RGB and Hexadecimal
RGB color describes a color as a combination of red, green, and blue light. The standard representation for RGB color is a series of 3 8-bit numbers; that is three numbers between 0 and 255. (This is not the only representation, -- since 255 is awkward to do math with, sometimes colors are represented as three floating-point numbers, so each color is in the range of 0.0 to 1.0. If you know about floating-point numbers, though, you'll know that this representation requires four times as much space).
A common way of representing numbers on a computer is hexadecimal. Like
the name implies, hexadecimal has 16 digits: 0 through 9, and then a, b, c,
d, e, f. a is equivalent to the decimal number 10.
The hexadecimal 10
is equivalent to the decimal number 16. The hexadecimal number
equivalent to the decimal number 255, so the entire range of 8-bit numbers
can be expressed in hexadecimal as
0x prefix is a common
designator of hexadecimal). When designating RGB colors, the red, green, and
blue channels are often concatenated to form one hexadecimal number. For
0x000000 is black,
0xff0000 is red,
0xffffff is white.
<body bgcolor=... tag takes as an argument a hexadecimal
representation of the color you want to use. Picking a color in RGB
isn't exactly intuitive. It's usually best to browse using an HSV color-picker
like the java applet below:
You can use any color picker though, as long as you know how to convert from what the color picker displays (usually 0-255) to what HTML wants as a tag (0x00-0xff). This is simply a matter of converting from base-10 (decimal) to base-16 (hexadecimal):
|Comments?||Last modified: Thu Jun 20 12:30:29 1996|


Geometry and textbook websites
- Complete textbook online and workbook: Ask your teacher for the username and password.
- There is a Homework Video Tutor for every section in the textbook. (The web code for that website is in the textbook, or ask your teacher for it.) Or practice with lesson quizzes, chapter tests, and/or vocabulary quizzes by clicking on the link at the top called "Go to Prentice Hall Mathematics: Geometry 2007/9 Student Home Page".
- Mr. Chou’s KnowGeometry.com website has great powerpoint notes for the 1st semester of Geometry
Free access to a live tutor online
Great websites for extra Algebra practice
Other math websites
Math in Real Life
- How maths makes the world go round - Note: "maths" in the british world = "math" in the USA
- 10 Jobs for Math Whizzes: "In 2008, mathematicians earned an average annual income of $94,960, according to the most recent data from the Bureau of Labor Statistics. Mathematician was also rated the best career, based on a January 2009 study of the best and worst professions by JobsRated.com..."
- When Will I Use Math - great website! check out the "Careers" and "Did you know?" > "Math in Real Life" > "15 Most Lucative College Degrees"; "Math Makes You Smarter"; "Medical School" (math majors have a higher acceptance rate into medical school than biology/science majors!)
- Imagining the Tenth Dimension - video
- Cracking the Scratch Lottery Code: How to beat the lottery and make $600 a day... although if you are a math/statistical consultant, you make more than that (six figures) and do more interesting work anyways.
- Vi Hart is a recreational mathemusician who loves to doodle and make videos and talk really fast. Check out her blog and videos on Mathematical Doodling -- who knew trees, stars, snakes, elephants, being sick, and doodling had math in them? :)


Applied Math & Physics
The Applied Mathematics and Physics Division employs an interdisciplinary team to address challenging technology problems for military, civilian, and commercial clients. Our team fills a niche between academic research and industrial application, teaming with leading universities to advance the state of the art while also teaming with major system integrators to put cutting-edge solutions into real-world products and system. Our staff contains a mix of scientist and engineers, with 80% at the M.S. level and 60% at the Ph.D. level in academic and professional fields including physics, mathematics, aerospace engineering, systems engineering, computer science, and operations research.
Our team works at the forefront of multi-sensor, MULTIINT, and multiple phenomenology data fusion. In a world where most sensor data that is collected “falls on the floor,” our algorithms and tools help connect the dots to create information out of data in an automated or semi-automated way. In addition to using tools from the traditional data fusion literature, we have experience developing techniques that apply concepts from machine learning, computer vision, and more “esoteric” fields such as differential geometry and topology to provide new capability to our customers and partners. We actively collaborate with leaders in these academic fields to feed our technology pipeline and have contractual relationships with system integrators to mature and transition technologies to real systems in the near-term.
- Guide – Decision Making Under Uncertainty: Mature software for a wide range of probability-based calculations including Bayesian networks, interacting-multiple models, unscented Kalman filtering, and solution of stochastic partial differential equations
- Adapt – Multi-sensor fusion framework: Software for modeling data fusion hierarchies for evaluating design of data fusion network and comparison testing specific algorithms
- Discover – Object Classification: A tool for classifying objects into types applicable to a wide range of sensor data including radar, electro-optical / infrared (EOIR), and LIDAR
Our team helps to develop the mathematical model and software tool by which the Federal Communications Commission analyzes incentive auctions for buying and selling bandwidth. Using techniques from operations research and working with world-class economists, the team helps the FCC design complex, combinatorial auctions that result in efficient use of the electromagnetic spectrum and benefits the U.S. taxpayer.
Papers Authored and Co-authored
Corporate fleets are constantly challenged to achieve targeted service levels at the lowest possible cost. DAC has developed large-scale optimization software for Fortune 500 companies in the United States. When implemented, DAC’s optimization software provides higher levels of customer service at reduced operation costs, allowing our customers to operate more efficiently in the face of rising fuel and labor costs.


Happy 4th of July Y'all!!
I'm hope your day is filled with friends, family, grilling, and relaxation!
I'm just stopping in today to share a little project I've been working on to celebrate the 4th. One of my favorite products, Multiplication Kaboom got a total over haul and I've in love.
There are new fonts, new backgrounds, and new frames!
Game Cards Included:
- KABOOM! cards
- 2 digit x 1 digit
- 3 digit x 1 digit
- 2 digit x 2 digit
Game play is simple. Each player draws a card from the deck or from the Kaboom container. (I prefer the cards that can be stored in baggies because they take up less space and I just don't have space to spare in my classroom.) If the player draws a multiplication card they must solve the problem. Other players solve as well to check their work.
If the players solves correctly, they get to keep the card. If not, it goes back in the deck. However...and this is where the fun comes in, if a player draws a Kaboom card, they loose all cards previously earned and must return all cards to the deck and start collecting again.
And... that's it! It's a great game and has been a huge success in my classroom! Once your kids know how to play, you can adapt it to practice any skill you want. I included some blank card pages for you and/or students to come up with their own problems to add to the game.
I don't know about you, but I need low maintenance, easy to set up, easy to explain math games in my classroom. Kaboom fits all my needs and my students love LOVE it!
The best part about this game is the game grows with them as their skills grow. I start with 2x1 digit multiplication for review and add game cards as we build our skills. It is also an easy game to differentiate for my students and a great tutoring activity!
To celebrate the 4th and the revisions,
Patriotic Kaboom will be 50% off July 4th & 5th!
Hello dollar deal!! :)
Click any of the images of Multiplication Kaboom above to download this game for $1 and add it to your collection for the summer or the upcoming school year. If you already own this product, please be sure to download the revised version for a more explosive and colorfully fun game!


