For each FFT bin (i.e. range of frequencies), the maximum value from all of the frames is kept as the feature. Continuing with the example above, we throw away 1/2 of every FFT (as it's simply a mirror image of the other half). We also throw away the bin at 0 Hz (as we filter out the DC bias anyway when we subtracted the mean), but we keep the Nyquist bin. As a result, we end up with 8 usable bins from each of our 16-point FFTs. For each bin, we find the maximum value from our 6 FFTs that we computed (in that particular bin). So, the number of features would be 8.


Math Master Puzzles & Riddles Latest version with unlimited money for android free download (com.divinecodeproductions.MathMaster)
With a blend of logical problems, Math Master Puzzles & Riddles will raise your IQ. Challenge yourself with several levels of math games to push your mental boundaries. It is well known that mathematics has helped humans perform basic tasks since its inception. The exercises that you will find in this game will help in training your brain!
— Math Master Puzzles & Riddles features:
1. Minimalist design with 2 themes (dark & light)
2. Weekly updates with new puzzles/riddles and features
3. 300+ unique puzzles/riddles/quizzes in different templates
4. Global ranking across all devices
5. You have the possibility to submit problems directly in the game
6. Improved stats to track your progress
7. Multilingual support (English, Romanian, French, Italian, Portuguese, Spanish and more will come soon)
8. We are always receptive to new ideas
9. We are always improving the game with new features
10. Play offline – no connection required
11. Detailed solution for each puzzle
12. Step by step instructions on how to play
— Benefits of playing puzzles & riddles games
1. Logical puzzles and math games will improve attention and focusing
2. Brain games, like IQ tests, improve memory and perception abilities
3. Educational games can help you discover your true potential in school and in everyday life
4. Brain games stimulate different cognitive functions, such as executive function and processing speed
5. Logical puzzles will manage stress in an enjoyable way
6. Problem solving and logic skills will be enhanced
7. Practice tool to crack competitive exams
8. You’ll develop new synapses with every new math quiz you solve
— Who can play?
Everyone! Basic and complicated mathematical procedures learnt in school can be used to answer any query. For complicated and cognitive solutions, addition and subtraction are usually sufficient. Some problems will require addition subtraction multiplication and division operations in a specific order. Cognitive riddles are the type of puzzles that will pique the interest of intelligent people. Players with excellent analytical thinking abilities perceive the pattern in logical puzzles and arithmetic games at a different level. Remember, you may face arithmetic operations like addition, subtraction, multiplication, division or statistics like average, mean, median or complex math concepts like sequence and series.
— How to play?
You’ll solve the numerical relationships in the geometrical figures and then fill in the missing numbers at the end. Chase the question mark and try to find a suitable answer that will be able to replace it. If your answer is good, you will receive another question and so on. For each correct answer, you will gain points. If you manage to solve multiple questions in a row, you will be rewarded with a bonus! Players with a strong analytical thinking ability will recognize the math pattern immediately.
— Submit problems in the game
You will find a separate page with all the details. First you will select the template, then you will fill the missing numbers and add specific hints and description of your algorithm. We will receive your problem and evaluate it. If it will be approved, you will find it in the next update having the author on top of it. Everything is simple and intuitive.
— Our goal
To become the biggest community where you can find all the interesting puzzles, riddles and quizzes! Math is universal so everyone is invited, no matter the age or language!
— Future updates
Math Master Puzzles & Riddles is constantly improving based on your feedback. We are always open to new ideas, so please don’t hesitate to share your thoughts with us! New languages and more interesting problems will come soon!
— Important mentions
This app will always be a free game so that anybody interested in math games or logical puzzles can play. You can always donate or watch ads in order to support the development!


Pressure applied at any point upon a confined liquid is transmitted undiminished in all directions (Fig.1).
This means that when more than one hydraulic cylinder is being used, each cylinder will lift at its own rate, depending on the force required to move the load at that point (Fig. 2).
Cylinders with the lightest load will move first, and cylinders with the heaviest load will move last (Load A), as long as the cylinders have the same capacity.
To have all cylinders operate uniformly so that the load is being lifted at the same rate at each point, either control valves (see Valve section) or Synchronous Lift System components (see Cylinder section) must be added to the system (Load B).
