up some money towards it…

SM3C Inverse Circular Functions. Inverse Circular Functions of Cosecant, Secant and Cotangent are introduced and then transformations of the their graphs are explored and implied domains and ranges are calculated. The video was produced for students undertaking Specialist Mathematics in Year 12 in the State of Victoria, Australia.

Hello reader,

I am an applicant to Oxford/Cambridge, and sometime after I was informed about whether I was accepted/rejected, I decided to write up on the application process for new applicants who may not know too much about Oxford and Cambridge. I have recently done some research, whipped up some numbers which I hope can help you in your application. Most people start off thinking about applying to Oxford and Cambridge (and other famous universities) without anything except their ambition to start with, and I hope this guide can provide you with useful information for your university applications.

This is written mostly from a Singaporean student’s perspective, although some information is applicable to applicants from other countries as well.

Overview of Oxford and Cambridge

Oxford and Cambridge are the oldest and 2nd oldest universities in the English speaking world respectively, and 2^nd and 3^rd oldest in the world…

Original lithograph (M. C. Escher, 1961)

This is the first entry of the “Infinity Series”, which will introduce you to the peculiar world of infinity. Today, we will be talking about something with finite area but with infinite perimeter. That sounds odd and impossible but I’ll show you how it can be done.

Draw a circle and an equilateral triangle inside it, with the three vertices of the triangle touching the circle.

Since we know that the circle has a finite area, the triangle inside must have finite area as well. At this point, we still have finite perimeter on the triangle as well.

Now, divide each edge of the triangle into thirds; draw an equilateral triangle using the middle piece of the divided edges as the base, as follows:

With careful observation, you can see that the perimeter has increased. Specifically, the perimeter increased by a third: instead of one…

There are countless situations in mathematics where it helps to expand a function as a power series. Therefore, Taylor’s theorem, which gives us circumstances under which this can be done, is an important result of the course. It is also the one result that I was dreading lecturing, at least with the Lagrange form of the remainder, because in the past I have always found that the proof is one that I have not been able to understand properly. I don’t mean by that that I couldn’t follow the arguments I read. What I mean is that I couldn’t reproduce the proof without committing a couple of things to memory, which I would then forget again once I had presented them. Briefly, an argument that appears in a lot of textbooks uses a result called the Cauchy mean value theorem, and applies it to a cleverly chosen function. Whereas I…

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Another Excellent blog for SAT Grammar and Reading Tips and Tricks.

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Silverturtle’s Guide to SAT and Admissions Success
Excellent strategy guide from a student covering SAT and College Admission. It also included an excellent SAT Grammar section study guide.

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College Board SAT Biology E/M Practice Tests and Information Website

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Good introductory SAT Biology study guide

So there’s a whole lot of posts, including one from this very blog, which give intuitive explanations of why a negative times a negative is a positive.

I haven’t seen nearly as much material for a negative divided by a negative. One can certainly appeal to the inverse — since $latex 1 \times -1 = -1$, $latex \frac{-1}{-1} = 1$. Google searching leads to answers like that, but I’ve found nothing like the multiplication picture above.

Can anyone explain directly, at an intuitive level, why a negative divided by a negative is a positive? Or is the only way to do it to refer to multiplication?

University of Idaho is offering a book club on What’s Math Got to Do with It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject by Jo Boaler.

The book club is free and includes a copy of the book! There is an option to participate for 1 credit offered through the University of Idaho for $60. Check out the flyer below and contact irmc@uidaho.edu to register.

American Mathematics Contest 10
The AMC 10 is a 25 question, 75 minute multiple choice examination in secondary school mathematics containing problems which can be understood and solved with algebra and geometry concepts.

American Mathematics Contest 12
The AMC 12 is a 25 question, 75 minute multiple choice examination in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts.

United States of America Mathematical Olympiad
The United States of America Mathematical Olympiad (USAMO) and the United States of America Junior Mathematical Olympiad (USAJMO) are six question, two day, 9 hour essay/proof examinations. All problems can be solved with pre-calculus methods. Approximately 270 of the top scoring AMC 12 participants (based on a weighted average of AMC 12 and AIME score) are invited to take the USAMO. Approximately 230 of the top scoring AMC 10 participants (based on a weighted average of AMC 10 and AIME…

My college algebra course boasts one of the driest textbooks on the planet. It’s one of those versions that has exercises from 1 to 99 for each section…brutal.   Can you relate?
The topics for college algebra are very standard and cover little more than what students should have encountered recently in their algebra 2 course. I therefore decided that this class would lend itself quite nicely testing out the theory that a high-level, rich question questioning can be facilitated from a traditional, drill-and-kill style textbook.