Effective June 23, 2019, USPS is having a second rate change requiring the use of dimensional weight to determine pricing for Priority Mail®, Priority Mail Express® and Parcel Select services.
They are also changing the way dimensional weight is calculated.
What is the purpose of Dimensional (DIM) Weight rates?
To charge for the space that a package occupies on a truck instead of just the physical weight alone.
How to Calculate Dimensional Weight?
Determine if the package is larger than 1 cubic foot (1,728 cubic inches); If it below, DIM pricing does not apply.
1) Measure the Length x Width x Height (In Inches) and divide by 166 (USPS Divisor)
For example: 15” x 15” x 22” = 4,950 cubic inches/166
1) If the Dimensional Weight is greater than the actual weight of the package, you must pay the Dimensional Weight instead.
A Couple of Cost Savings Tips:
1. Use the smallest box possible that will fit your item(s).
2. Use USPS Flat Rate boxes. You pay the flat rate and the DIM weight does not apply.
For more information on the upcoming price change: https://pe.usps.com/PriceChange/Index


To understand what 1 trillion to the 10th power means, it’s essential to comprehend the concept of exponentiation. When we raise a number to a certain power, we are essentially multiplying that number by itself multiple times. In this case, you want to find out what 1 trillion (1,000,000,000,000) raised to the 10th power is.
Mathematically, it can be represented as follows:
Calculating this value can be quite overwhelming, as it involves multiplying 1 trillion by itself ten times. However, modern computers and calculators can handle such large calculations with ease.
1,000,000,000,000^10 is an incredibly massive number, and it’s challenging to grasp its magnitude without the help of scientific notation or comparison.
To put this number into perspective, consider that:
1,000 is 10 to the 3rd power (10^3).
1,000,000 is 10 to the 6th power (10^6).
1,000,000,000 is 10 to the 9th power (10^9).
So, 1 trillion (1,000,000,000,000) is 10 to the 12th power (10^12).
Raising a number to the 10th power means you’re multiplying it by itself 10 times. In this case, 1 trillion to the 10th power is:
This results in an astronomical number, often expressed in scientific notation. Scientific notation allows us to represent very large or very small numbers in a more manageable form.
1 trillion to the 10th power in scientific notation is:
1 x 10^12 to the 10th power, which simplifies to:
1 x 10^120
So, 1 trillion to the 10th power is equal to 1 followed by 120 zeros:
This number is so incredibly large that it’s challenging to comprehend. It exceeds the number of atoms in the observable universe by many orders of magnitude. It’s a number that doesn’t often appear in practical calculations and is more of a mathematical concept to help us understand the power of exponentiation and the vastness of numbers.
Certainly, let’s explore further the magnitude and some context for a number as colossal as 1 trillion to the 10th power, or 10^120.
Cosmic Scale: To put this number in perspective, it’s helpful to look at it in the context of the universe. The observable universe contains an estimated 2 trillion galaxies, and each galaxy can have billions to trillions of stars. Even with such an astronomical number of stars and galaxies, 1 trillion to the 10th power (10^120) is vastly greater.
Particle Physics: At the other extreme of scale, consider the tiniest particles in the universe. A grain of sand contains roughly 1 sextillion (10^21) atoms. Even if you were to create a stack of grains of sand, each containing as many atoms as there are in our real universe, you would still fall woefully short of 10^120.
Combinatorics: The number 10^120 is also much larger than the total number of unique chess games that can ever be played. The estimated number of possible unique chess games is on the order of 10^120 itself, which demonstrates the staggering complexity of the game of chess.
Computational Limits: From a practical standpoint, modern computers, no matter how advanced, would struggle to process or store a number as enormous as 10^120. The sheer magnitude of this number is far beyond what current technology can handle.
Mathematical Utility: Numbers like 10^120 are often encountered in abstract mathematics and theoretical physics when dealing with cosmological or quantum-scale phenomena. They are used to represent the extreme boundaries of what can be conceived mathematically.
In essence, 1 trillion to the 10th power is a number that stretches the limits of human comprehension and practical utility. It serves as a mathematical concept that helps us appreciate the vastness and complexity of the universe and the mathematical structures used to describe it. While we might not encounter such numbers in our day-to-day lives, they play a crucial role in fields like cosmology, theoretical physics, and advanced mathematics, helping us explore the furthest reaches of our understanding of the universe and its underlying principles.


Clns 82/511 - 1 - notes on statistics for physicists, revised jay orear preface these notes are based on a series of lectures given at the radiation laboratory in the. / currency data for which no sources are mentioned are compiled by the bank of japan all times are japan standard time unless noted otherwise notes. Statistical notes our staff has put together the packages below please allow up to 15 seconds for the notes to load in your browser. 129 quotes have been tagged as statistics: benjamin disraeli: ‘there are three types of lies -- lies, damn lies, and statistics’, scott dikkers: ‘statis. Chapter 1 probability theory 11 experiments and random events deﬂnition 111 in probability theory, random experiment means a repeatable process that. Notes for bock, velleman, de veaux third edition (in the order i plan to cover them) unit 1: exploratory data analysis chapter 2: data chapter 3: displaying and. United nations, department of economic and social affairs statistics division, demographic and social statistics branch 30 august 2004 briefing note on the collection. Notes for probability and statistics john kerl february 4, 2009 abstract this is my primary reference for probability and statistics: i include what i feel to be the most.
Notes on note-taking: review of research and insights for students and instructors michael c friedman harvard initiative for learning and teaching. This section provides the schedule of lecture topics and lecture notes for each session of the course subscribe to the ocw order statistics and expectations. Quiz: introduction to statistics method of literature notes test prep study guides student life. Lecture notes on probability, statistics and linear algebra c h taubes department of mathematics harvard university cambridge, ma 02138 spring, 2010.
This web site contains notes and materials for an advanced elective course on statistical forecasting that is taught at the fuqua statistics resources on the. Free statistics lectures welcome to statisticslecturescom, your top quality resource for help in statistics here you can view hundreds of free text and video-based. Notes on statistics by se van bramer march 8, 1995 1 terminology a indeterminate (random) error: evaluate with statistics b determinate (systematic) error. Homework help in statistics from cliffsnotes need homework and test-taking help in statistics these articles can help you understand the advance math concept.
Introductory statistics notes jamie decoster department of psychology university of alabama 348 gordon palmer hall box 870348 tuscaloosa, al 35487-0348. The history of statistics can be said to start around 1749 although, over time, there have been changes to the interpretation of the word statistics. 7 chapter 1 probability review basic combinatorics number of permutations of ndistinct objects: n not all distinct, such as, for example aaabbc: 6.
Notes on probability theory and statistics antonis demos (athens university of economics and business) october 2002. 1 a note on us trade statistics august 22, 2014 this note, prepared by the staff of the united states international trade commission (usitc.
A frequency distribution is a tabular arrangement of data whereby the data is grouped into different we note that the values range from 0 to statistics content. Notes on sampling and hypothesis testing allin cottrell∗ 1 population and sample in statistics, a population is an entire set of objects or units of observation of.
Lectures on statistics william g faris note the important identity in statistics the sample mean is used to estimate the population mean. Ii preface the title given these notes, and the course numbered statistics 601 at iowa state university, is advanced statistical methods one might reasonably won. Contents 5 these notes in english will closely follow mathematische statistik, by hr kunsc¨ h (2005), but are as yet incomplete mathematische statistik can be used.