The amount of force a hydraulic cylinder can generate is equal to the hydraulic pressure times the “effective area” of the cylinder (see cylinder selection charts).
An RC-106 cylinder with 14,5 cm2 effective area operating at 700 bar will generate what force? Force = 7000 N/cm2 x 14,5 cm2 = 101500 N = 101,5 kN
An RC-106 cylinder lifting 7000 kg will require what pressure? Pressure = 7000 x 9,8 N ÷ 14,5 cm2 = 4731,0 N/cm2 = 473 bar.
An RC-256 cylinder is required to produce a force of 190.000 N. What pressure is required? Pressure = 190.000 N ÷ 33,2 cm2 = 5722,9 N/cm2 = 572 bar.
Four RC-308 cylinders are required to produce a force of 800.000 N. What pressure is required? Pressure = 800.000 N ÷ (4 x 42,1 cm2) = 4750,6 N/cm2 = 476 bar.
Remember, since four cylinders are used together, the area for one cylinder must be multiplied by the number of cylinders used.
A CLL-2506 cylinder is going to be used with a power source that is capable of 500 bar. What is the theoretical force available from that cylinder? Force = 5000 N/cm2 x 366,4 cm2 = 1.832.000 N = 1832 kN
Use this formula to determine either force, pressure or effective area if two of the variables are known.
Cylinder Oil Capacity
The volume of oil required for a cylinder (cylinder oil capacity) is equal to the effective area of the cylinder times the stroke*.
An RC-158 cylinder with 20,3 cm2 effective area and 200 mm stroke requires what volume of oil? Oil Capacity = 20,3 cm2 x 20 cm = 406 cm3
An RC-5013 cylinder has an effective area of 71,2 cm2 and a stroke of 320 mm. How much oil will be required? Oil Capacity = 71,2 cm2 x 32 cm = 2278,4 cm3
An RC-10010 cylinder has an effective area of 133,3 cm2 and a stroke of 260 mm. How much oil will it require? Oil Capacity = 133,3 cm2 x 26 cm = 3466 cm3
Four RC-308 cylinders are being used, each with an effective area of 42,1 cm2 and a stroke of 209 mm. How much oil will be required? Oil Capacity = 42,1 cm2 x 20,9 cm = 880 cm3 for one cylinder. Multiply by four to obtain the required capacity: 3520 cm3
* Note: these are theoretical examples and do not take into account the compressibility of oil under high pressure.
TECHSYSTEM Krzysztof Perszke
Realizuje projekt dofinansowany z Funduszy Europejskich


*Note: All of the links on this page will take you out of the McMinnville School District website.
Limitless and leveled math practice for all levels. Grades Pre K- 12
This is a great site for worksheets, flashcards, and games
for basic math operations. Work on addition, subtraction,
multiplication, and division. Includes fractions and decimals! Grades K-6
Students choose a math skill from a table of contents and a mathematic
question and animation based on a basketball
game appears. Click the results option to get a score.
Interactive math activities for students with games, calculators, etc. Grades 3-8.
PBS Kids with math games.
This site contains a variety of edu-taining activities covering areas of math, language arts, social studies, and science. Grades k-6
Create bar, line, area, or pie graphs online!
You submit the data and the website does the rest! Grades 2-8.
High School Math Resources
Higher level mathematics from Pre Algebra and up.
I Know That Math Games
This a great site with FANTASTIC graphics to help
with basic skills of addition, subtraction, multiplication, and division.
Site that uses Virtual Manipulatives for k-12.
math skills aligned to Oregon Standards
Online resource for grades k-12.
Math for Everyone
This is a link to be shared and enjoyed
by all grade levels, K-12.
Math Glossary provides illustrated and animated definitions and
explanations for math terms to help
students with homework. Grades K-6.
This site includes math games to practice
math skills from k-12. Games include
Battleship, Jeopardy, and Concentration.
Interactive practice to help students
visualize fractions. Grades 3-8.
Who Wants To Be a Mathonaire?
Online game to practice math skills based on
the TV show "Who Wants To Be A Millionaire?"