Previously, I recall that Operations on Functions was a particularly awful topic for both me and my students.  The textbook presents this concept in exactly the way you might think:

f(x) = [expression involving x]  and g(x) = [similar expression involving x]

Find f(x) + g(x), f(x) – g(x), f(g(x), f(x) *g(x), f(x)/g(x)…f(snoozefest)…you get the point.  It’s boring, they’ve done it before, and there’s…

I found this discussion on reddit “How to avoid careless mathematical errors?“:

Hi //math.

I am a high school student who happens to be VERY good at math, but who consistently fails to get As on tests due to careless errors. Most of the time, they come from forgetting a 0 after a decimal place, multiplying instead of dividing, putting a decimal point in the wrong place, or just factoring wrong. I actually had to drop a Precalc Honors class because I got Ds on tests from the sheer number of stupid mistakes I made, despite understanding the material very well.

I assume that this occurs because I work quickly, but if I work slowly, I run out of time on the test. Additionally, my handwriting is horrible, but there’s really nothing I can do about that. And even when I check my answers after finishing, I still…

這是一個測試網誌。

數學世界並不如一般人想象中的理性、不可抗逆。相反，很多表面上合理的論証，卻會引申出非常荒謬的結論。

其中一個我十分喜愛的算式如下：

最先論証這個看似荒謬的算式的人是印度著名數學家斯里尼瓦瑟．拉馬努金 (Srinivasa Ramanujan)。他提出的論証中牽涉到冪級數 (Power Series) 的運用。因此，我將會用一個比較易懂的方法，嘗試去証明這條算式是正確的。

首先設以上三項算式為S1、S2及Ｓ。

S1的答案比較易懂，從算式中可見，S1必定為 或 1 這兩個可能答案：

我們取 及 1 的平均數 1/2 為S1的答案 (假如取其平均數這做法令你感到很不安的話，事實上我有另一個論証方法可以証明 1/2 是最合理並最接近的答案，容後分解)。

然後將S2乘 ２ ，算式如下：

兩列算式的數字各自相加後得出的結果如下：

很眼熟吧？沒錯，從以上計算中可以歸納出 2 x S2 = S1

而由於S1 = 1/2，代入以上算式可以得出 S2 = 1/4

之後，我們進行 S – S2 這一操作，運算過程如下：

奇妙的事情就在這裡開始發生了。剛才我們証明了S2 = 1/4，因此

S – 1/4 = 4*S

3*S = -1/4

S = -1/12

亦即証明：

難以致信吧？假如你無法接受這個違反常識，卻又看似合理的論証結果，我可以很榮幸地告訴你，

你的質疑是非常合理。

因為上述所有推算過程中都犯下了一個數學世界的禁忌，就是嘗試對無限 (Infinity) 進行操作。

真相是，無限本身是一個概念，而不是一個數字。因此，假如一意孤行地對無限進行加減乘除等操作時，便會出現如上述般荒誕的結果，就像整個數學系統當機了一樣。

不過，並非所有無限都是不可操作的。例如收斂級數 (Convergent Series) 便是一個有求和答案的無窮數列。

(credit: Numberphile)

Kennesaw- Who says that Liberal Arts is dead?  One day in 1637, a lawyer and amateur mathematician named Pierre de Fermat scribbled a curious note in his journal: “The equation xⁿ+ yⁿ = zⁿ, where x, y, and z are positive integers, has no solution if n is greater than 2… I have discovered a most remarkable proof, but this margin is too narrow to contain it.”

In his spare time, Fermat studied languages, classical literature and natural science.  He also discovered the fundamental principle of analytic geometry. His methods for finding tangents to curves and their maximum and minimum points led him to be regarded as the inventor of the differential calculus. Through his correspondence with Blaise Pascal he was a co-founder of the theory of probability.

It took mankind over 350 years to prove Fermat’s last theorem.

Spend time today encouraging…

I have been doing a lot of thinking about place value lately. Yes, I need a life outside my standard state-issue gray cubicle. Nonetheless, I have become caught up in the beauty of the Hindu-Arabic number system. We also know it as the Base-Ten numeration system. It is beautiful and elegant. Let me elucidate (I love that word!)

One of the best t-shirts I have ever seen had this quote on the front, “There are only 10 kinds of people in the world; those who understand binary and those who don’t.” Some of you might be laughing right now while others are scratching their heads wondering what is wrong with me. Okay, here’s the joke. Binary is a base-2 numeration system. It has two digits – 0 and 1. Place value in binary is determined by powers of 2. So, in the units place you can have 0 and…

Note: I originally published this entry on a Google Sites page in September 2013, but I have moved it hear as I hope to make better use of the $latex \LaTeX$ support offered by wordpress.com

 

I completed the first year of my maths degree in June 2013. I studied Analysis in all three terms (sequences and series in the first term, continuity and differentiability in the second, and Riemann integration in the third) and feel I have learnt a lot. Occasionally, amongst all this theory, I must admit to sometimes having lost touch with the original questions that provoked its development.