A mathematical glimpse into spherical harmonics
The story of Joseph Fourier (1768-1830)
If you’ve been involved in physics or mathematics at some point in your life, you probably heard of a famous scientist by the name of Jean-Baptiste Joseph Fourier. Born in France in 1768, he lived in a very fertile time for French Science. He saw the creation of the prestigious « Ecole Polytechnique » and interacted with fellow contemporaries such as Gaspard Monge, Joseph-Louis Lagrange, Pierre-Simon de Laplace, Siméon Denis Poisson, or Adrien-Marie Legendre.
In 1798, Fourier was asked by Napoleon to accompany him as a scientific adviser on his Egyptian expedition. After a few years spent as an adventurer in the faraway land, he came back to France in 1802 and was appointed by Napoleon Prefect of Isère, in Grenoble, a city in the French Alps. Fourier accomplished his duties as a governor of the region, and meanwhile he always continued his scientific career. By an original twist of fate, it was in Grenoble that he met with a gentleman by the name of Jean-François Champollion… who later deciphered Egyptian hieroglyphics. The cold temperatures in Grenoble made a sharp contrast with Fourier’s Egyptian experience, and finally ended up taking a toll on his health. Coincidence or timely matter, Fourier became interested in topics such as the evolution of temperature inside matter, and heat flow theory. In 1822, he published his work as « The Analytical Theory of Heat ». Not only did Fourier solve many mysteries about heat flow but also did he invent, in the course of his work, an invaluable mathematical tool for scientists. This is what we call today « Fourier series » (and « Fourier transform »). Fourier discovered that he could model a mathematical signal by an infinite sum of trigonometric functions. This is a profound way to decompose and solve mathematical problems. As the car manufacturer and industrialist Henry Ford said many years later (with no relation whatsoever to Fourier), « Nothing is particularly hard if you divide it into small tasks ». Transforming something complicated into something simple is something we definitely love at Stellar. So, Fourier figured out that he could transform a big complex signal into an infinite sum of small, basic, well known signals. How smart that was! Using Fourier’s technique, many problems in maths and physics can be reduced to determining the value of coefficients that multiply basic trigonometric functions. This is a major contribution to science, with countless applications in every field, from heat flow in the 19th century to the MP3 or JPG compression algorithms that make your music and pictures fit into the memory of your iPhone in the 21st century.
« Fourier Series » in a nutshell
Not every student in science today is aware that Fourier travelled to Egypt with Napoleon. But almost everyone of them knows about of Fourier series. We can safely bet that the current reader doesn’t need any introduction either. So, what’s the principle in a nutshell? A periodic signal can be decomposed in the sum of a mean (a constant value, or harmonic of order zero), plus a sine and a cosine at the period of the signal (harmonic of order one), plus a sine and a cosine at half the period (harmonic of order two), etc. etc. until infinity. The goal of the game is to determine what coefficient needs to be put in front of each harmonic, in order to properly reconstruct the signal. It is as simple as that.
A look inside Fourier’s mind
Put your new glasses on: From 2D to 3D
Sines and cosines used in the original Fourier series are well designed when accounting for 2-dimension signals. If you can plot a signal as y in function of x on your board (or tablet), then Fourier series will do the trick. That’s fine. However, could we just think out of the box for a minute (out of the plane, rather), and extend that concept to three-dimensional signals? Last time we spoke, we introduced something called the « geoid ». In our post « Is the Earth flat? », we considered the equipotential of Earth’s gravity field, which is a 3-dimensional surface, roughly a sphere, more accurately an ellipsoid, and more realistically a « potatoid » (literal transcription of the slightly colloquial, and definitely geeky French « patatoïde »). As you saw in the previous blog post illustration, the geoid has bumps and hollows. Peaks and valleys. Well, with some sort of trigonometric functions, we could imagine that it would be possible to make a similar mathematical decomposition. And render bumps and hollows with sines and cosines at proper wavelengths, this time along latitude and longitude. Trying to mimic the Fourier series process, we could invent a set of (infinite) basic trigonometric functions, this time depending on radius, latitude and longitude instead of abscissa and ordinate, and project any spherical-looking shape on this set of functions. Well, easy! Just think about it and it’s done! Look at the set of spherical harmonic functions smiling at you:
We’ll deep diver into how « easy » that is in the next paragraph, but let’s skip the maths for one or two more minutes. With this set of functions, you now have the same property than with Fourier harmonics: you can decompose any object that looks « kind-of-spherical » into an infinite sum of basic functions, as long as you multiply each basic function by the right coefficient.
A closer look
Let’s give it a closer look now. In Fourier series, you had an « order » for each harmonic. The order would define the period (or wavelength) along the x axis. In spherical harmonics, you need not one, but two numbers, in order to take into account the two directions of waves: North-South waves in the direction of Earth’s meridians (latitude), and East-West waves in the direction of Earth’s equator (longitude). This is why you now need a « degree » and an « order » to classify each spherical harmonic function. Let’s have a look at the mathematical convention. The degree « n » is the total number of waves. The order « m » is the number of waves in longitude. The number of waves in latitude is thus « n – m ». You guess that with this definition, m is always smaller than n.
Let’s continue our quick analysis. When m is equal to zero, you have no wave in longitude. You only have waves in latitude (n waves, since n=n-m since m equals zero). We call them zonal harmonics. Theses functions with dependency in latitude only look like horizontal rings on top of another.
Zonal coefficient (n=9, m=0)
If m is equal to n, then n-m equals zero, and there’s no room left for waves in latitude. There are only waves in longitude (n or m, since n-m=0). We call them sectorial harmonics. These functions with dependency in longitude only look like vertical « orange quarters ».
Sectorial coefficient (n=9, m=9)
If m is strictly greater than zero and strictly smaller than n, then you have a mix of waves in both direction (North-South and East-West). We call them tesseral harmonics. So here we are. We have our set of functions, with our three main types: zonal, sectorial and tesseral. We have an unlimited number of possibilities for the degree n, and for each degree n, we have n+1 possibilities for the order m (from 0 to n). And then we have an unlimited number of combinations of all those harmonics. All we need to do now, in order to decompose a spherical shape into spherical harmonics, is to find the right coefficient that goes in front of each fonction.
Tesseral coefficient (n=9, m=6)
Back to the geoid
So let’s go back to our beloved « geoid ». Our application for gravity field is very simple. We want to describe the geoid as an infinite sum of harmonic functions. The goal of the game is to determine the right coefficient in front of each spherical harmonic function. It works like magic.
So, let’s go ahead and make the decomposition. Let’s take into account the first effect that we mentioned in our « Is the Earth flat? » piece: the fact that the Earth is not flat but spherical. That would be given by the first harmonic function, a.k.a. « the mean », or the sphere (n = 0 and m = 0, no wavelengths along any direction). Then, let’s take into account the second effect that we mentioned in our post, the fact that the Earth is actually not spherical, but rather flattened at the poles. There must be a coefficient for that, yes? Well, if you take degree 2 and order 0, for example, what do you get? A « zonal » coefficient, with two wavelengths in latitude (along the whole circle): one hollow at the North pole, one bump at the equator, one hollow at the South pole, and then agin one bump at the equator. Perfect, this is exactly what we wanted. We call it « J2 » in colloquial space mechanics vocabulary. Then, what about the flattening of the equator (along the longitude direction)? Well, then the (sectorial) harmonic function of degree 2 and order 2 is the one that perfectly fits the bill. You get the point. It can go on and on. For example, if you want to account for finer details and smaller wavelengths (such as the Alps mountains that were dear to Fourier), you need higher degrees and orders. How high? For example, a spherical harmonic function of degree 200 has a wavelength of 40 000 km / 200 (circumference of the Earth divided by the number of waves), i.e. 200 km (100 km large for the bump and 100 km for the hollow). Degree 200 is about the best we can do today with space geodesy missions. Of course we can do much better with in-situ measurements campaigns, but we don’t get a full global coverage of the Earth. So by now, we’re certain you get the idea behind spherical harmonics. Let’s continue our exposé with a little practice and conclude with a little maths.
Practice, practice, practice
There’s a famous quote among professional jazz musicians that says: « One day without practice, and you will notice. Two days without practice and critics will notice. Three days without practice and everyone will notice ». Needless to say that this wisdom applies to every single field of arts, crafts and science. It’s as true for jazz musicians who fly every summer from New Orleans (Louisiana), to perform in the small village of Marciac (Gers), as it is for geodesists who calculate satellite orbits all year long at the Space Center of Toulouse (Haute-Garonne). So, let’s put the practice quote into practice. Given that the Alps are roughly 1200 km wide (from East to West), and 300 km large (from North to South), what is the degree and order of the spherical harmonic required to account for them? Everyone who sends us a correct answer is eligible for a free coffee in our office! As a side note, we shall add two comments to that question. First: the coefficient in front of the spherical harmonic function can be positive, which allows us to account for either a bump or a hollow. Second: you might notice that we need an additional parameter which we didn’t mention until now (but will in the next paragraph), in order to phase the bump at the right place.
A little bit of maths
Maths. We love maths. One reason is because it allows us to say a lot of things with very few words. We could basically replace almost all of our previous speech by the single following line:
This is the formula of the gravity potential in function of latitude phi, longitude lambda and radius r. This formula may look impressive at first sight, and somehow hard to digest, but it’s actually very simple. It’s an infinite sum of spherical harmonics. n is the degree (from 0 to infinity), m is the order (from 0 to n). Cnm and Snm are constants (the coefficients that we care about). Let’s have a look at what is written inside the square brackets. The dependency in longitude is a basic sine and cosine : that makes m waves in longitude around the equator. Notice that you need both a sine AND a cosine in order to phase the bump correctly. The dependency in latitude is a little more tricky. A simple sine and cosine would not accomplish all the characteristics we want to build a set of orthonormal basis functions. Instead, the dependency in latitude is a Polynomial of the sinus of the latitude. Finding the right polynomial was the work of Adrien-Marie Legendre, and they are named after him. Legendre Polynomials are calculated using recursive formulas:
Just replace x by sin(phi) and you’re good to go. The dependency in longitude multiples the dependency in latitude. You just built a basis of orthonormal functions on which the decomposition is unique. Therefore, the only thing you now care about is the value of the set of coefficients Cnm and Snm. Find the right set of coefficients, and you have your description of the geoid.
This is it for our brief introduction to spherical harmonics. We hope you enjoyed it. There’s a lot more to dig into for curious minds and for people who will be involved with us in space geodesy and gravity missions. The main takeaway is that spherical harmonics are a smart and convenient way to describe a spherical shape like the geoid: simply by a set of coefficients. And it was all inspired by the work of Joseph Fourier, who first had the idea of such a decomposition with basic sines and cosines for 2D signals.
Well, who would have thought that, suffering from the cold in the Alps, after coming back from an expedition to Egypt with Napoleon, would yield such fertile results, that would be used two hundred years later by geodesists all around the world??
Any comments? Questions? We love to hear from you.