Interactive flash and shockwave games to
practice basic math facts. Also provides fraction practice


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Book now to arrange a private tour or to attend our Open Morning on Friday 10th May, 2024
We aim to provide a high quality curriculum for the teaching and learning of mathematics that will:
- extend each pupil to his or her fullest potential, build upon on previous knowledge and recognise individual capabilities
- foster a positive attitude towards mathematics and an awareness of the fascination of the subject
- develop pupils competence and confidence in mathematical knowledge, concepts and skills
- provide pupils with a broad range of mathematical skills and understanding in order that they can apply these across the curriculum and in real life
- enable pupils to solve problems, to reason, to think logically and to work systematically
- provide opportunities for pupils to work with initiative, both independently and in cooperation with others
- develop pupils ability to communicate their ideas clearly
- encourage a deeper understanding of mathematics through a process of enquiry and experiment
Head of Department – Mr S Lawry
In the Prep Department, mathematics lessons take the form of a 60 minute lesson. Every fortnight the pupils experience at least the following number of lessons:
Years 3 to 6: 8 lessons
Years 7 to 8: 7 lessons
In Years 3 to 6 an additional set group is made, enabling set sizes to be kept low, particularly for those who find mathematics a challenge. The Learning Support Department also provides support in some of these lessons.
In years 3 to 6, the children use the Abacus Scheme, which follows the objectives from the recent 2014 National Curriculum. The depth of material provided allows teachers to offer additional support or extension activities as required. More able children are often stretched with strategies from the year above when deemed appropriate. They also use additional challenge books that allow them to apply their knowledge in complex problem solving exercises.
In Years 7 and 8, children follow the Common Entrance Syllabus and the top sets work from plans a year in advance. In all years, children use Mathematics for Common Entrance books, alongside other materials as required. The 13+ maths syllabus can be found on the ISEB website. The Mathematics Department makes regular use of ICT resources to support the children’s learning. An expanding list of revision materials and links can be found on the maths section of Firefly.
Trips and Events
Mathematical challenges and visiting speakers form an important part of the curriculum as they help foster an enjoyment in mathematics. Pupils are entered for numerous regional and national mathematical competitions and challenges.
Biannually we hold a World Maths Day where all children from reception to Year 8 compete against other children from around the world. In addition we hold an annual STEM Week (Science, Technology, Engineering and Mathematics), where pupils have the opportunity to explore and make further links between these subjects. Every three years, we hold a maths evening, where parents are invited to attend a range of workshops on modern pedagogy and teaching methods in mathematics.


NCERT Exemplar Problems Class 11 Mathematics Chapter 9 Sequence and Series
Short Answer Type Questions:
Q1. The first term of an A.P. is a and the sum of the first p terms is zero, show that the sum of its next q terms
Sol: Let the common differeence of the given A.P be d
Q2. A man saved Rs. 66000 in 20 years. In each succeeding year after the first year, he saved Rs. 200 more than what he saved in the previous year. How much did he save in the first year?
Sol: Let us assume that the man saved Rs.a in the first year.
In each succeeding year, an increment of Rs. 200 is made. So, it forms an A.P. whose
First term = a, Common difference, d = 200 and n=20
=> 6600 = 2a + 19 x 200 => 2a = 2800
∴a = 1400
Q3. A man accepts a position with an initial salary of Rs. 5200 per month. It is understood that he will receive an automatic increase of Rs. 320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?
Sol: The man gets a fixed increment of Rs. 320 each month. Therefore, this forms an A.P. whose
First term, a = 5200 and Common difference, d = 320
Q5. A carpenter was hired to build 192 window frames. The first day he made five frames and each day, thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?
Sol: Here, a = 5 and d = 2
Q6. The sum of interior angles of a triangle is 180°. Show that the sum of the interior angles of polygons with 3, 4, 5, 6, … sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
Sol: We know that, sum of interior angles of a polygon of side n is (n – 2) x 180°.
Q7. A side of an equilateral triangle is 20 cm long. A second equilateral triangle is inscribed in it by joining the mid-points of the sides of the first triangle. This process is continued for third, fourth, fifth, triangles. Find the perimeter of the sixth inscribed equilateral triangle.
Sol: Let the given equilateral triangle be ∆ ABC with each side of 20 cm.