Consider the following naïve approach to infinite series. What is the answer if I add 1 and -1 alternately forever? That is

$latex \displaystyle{1+(-1)+1+(-1)+ … = ???}$

If I bracket the terms in pairs starting with the first and second terms, the answer appears to be 0:
$latex…

Pi(Π) number is irrational and equal to 3.14159265… .                                    

Let us draw a circle with radius ‘r’

$latex \int_0^{2\pi} r \mathrm{d}\theta = {2\pi} r &s=2$

So, circumference of the circle is equal to $latex {2\pi} r $

If the value of circumference is divided by value of diameter, the result is $latex \pi $

Area of quarter circle = $latex \frac{\pi r^2}{4}&s=2$

Area of the square = $latex {r^2}&s=2$

Probability of putting a dot on quarter circle is shown below.

$latex \frac{\frac{\pi r^2}{4}}{r^2} = \frac{\pi}{4}&s=3$

The area of circle is divided by total area.

In order to achieve $latex \pi $ value the result is multiplied by 4.

We can use Monte Carlo method for approximation $latex \pi $ value. For example, 10000 dots will be put on picture above. Each dot…

 

This blog was my very first venture into blogging on the fabulous Primary English blog.   I’m very grateful to them for publishing it last May which led to me thinking seriously about starting my own blog.  Their site is well worth a visit and they also have some amazing pinterest boards on all sorts of themes.

Here is what I blogged back in May:

As a maths leader, I quite often have the privilege of doing planning trawls and looking at weekly and medium term planning from other teachers.  I’m often very impressed by the thought and detail that goes into these.  But there’s one section that seems very rarely to be given much thought.  If your weekly or medium term planning format is anything like mine, there’s a small section headed ‘cross-curricular links’, and I hardly ever see it filled in, except perhaps with the suggestions…

Link: https://www.coursera.org/course/mathematicalmethods

Coverage: the official description is thorough. The caveat is that all topics are discussed without going too much into subtleties.

Potential audience: people with moderate knowledge of multivariable calculus, linear algebra and a bit of R programming experience who want to see how this knowledge may be applied to finance.

Format:

• a lecture; slides of lectures can be downloaded
• in-video questions
• a problem set; answers are provided for most problems
• a quiz; usually, it’s easier than the problem set

Note that only quizzes are graded.

Workload: 2-3 hours assuming you know calculus & linear algebra. If you don’t, then it’s hard to say.

Misc:

1. there is no Statement of Accomplishment
2. there is no course staff to help you, i.e. you should rely only on the assistance from your fellow courserians
3. there are some misprints in materials. Hope Dr.Konis will have them fixed before the next session.

To sum…

Since I decided to call this blog martingalemeasure it seems only fitting that the first post should be about probability; martingales in particular. In my favorite introductory book on measure theoretical probability, “Probablity with Martingales” by David Williams, we find an exercise in Chapter 10, which I paraphrase here:

Suppose a monkey is typing randomly at a typewriter whose only keys are the capital letters $latex A$ through $latex Z$ of the english alphabet. What is the expected (average) time it will take for the monkey to type the word $latex ABRACADABRA$?

This is not an easy problem. In fact it’s not entirely obvious that the average time is even finte! Williams expects the reader to solve it using the beautiful theory of martingales and in particular Doob’s optional-stopping theorem. We will calculate the result below, but stop short of a proof. (There are many proofs of this result…

Look Like a Genius

Calculating the day of the week for historical date is impressive, but it isn’t as hard as it seems. Several methods have been shown to work and be performable by average humans. This is an improvement to one of the simplest methods.

The Doomsday Rule

Paraphrased from wikipedia:

The Doomsday rule or Doomsday algorithm is a relatively simple way to calculate the day of the week of a given date. It was devised by John Conway, drawing inspiration from Lewis Carroll’s work on a perpetual calendar algorithm. The algorithm takes advantage of the fact that several easy-to-remember dates fall on the same day each year; for example, 4/4, 6/6, 8/8, 10/10, 12/12, and the last day of February are all the same day of the week.

Conway’s original algorithm required dividing by 12 and 4, and remembering several intermediate values. It’s achievable, but not…

When I first started teaching, I was always looking for the correct answer on a math problem. I would mark it wrong or right, there was no gray area.  I began to change my thinking a bit when I noticed my students weren’t really growing.  I knew that I needed to do something differently, so I began to start looking at the process of their thinking so that I could give direct feedback to help them get better.

If you think about it, we do the same thing in reading.  We don’t expect students to become perfect readers overnight, so we give them reading strategies to become better. We look at their fluency, comprehension, how they monitor and self correct…we intervene and give feedback to help them.