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Now, take advantage of plenty of learning material 24 hours a day, 7 days a week at the Math Kangaroo Media Library. Login and click "Video Solution" tab to enjoy the creative thinking behind the solutions to all the fun math problems. Click here to learn more.
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Combine numbers using math operators to reach the target number:
Combine letters to make words!
Merge tiles to reach 2048!
In "Digits", every puzzle presents you with a target number and six unique starting numbers. Your task is to employ these initial digits in mathematical calculations, aiming to craft the elusive target number. With the constraint that each of the six numbers can only be used once, the answer to each calculation morphs into a fresh number, making this game a captivating fusion of math, strategy, and creativity as you strive to conquer each numeric challenge.
Begin by breaking down the target number into its prime factors or identifying any recognizable patterns or relationships within it. This can give you insights into how to create or approach that number using the starting numbers.
Apply the operations to the starting numbers to create intermediate results. These intermediary numbers can serve as stepping stones toward your target. Keep track of your calculations and intermediate results. Since you can undo operations, be prepared to backtrack and adjust your approach if you hit a dead end or realize a more efficient path to your goal.
Keep playing and mixing up the numbers until you hit the target. Don't be afraid to try different combinations and approaches. With practice, you may develop a knack for recognizing efficient strategies and discovering new ways to reach the target number.


Math is enjoyable for some folks but children might not find it so. These younger ones have favored pastimes that they like doing which keeps them entertained. Why shouldn’t math also be entertaining? Modern working day kids devote most of their time observing videos and enjoying personal computer game titles.
You can’t don’t forget the 1000′s of algebra principles and equations with out truly understand what every equation truly means. What is the cause driving every single algebra notion.
Before you can change them constructive, you have to take away the unfavorable views in your head 1st. The only way to do this is by means of subliminal programming. You can do this by in search of subliminal resources created to get rid of the anti-math ideas lurking in your unconscious and replace them with good kinds.
First you will want to appear up with some entertaining math reality games. I have discovered that performing a little bit of on the internet analysis in progress will help you to uncover some on-line ideas on how you can aid your little one learn their math facts. Following you will want to print out and hang up some math fact charts all around the area. Start off with the easy ones and perform your way up via the checklist. Don’t forget that the much more entertaining that your little one has the simpler it will be for them to learn the math information. After they have been able to grasp these information they will have a a lot easier time obtaining through their math perform.
You can enrich your kid’s finding out and help him advance at his possess rate so he masters each and every subject. There are courses supplied on the web that will grade and give guidance as your child performs by way of their plan. You will uncover software programs set up as video games that guide a youngster via different mathematical responsibilities. There are on the internet web sites that instruct math via sports activities and other subjects that are fascinating to young children and teenagers.
Math can help you create artwork. Not only will you want to get the appropriate measurements and colour mixtures, math is used in virtually each and every factor of art. Key universities offer an complete course called “Mathematics in Artwork and Architecture”. Techniques like tiling, tessellation, standpoint, sample, and symmetry are some of the concepts utilised in artwork that demand the use of math. Make oneself and your planet more lovely. http://support.acadsoc.com/.
Math online games are a amazing way to stimulate your youngster’s capabilities. The interaction offered makes your mind perform in various ways, leading to it to produce and appear at problems in a new gentle.
These on the web tutorial internet sites that instruct math can also demonstrate you how to enhance the way you handle math in day-to-day programs. One area that affects a lot of individuals is counting change. Numerous individuals can not give adjust back again without the use of a register. These applications can provide you with a way to overcome this.


The Math Success Center, located in MA 130, is a lab where you can study with friends, form study groups, and have your math questions answered by highly-qualified peer tutors. Current math textbooks as well as math resources are available for use in the Center. The Math Success Center is also equipped with computers that support PLATO, MyMathlab, and Webassign for your convenience.
The Math Success Center will be open the following hours:
Monday - 9:30 a.m. to 6 p.m.
Tuesday - 9:30 a.m. to 6 p.m.
Wednesday - 9:30 a.m. to 6 p.m.
Thursday - 9:30 a.m. to 6 p.m.
Friday - 9:30 a.m. to 1:30 p.m.
Click here to view the Math Success Center Tutor Schedule for Fall Semester.


Hi everyone and welcome back! I hope you had a lovely summer break with family and friends! 2021 promises to be another great year of learning for your music journey. We have our annual studio concerts and mathematical music theory workshop(s) this year. Most of you are preparing for examinations and so the concerts are a great way to test drive your work with family, friends and new acquaintances before the other big day of the exam. There’s plenty more news to share with you but for now please send in your new Studio Policies signed and your Student Information Sheet filled out so that I can update your student files. Every year I feel very proud to see how much each of you has grown in learning, practical performance skills and deepened knowledge of your interactions with music, mathematics and acoustics. We hear so much about student engagement in learning and through your studies in mathematical music theory each of you demonstrate genuine engagement through representing, discussing and applying your embodiment of acoustics, music and mathematics. Congratulations to you all and looking forward to teaching and learning with you this year.
Congratulations students on another year of excellent learning, performances and dedication to your music studies. Each of you in your own way achieved new levels of understanding and application of mathematical music theory. I observe so many benefits to your learning through mathematical music theory in your performances and music theory examination preparation, band achievements, school invitations to perform at special events and excellent results for school music examinations. As you all grow in confidence, I can’t help feeling pride in each one you and your efforts this year. Great work!
Also, a special thank you to our wonderfully supportive families who encourage regular practice and to the parents and carers who are also joining in with the lessons especially with our youngest students! Thanks also to the families who have been working together to perform music at home in ensembles. I look forward to working with all you in 2021 to continue to develop your personalised projects.
Next year we will have our annual concerts and mathematical music theory workshops – see the Events page for updates.
Wishing everyone a very happy Christmas and a blessed New Year 2021!
29 November 2020
Congratulations on a great effort at Sunday’s concert! I was very proud of everyone and each of you made the afternoon a very memorable occasion. Your love of music and learning music through mathematical music theory was reflected through your wonderful performances. Thank you for sharing your musical journey with us through performing your music at An Afternoon of Music. Also, families and friends your enthusiasm and support for our students’ music education is a constant source of inspiration for me as their teacher. I love talking with you all about the benefits you observe to your children for learning music through a psychoacoustic approach to mathematical theory. Finally, keep up the good work visualizing beat-class theory through ski-hill and cyclic graphs to annotate scores and learn about performance.


Top 10 Tips to Overcome Math Phobia and Excel this Board Exams Season
Identify the Source of Anxiety
View math not as a source of fear but as a puzzle to be solved
Begin with basic concepts you feel comfortable
Practice is key to overcoming math anxiety.
strive to understand the underlying concepts and how they connect.
Understand the practical applications of math in everyday life.
Use flashcards to practice basic facts and formulas.
Develop Problem-Solving Strategies
Be patient with yourself, celebrate your progress


Particularly those Algebra questions that involves complex manipulations such as factoring.
Allow me to share with you some of the best online calculators which you can use as a guide to solve any quadratic equations.
But before that, I have a few pointers for you~:
if you are using it for homework, you should attempt the algebraic problems at lest once, by your own effort. Homework is meant to be practice not copy.
|You have been warned|
First, the answers they provide may not always be correct. Second, you wont be able to learn anything...
Third, I'm sure you wont be able to use these calculators during your class test/exam.
Therefore! Be warned~ Do not over rely on these calculators to do your HOMEWORK~
Else, you may end up having trouble dealing with even the simplest polynomials.
Though, there are plenty of these powerful calculators online, below are the better few which I tested and recommend.


- GeoGebra in the teaching and learning of Mathematics in the different modalities and educational levels
This thematic group will include communications and workshops that present experiences and reflections on the use of GeoGebra in the classroom.
- GeoGebra and research.
Thematic group will be dedicated to studies or research work whether qualitative or quantitative around GeoGebra or its use in mathematical education.
- GeoGebra in teacher training in Mathematics.
This thematic group will include all aspects related to teacher training, both
specific to GeoGebra, and its integration into other aspects of teacher training
- GeoGebra in STEAM
Experiences and educational proposals that address the use of GeoGebra as another resource in other areas or in STEM or STEAM projects will be acepted as submisions to this thematic group.
- The future of GeoGebra
All aspects related to the future that users demand for this software will be included in this thematic group, namely, the use of GeoGebra in augmented reality, artificial intelligence, automatic demonstration, etc.
- GeoGebra in other fields not necessarily related to teaching.
Clearly, the use of GeoGebra goes beyond the educational field, so this nucleus will collect all the actions related to the use of GeoGebra in other fields, such as scientific dissemination, mathematical tourism, 3D printing for fields such as medicine, engineering, design, etc.


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I placed an order to sell 511 KARMATOKEN at price 0.00139 XCP per KARMATOKEN.
The highest bid was at 0.00140 XCP per KARMATOKEN and the amount was higher than the 511 I wanted to sell.
The order matched but only for 510.99999286 KARMATOKEN. I am left with 0.00000714 KARMATOKEN unsold.
Is there a bug in Counterwallet ???.
In the GUI I specified numbers of tokens to sell and the price, but CW needs to convert these to implied numbers of tokens to give and take.
I guess the CW makes some simple divisions etc, and somewhere inverts these, and then a rounding error may apply.
If I do the simple math, I should want exactly 0.71029 XCP. See there’s less than eight digits, so no rounding would be needed here.
Since the match is at a higher price, I did receive more XCP, as expected, 0.7154051 XCP.
Possibly the bug is with the Counteparty core code ???
Blockscan claims I am selling 511 KARMATOKEN (0.00000714 KARMATOKEN remaining). Hopefully this is a rounding error on Blockscan’s part. These remaining tokens are not left in the order book. Either the core code fails to match the remaining 0.00000714 tokens (it’s a bug, both the buyer and the seller would want this to match) and it cancels the order for the leftover OR as speculated above, the bug lies within Counterwallet.