With problem solving it can be the same way.  We can take a look at the work a student writes down and see…

A really interesting and creative video about Pythagoras and the Pythagorean Theorem. Showing us how maybe Pythagoras wasn’t as awesome as we thought he was.
An extremely well done and well thought out video, about math, history and beans.

Use a checklist to moniter my students’ understanding of fractions.

We have a unit test coming up at the end of the week on fractions. All throuout this week we will be working in small groups with students and I want to ensure that I am targeting my instruction to the concepts students need to review.

Reflection: I had some difficulties with this goal, but I definitely want to try this again another time. First, it was difficult for me to gauge the entire class in the short time I was using my checklist. For various reasons, I was only able to use it through portions of two lessons. The main problem I could see was that, since this was the end of the unit, I was not able to gather data on everything we had covered. Although, that is what I was trying to do with my ten item…

Prime numbers are numbers can only be divided by itself and 1.  They are greater than 0. In this text, we will see how to test whether a number is a prime number or not. Furthermore, we will find prime numbers up to N. ( N is a number entered by user. )

For example,  2 can be divided by 2 and 1. So it is a prime number.

Prime numbers: 2, 3, 5, 7, 11 ,13, 17, 19, …

How to test a number ?

“A” is a positive number greater than 1. Let X be a number different than number A and 1. If we can find such a number divide A with remainder 0, A is not a prime number.

• X can be preferred prime numbers less than square root of A. This make our program slightly faster.

A : 29 and Square root of A : 5.385164

A signal processing snippet.  Let $latex x(t)$ be a bandlimited signal restricted to the interval $latex [T_1, T_2]$, so that in particular

$latex x(t) = \displaystyle{\sum_{n=0}^{N-1} x(n \Delta t) \text{sinc}\left(\frac{t – n\Delta t}{\Delta t}\right)} 1_{T_1 \leq t \leq T_2}$

Here, as opposed to previous entries, we have defined $latex \text{sinc}(t) = \sin(\pi t)/(\pi t)$ (I have had a change of heart). Then the Fourier transform of $latex x(t)$ is

$latex X(f) = \displaystyle{\sum_{n=0}^{N-1}} x(n\Delta t) e^{-i 2\pi f n \Delta t} R(n \Delta t – T_2, n \Delta t – T_1, f – 1/2\Delta t, f + 1/2 \Delta t) \,\Delta t$

where we define

$latex R(t_1, t_2, f_1, f_2) = \dfrac{(\text{Ei}(i 2\pi f_2 t_2) – \text{Ei}(i 2\pi f_2 t_1)) – (\text{Ei}(i 2\pi f_1 t_2) – \text{Ei}(i 2\pi f_1 t_1))}{i 2\pi}$

and $latex \text{Ei}$ is the exponential integral, which for imaginary arguments is

$latex \text{Ei}(it) = i \dfrac{\pi}{2} -\displaystyle{\int_{t}^{\infty}} \dfrac{e^{i…

Why is this called 2=1, one might ask. Here’s “proof” for something that’s wrong. If you’re able to appreciate these things, try finding the mistake.

a  =  b

a²  =  a b

a² + a²  =  a² + a b

2 a²  =  a² + a b

2 a² – 2 a b  =  a² + a b – 2 a b

2 a² – 2 a b  =  a² – a b

2 ( a² – a b )  =  1 ( a² – a b )

2  =  1

This will be the first and only geeky post, I promise.

Symbols and conventions are both ubiquitous and necessary in our every day life.  They enable our culture, on the one hand. On the other hand, they limit our imagination and we need to challenge them sometimes. This is a conflict I find fascinating and would like to base…

I absolutely love this idea. I’ve done two other worksheets with this same idea (click here and click here to see them) and I’ve found it to be really successful. It’s super important to get kids familiar with the hundreds chart. Hopefully children will be so familiar with it that they can make their own hundreds chart on a piece of scratch paper during a test (since kids don’t get a printed hundreds chart on their standardized tests).Click here to see my third 10 more that/10 less than hundreds chart worksheet: hundreds chart- more than- less than OWL

So a friend of mine got a little riled up about this video:

You know, I write and read poetry, and there’s this thing that happens when I talk about poetry, a thing that I know also happens all the time when people who write and read math talk about math. People say, “I don’t like poetry.” Or sometimes, more charitably, “I don’t understand poetry.” Sometimes — if they like me — they think my interest in poetry is adorable. But they don’t want to talk about it with me. And meanwhile, I’m thinking, what do you mean you don’t like poetry? Poetry is a big thing! It’s like saying you don’t like music! Or food! There’s so much of it, I’m sure we could find something you would like.

Well. That’s poetry. Because you know what my reaction was to my friend’s curiosity about that video up there?…