History of Math Inventions
Mathematics, as a science, has quite a number of branches. The main ones are algebra, calculus, and geometry.
Note that this discipline is not an invention; in fact, laws of science and discoveries are not considered inventions. Inventions are material processes, whereas a discipline like mathematics has a history behind it. Math also has solid relationships with inventions, and mathematical instruments are actually regarded as inventions.
Mathematics is an organized science that did not exist up to around 600 to 300 B.C. This was the classic Greek period, and prior civilizations significantly impacted the formation of math.
This article will give a comprehensive rundown of math’s history.
What is Mathematics
Let’s begin by understanding what math is (and why you have to deal with math homework everyday), beyond what we already know. Well, mathematics is the study of the relationship between properties, magnitudes, and quantities. Math also studies the logical operation of the entities mentioned above.
Historically, it was considered as the science of quantity; quantity representing magnitudes (geometry), numbers (arithmetic), or the generalization for the two (algebra).
For some, mathematics boils down to a search for patterns.
In the 19th Century, the mathematical scope broadened to include mathematical logic and, as a result, referred to as the science of drawing necessary conclusions. Even so, there are many that think this definition is too restrictive.
Today, mathematics as a discipline covers analysis (calculus), geometry, number theory, algebra, mathematical logic, and set theory, as well as applied mathematics like statistics and probability theory.
There’s also a perplexing array of specialized mathematical fields of study that include;
- Order theory
- Sheath theory
- Group theory
- Differential geometry
- Graph theory
- Knot theory
- Fractal geometry
- Complex analysis
- Catastrophe theory
- Game theory
- Singularity theory
- Chaos theory
- Model theory
- Control theory
- Functional analysis
- Measure theory
- Category theory
- Complexity theory
As we’ve already seen, the history of mathematics as about as old as humanity itself. It has been a fundamental discipline since antiquity. Through math, there have been endless advancements in engineering, science, and philosophy.
Over the years, math has evolved from simple counting, calculation and measurement, the systematic study of motions, shapes, and physical objects, through imagination and logic, application of abstraction, to the complex and abstract discipline we know today.
The history of math is a long and impressive one because it begins from early man to the agrarian revolution, through the revolutionary developments of ancient times.
This history is believed to have began from Mesopotamia and Egypt during the agrarian revolution. This gospel was then carried to Ancient Greece during their revolutionary development.
Later, the baton was passed to the East, specifically to India, China, as well as the medieval Islamic empire.
From here, mathematical innovation moved back to Europe during the renaissance. As you know, some of the greatest artists and painters of all time come from this middle age.
In the 17th and 18th Centuries, Europe set the stage for the increased complexity in math as are witnesses in the 19th and 20th Centuries. The 20th Century saw quite audacious and devastating discoveries.
This drastic change was as a result of a new wave of revolutionary developments in mathematical abstraction. As the story of mathematics unfolds, we realize that it’s a continuity of several linked sections in different centuries. It’s like different chapters of a book with each proceeding one linking back to the last.
Mathematics is a fundamental part of our lives today so we’re grateful that it was formed. Always remember that this is not an invention, but rather a stand-alone invention that has relationships with other inventions and sciences. Through mathematics, there has been many innovations in the fields of engineering, philosophy as well as science.


\[ x^6 + y^6 + z^6 - 1 \]
- degree 6
- not really a cube
- basically a sphere
The `cube' is not really a cube. It has no corners, but instead is smooth. Taking the power of the variables to infinity produces a cube, and is related to the various metrics for distance.
Random projection is all good, and no need for a custom bounding sphere. Just let 'er rip.
Funnily enough, if you decompose a sphere you get a regular polyhedron, but you don't with this.


Using Mass Percent in Calculations Sometimes you may want to make up a particular mass of solution of a given percent by mass and need to calculate what mass of the solute to use. Using mass percent as a conversion can be useful in this type of problem. The
5. In a chemical analysis to test the purity of a bottle of sodium bromide, a solution containing 1.17 g of sodium bromide was reacted with an excess of dimercury (I) acetate solution. The dry precipitate had a mass of 2.73 g. Calculate the percent yield for the
Solution for Calculate the nuer of photons of light with a wavelength of 3000 pm that provide 2 J of energy. Ask a Homework Question! Subscribe to bartleby learn and ask 30 homework questions each month. Subject matter experts are on standby 24/7 when you
50.0 cm3 of a 1.00 mol dm–3 solution is formed, and the temperature rise, ûT, is 15.0 C. (i) Calculate the energy transferred, in joules, for this process using: Energy transferred in joules = volume of solution formed u 4.2 u ûT (1) (ii) Calculate the enthalpyH in kJ.
Start with the equation for the reaction of interest. The coefficient for each reactant enables you to calculate the ratio of the ingredients that are needed. Compare the amount you have of each reagent to the amount required. The one that falls b
calculate the mass percent of the following solutions 1.25 g CaCL2 in 35.0 g H2O I am not sure if i have the correct set up (1.25g CaCl2/1.25 g CaCl2 + 35.0 g H20)*100 asked by ashley on May 14, 2014
10/7/2015· In this video we will discuss solution composition. More specifically we will discuss one way of looking at solution composition called mass percent. We will
50 g / 100 g = 0.5 In order to get the mass of CO 2 we need to times the RFM of CO 2 by 0.5 (or divide by 2) because of the ratio of CaCO 3 to CO 2: 44 g x 0.5 = 22 g 22 g of CO 2 are obtained from decomposition of 50 g of CaCO 3.
Let''s address the question for both percent concentration by mass and for percent concentration by volume. Percent concentration by mass is defined as the mass of solute divided by the total mass of the solution and multiplied by 100%. So, #c% = m_(solute)/(m
Calculate the actual mass of silver that can be produced in this reaction if 100.0 g of silver chloride is converted to silver metal. 2AgCl(s) y asked by Anonymous on June 1, 2008 Chemistry help What mass of silver chloride can be prepared by the
14. When 1.5 g of a non-volatile solute was dissolved in 90 g of benzene, the boiling point of benzene raised from 353.23 K to 353.93 K. Calculate the molar mass of the solute. (Kb for benzene = 2.52 kg mol-1) Answer: 15. Calculate the freezing point of the 2 H 6
Balance the reaction of AgNO3 + CaCl2 = AgCl + Ca(NO3)2 using this chemical equation balancer! Instructions To balance a chemical equation, enter an equation of a chemical reaction and press the Balance button. The balanced equation will appear above.
(i) Calculate the amount, in moles, copper(II) carbonate will react with 0.200 mol hydrochloric acid. (1) (ii) Calculate mass, in grams, this amount of copper(II) carbonate. (2) (iii) Calculate ofcarbon gas at room and that will be in reaction. (The vohlme of
In this example, a set of sodium chloride standards was prepared with mass percent compositions ranging from 5 – 25% of solute in solution. After plotting the data, apply a linear trend-line, and determine the equation, where y is density and x is mass percent composition.
(d) Calculate the mass of Ca(OH)2 that can be formed from the reaction of 10.0 g of Ca3N2 with an excess of H2O. Solution: moles Ca3N2 = 10.0 g Ca3N2 x = 0.06743 mole 1 mole Ca3N2 148.3 g Ca3N2 moles Ca(OH)2 = 0.06743 mole Ca3N2 x = 0.2023 mole 3
19) Write the name for Ca3(PO4)2. A) calcium (III) phosphite B) calcium phosphate C) tricalcium phosphorustetraoxide D) calcium phosphite E) calcium (II) phosphite 19) 20) Determine the name for CoCl2·6H2O. Remeer that Co forms several ions. A) cobalt (I
Sum = molar mass = 18.0152 g H 2O per mole Chapter 3 Calculation of Molar Masses Calculate the molar mass of the following Magnesium nitrate, Mg(NO 3)2 1 Mg = 24.3050 2 N = 2x 14.0067 = 28.0134 6 O = 6 x 15.9994 = 95.9964 Molar mass of Mg(NO 3)2
mass = volume x density 2. Determine concentration in percent by mass of the solute in solution. Change to the decimal equivalent. 3. Calculate the molar mass of the compound, MM. 4. Multiply mass (step 1) by mass % (step 2) and divide by the whole 5.
2/12/2012· Mass = 10g Mr = 100.08 Then use Moles = mass / molecular mass Then, use the gas volumes calculation - moles = volume / 24 You must rearrange the formula to give volume = moles x 24 This will then give you the volume of CO2.
9/10/2017· 10 g sample of a mixture of CaCl2 and NaCl is treated to precipitate all the calcium as CaCO3. this CaCO3 is heated to convert all the Ca to CaO and final mass of CaO is 1.68g the mass % of CaCl2 in original mixture is ?
15/3/2015· Silver Nitrate + Calcium Chloride - Balanced Molecular and Net Ionic Equation - AgNO3 + CaCl2 - Duration: 5:28. The Organic Chemistry Tutor 17,739 views
10.5 M 21.0 M 10.5 M [Na +] = 21.0 M, [SiO 3 2-] = 10.5 M 2. What is the concentration of each ion in the solution formed when 94.5 g of nickel (III) sulphate is dissolved into 850.0 mL of water? Molarity = 94.5 …
Calculate the minimum mass of iron needed to displace all of the copper from 50 cm3 of copper(II) sulfate solution. The concentration of the copper(II) sulfate solution is 80 g CuSO4 per dm3. Relative atomic masses (Ar): O = 16; S = 32; Fe = 56; Cu = 63.5 .
Limiting reagent is the reactant which limits the progress of a chemical reaction. If two atoms must coine at a 1 to 1 ratio, but there is an unequal amount of 1 atom, then the reaction will stop when the atom with less quantity runs out.
6/3/2017· Here''s how you can do that. Your strategy here will be to use the molarity and volume of the solutions to figure out how many moles of salt were dissolved to make the solution. Once you know that, you can use the chemical formula of the salt to figure out how many moles of chloride anions, "Cl"^(-), will be produced in each solution. I''ll show you how to solve points A) and B) and leave the


How the budget calculator does calculations...
The budget calculator is designed to help you make estimates of your income. An estimate is not accurate but a very good guess. It gives you a rough idea of how much money you have. If the amount of money you get and the amount of money you owe are very similar, you will have to calculate a more accurate figure.
To estimate how much you get a week you may have to round your pay per hour to an easier number.
If you earn £8.91 an hour, you would estimate this at £9. If you worked 21 hours in a week, you would estimate this at 20 hours. An easy way to work out how much you get a week is to multiply 9 by 10, which is 90 twice, giving a figure of £180. (9 x 10 = 90) (90 x 2 = 180)
If your rent is £48 you would estimate this at £50, and if you put £6 on your electricity key you would estimate this at £5. This means you would owe £55. (50 + 5 = 55)
The calculator assumes a week has 7 days and the length of a month is the average number of days in a month. (365 ÷ 12 = 30.42)
It isn’t perfect but it is quick!


Boethian Number TheoryA Translation of the De Institutione Arithmetica (Studies in Classical Antiquity)
- 200 Pages
- January 1983
- 0.34 MB
- 1942 Downloads
History of mathematics, Mathematics, Arithmetic, History & Philosophy, Early works to 1900, Mathematics / History, Mathematics-Arith
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Boethian Number Theory: A Translation of the de Institutione Arithmetica (with Introduction and Notes) (Studies in Classical Antiquity): Masi, Michael: : Books. See Cited by: 4.
Boethian Number Theory book Number Theory: A Translation of the De Institutione Arithmetica (Studies in Classical Antiquity) by Michael Masi () Paperback out of 5 stars 1 rating See all 4 formats and editions Hide other formats and editions5/5(1).
Boethian Number Theory: A Translation of the De Institutione Arithmetica (with Introduction and Notes) Volume 6 of Studies in classical antiquity, ISSN Authors.
Anicius Manlius Severinus Boethius, Boethius, Michael Masi. Editor.
Details Boethian Number Theory PDF
Clear rating. 1 of 5 stars 2 of 5 stars 3 of 5 stars 4 of 5 stars 5 of 5 stars. Boethian Number Theory: A Translation Of The De Institutione Arithmetica (With Introduction And Notes) by. Boethius, Michael Masi (Translator) really liked it avg rating — 2 ratings — published Publication Date: 1 January ISBN: Cited by: 7.
Boethian Number Theory的话题 (全部 条) 什么是话题 无论是一部作品、一个人，还是一件事，都往往可以衍生出许多不同的话题。. Modern number theory, according to Hecke, dates from Gauss's quadratic reciprocity law. The various extensions of this law and the generalizations of the domains of study for number theory have led to a rich network of ideas, which has had effects throughout mathematics, in particular in algebra.5/5(1).
number theory: free download. Ebooks library. On-line books store on Z-Library | B–OK. Download books for free. Find books. Buy Boethian Number Theory: A Translation of the De Institutione Arithmetica (Studies in Classical Antiquity) by Masi, Michael (ISBN: ) from Amazon's Book Store.
Everyday low prices and free delivery on eligible orders. mation about number theory; see the Bibliography. The websites by Chris Caldwell  and by Eric Weisstein  are especially good.
To see what is going on at the frontier of the subject, you may take a look at some recent issues of the Journal of Number Theory which you will ﬁnd in any university library. "The Boethian Wheel" is a theme throughout the book, which won the Pulitzer Prize for Fiction in  Peter Glassgold translated Boethius's poems on the consolation of philosophy "out of the original Latin into a diverse historical Englishings diligently collaged," (Sun and Moon Press, ).Born: c.
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ADRome, Kingdom of Odoacer. Boethian Number Theory: A Translation of the De Institutione Arithmetica (Studies in Classical Antiquity) (Studies in Classical Antiquity) Boethius Out of Stock. Another interesting book: A Pathway Into Number Theory - Burn [B.B] The book is composed entirely of exercises leading the reader through all the elementary theorems of number theory.
Can be tedious (you get to verify, say, Fermat's little theorem for maybe $5$ different sets of numbers) but a good way to really work through the beginnings of. This book is written for the student in mathematics. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the methods that are used.
We have avoided that style which gives a systematic development of the apparatus and have used instead a freer style, in which the problems and the methods of solution are closely interwoven. Download PDF Boethius The Principles Of Music ebook full free. Boethius The Principles Of Music available for download and read online in pdf, epub, mobi.
Boethian Number Theory. Release on This book offers an important new perspective on the Western tradition of musical aesthetics through an examination of Anicius Boethius and. Facts is your complete guide to Number Theory, An Introduction to Mathematics.
In this book, you will learn topics such as as those in your book plus much more. With key features such as key terms, people and places, Facts Number Theory (Dover Books on Mathematics) - Kindle edition by Andrews, George E. Download it once and read it on your Kindle device, PC, phones or tablets.
Use features like bookmarks, note taking and highlighting while reading Number Theory (Dover Books on Mathematics)/5(56). Boethius's Arithmetic was based on the work of Nicomachus and, although this gave it a strange character by the mathematical standards of today, it taught medieval scholars about Pythagorean number theory.
Boethius was one of the main sources of material for the quadrivium, an educational course introduced into monasteries consisting of four. Boethian Number Theory: A Translation of the De Institutione Arithmetica (Studies in Classical Antiquity) by Michael Masi ()5/5.
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Description Boethian Number Theory PDF
Boethian number theory: a translation of the De institutione arithmetica, with introduction and notes in SearchWorks catalog. prerequisites for this book are more than the prerequisites for most ele-mentary number theory books, while still being aimed at undergraduates.
Notation and Conventions. We let N = f1;2;3;gdenote the natural numbers, and use the standard notation Z, Q, R, and C for the rings of integer, rational, real, and complex numbers, respectively.
Elementary Number Theory (Dudley) provides a very readable introduction including practice problems with answers in the back of the book. It is also published by Dover which means it is going to be very cheap (right now it is $ on Amazon).
It'. (13) De top. diff. is a fairly short treatise, but it is Boethius's advanced book on the subject; it is written in a concise, even crabbed style, and it clearly presupposes acquaintance with the subject matter.
In contrast, ICT is Boethius's elementary treatise on the Topics. Boethian Number Theory. Number Theory is more than a comprehensive treatment of the subject. It is an introduction to topics in higher level mathematics, and unique in its scope; topics from analysis, modern algebra, and discrete mathematics are all included.
The book is divided into two parts. Part A covers key. Boethian Number Theory A Translation of the De Institutione Arithmetica (with Introduction and Notes) Volume 6. This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory.
Topics include: Compositions and Partitions; Arithmetic Functions; Distribution of Primes; Irrational Numbers; Congruences; Diophantine Equations Author: Leo Moser. Boethian number theory: a translation of the De institutione arithmetica (with introduction and notes).
ANALYTIC NUMBER THEORY | LECTURE NOTES 3 Problems Siegel's Theorem * Some history The prime number theorem for Arithmetic Progressions (II) 2 38 Goal for the remainder of the course: Good bounds on avera ge Problems The Polya-Vinogradov Inequality Problems Further prime.
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued mathematician Carl Friedrich Gauss (–) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of.
Boethian Number Theory: A Translation of the de Institutione Arithmetica (with Introduction and Notes) (Studies in Classical Antiquity) by Michael Masi ISBN (). Algebraic number theory involves using techniques from (mostly commutative) algebra and nite group theory to gain a deeper understanding of the arithmetic of number elds and related objects (e.g., functions elds, elliptic curves, etc.).
The main objects that we study in this book are number .Neuware - From the reviews: '.a fine book. treats algebraic number theory from the valuation-theoretic viewpoint. When it appeared in it was a pioneer. Now there are plenty of competing accounts.He wrote a very inﬂuential book on algebraic number theory inwhich gave the ﬁrst systematic account of the theory.
Some of his famous problems were on number theory, and have also been inﬂuential. TAKAGI (–). He proved the fundamental theorems of abelian class ﬁeld theory, as conjectured by Weber and Hilbert. NOETHER.
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Researchers at the Department of Philosophy, Linguistics and Theory of Science at the University of Gothenburg, Sweden, have created a computer program that scored up to 150 on specific portions of an IQ test: identifying patterns in pictures and number sequences. The research group has recently started collaborating with the Department of Psychology at Stockholm University, with a goal to develop new IQ tests with different levels of difficulty. IQ tests include progressive matrices, which test the ability to see patterns in pictures, and number sequences, which test the ability to see patterns in numbers.
For Claes Strannegård, researcher at the Department of Philosophy, Linguistics and Theory of Science, this was a reason to try to design “smarter” computer programs. “We’re trying to make programs that can discover the same types of patterns that humans can see,” he says.
The research group have integrated a mathematical model that models two types of human-like problem solving. They developed two programs. One solves progressive matrices. It scores IQ 100 and has the unique ability of being able to solve the problems without having access to any response alternatives. Another program specializes in number sequences and is able to ace the tests, implying an IQ of at least 150 for that class of problems.
“Our programs are beating the conventional math programs because we are combining mathematics and psychology,” says Strannegård. Our method can potentially be used to identify patterns in any data with a psychological component, such as financial data. But it is not as good at finding patterns in more science-type data, such as weather data, since then the human psyche is not involved.” We have developed a pretty good understanding of how the tests work. Now we want to divide them into different levels of difficulty and design new types of tests, which we can then use to design computer programs for people who want to practice their problem solving ability,” says Strannegård.


How Can Parents with Math Anxiety Help Their Kids Learn Math?
When I was in school, learning math meant nerve-wracking timed practice drills and intimidating worksheets with endless rows of numbers. My dislike of math turned to math anxiety in fourth grade, when the teacher tested us on multiplication facts orally, in front of the class. I would forget everything I had studied and return to my seat, humiliated by my failure. I never did memorize the multiplication tables. After a while, I began to believe that I was just “bad at math.” Like an estimated 10 to 20 percent of adults, I have math anxiety.
If you have math anxiety and are considering an online school for your child, you may worry about helping your child succeed in math. You’re smart to think about it, because studies prove that parents may unknowingly pass math anxiety to their kids, negatively impacting a student’s achievement in math. Fortunately, with some effort and the tips below, you can overcome this obstacle so that you and your child can succeed in mathematics together.
Your Best Resource
When beginning virtual school, even parents who don’t have math anxiety wonder if they’ll be able to support their children’s mathematics learning. After all, it may have been many years since you last tackled a word problem or solved for x! Luckily this concern is easily put to rest, because with this educational method, unlike traditional homeschooling, a virtual school family has access to a wonderful resource: real, live teachers.
For example, at Connections Academy® online public schools and International Connections Academy online private school, certified online teachers meet periodically with students in the virtual classroom to teach new concepts or to work with small groups of students who need assistance with more challenging skills. They are available by phone or email to help parents, Learning Coaches, and students when support is needed. These experienced online teachers can answer questions, provide feedback on schoolwork, and connect families to additional study resources.
Adjust Your Math Attitude
Parents can also help their children do well at math by encouraging them. But if you have math anxiety, this may require some big adjustments to your own attitude. Here are some thought-provoking hints to help you present a more positive outlook about learning math:
- Choose a growth mind-set. When talking with your child, remind him or her that no one is “born a math person.” People become good at math by making the effort needed to learn. Your child (and math-anxious adults like you!) can get better at math by practicing math skills.
- Recognize mistakes as part of learning. Some kids (and adults) are perfectionists who hate to be wrong! But educators have realized that asking children to solve problems through trial and error can be a valuable teaching approach. Encourage your child to persevere! After the problem is solved, praise your child for the effort made. Ask your child to explain the reasoning used, or what he or she learned from the attempts that didn’t work.
- Consider more than one approach. Today, math educators sometimes encourage students to suggest two or more ways to solve a math problem. Asking open-ended questions like this invites students to pursue their own natural curiosity. Answering the questions involves creative and critical thinking, which makes math more interesting and also encourages discussions that enhance math fluency. Give it a try with your child!
- Let your kids see you do your “homework.” Be sure to let your children witness how you use math in real life. Whether you are balancing a checkbook, estimating how much a home-improvement project will cost, or adding fractions while doubling a recipe, you can turn that experience into a teaching moment.
- Watch your words! Any time you say, “I’m so bad at math,” “I hated math class,” or “Math is so hard!”, you can have a negative effect on your child’s attitude—and achievement. It’s okay to tell your child that you struggled with math in school, but try to balance this with examples of how your worked hard and overcame a problem, or how you’ve learned to manage mathematical tasks like budgeting or calculating interest rates in your adult life or career. Kids notice when their parents are lifelong learners!
Helping your child succeed in a tough subject that gave you difficulties when you were younger may have its challenges, but the rewards are worth it. With a shiny new attitude about math and the support of your online math teacher, you can help your child build strong math skills while developing confidence and a can-do attitude that will last a lifetime.
How have you overcome math anxiety or supported your child’s math fluency? Share your brightest ideas in the comments!