By joining the mid-points of this triangle, we get another equilateral triangle of side equal to half of the length of side of ∆ABC.
Continuing in this way, we get a set of equilateral triangles with side equal to half of the side of the previous triangle.
Perimeter of first triangle = 20 x 3 = 60 cm;
Perimeter of second triangle = 10 x 3 = 30 cm;
Perimeter of third triangle = 5×3 = 15 cm;
Q8. In a potato race 20 potatoes are placed in a line at intervals of 4 m with the first potato 24 m from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
Sol: Distance travelled to bring first potato = 24 + 24 = 2 x 24 = 48 m
Distance travelled to bring second potato = 2(24 + 4) = 2 x 28 = 56 m
Distance travelled to bring third potato = 2(24 + 4 + 4) = 2 X 32 = 64 m; and so on…
Clearly, 48, 56, 64,… is an A.P. with first term 48 and common difference 8. Also, number of terms is 20.
Total distance run in bringing back all the potatoes,
Q9. In a cricket tournament 16 school teams participated. A sum of Rs. 8000 is to be awarded among themselves as prize money. If the last placed team is awarded Rs. 275 in prize money and the award increases by the same amount for successive finishing places, how much amount will the first place team receive?
Sol: Let the first place team get Rs. a as the prize money.
Since award money increases by the same amount for successive finishing places, we get an A.P.
Q11. Find the sum of the series (33 – 23) + (53 – 43) + (73 – 63) + … to
(i) n terms
(ii) 10 terms
Sol: Given series is: (33 — 23) + (53 – 43) + (73 – 63) + … n terms
Q12. Find the rth term of an A.P. sum of whose first n terms is 2n +3n2
Sol:Sum of k terms of A.P., Sn = 2n + 3n2
Long Answer Type Questions
Q13. If A is the arithmetic mean and G1 , G2 be two geometric means between any two numbers, then prove that 2A =
Sol: Let the numbers be a and b.
Q15. 1f the sum of p terms of an AP. is q and the sum of q terms isp, then show that the sum ofp + q terms is —(p + q). Also, find the sum of first p — q terms (where, p > q).
Sol: Let first term and common difference of the A.P. be a and d, respectively. Given, Sp = q
Q16. If pth, qth and rth terms of an A.P. and G.P. are both a, b and c, respectively, then show that ab-c bc-a-ca-b = 1.
Sol: Let A and d be the first term and common difference of A.P., respectively. Also, let B and R be the first term and common ratio of G.P., respectively.
It is given that,
Objective Type Questions:
Q17. If the sum of n terms of an A.P. is given by Sn = 3n + 2n2, then the common difference of the A.P. is
Sol: (d) Given, Sn = 3n + 2n2
Q18. If the third term of G.P. is 4, then the product of its first 5 terms is
(d) none of these
Sol: (c) Let a and r the first term and common ratio, respectively.
Q19. If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is
Sol: (a) Let the first term and coiqmon difference of given A.P. be a and d, respectively.
Q20. If x, 2y and 3z are in A.P. where the distinct numbers x, y and z are in G.P., then the common ratio of the G.P.is
Sol:Since x, 2y and 3z are in A.P., we get
Q21. If in an A.P., Sn = qn2 and Sm = qm2, where Sr denotes the sum of r terms of the AP, then Sq equals
Q22. Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
Sol: (b) Let first term be a and common difference be d.
Then, S2n = 3Sn
Q25. If tn denotes the nth term of the series 2 + 3+ 6+11 + 18+…, then t50 is
(a) 492 – 1
(d) 492 +2
Q26. The lengths of three unequal edges of a rectangular solid block are in G.P. If the volume of the block is 216 cm3 and the total surface area is 252 cm2, then the length of the longest edge is
(a) 12 cm
(b) 6 cm
(c) 18 cm
(d) 3 cm
Sol: (a) Let the length, breadth and height of rectangular solid block be a/r, a and ar, respectively.
Fill in the Blanks
Q27. If a, b and c are in G.P., then the value of is equal to _________
Sol: Given that, a, b and c are in G.P.
Q28. The sum of terms equidistant from the beginning and end in an A.P. is equal to _____.
Sol:Let a be the first term and d be ihe common difference of the A.P.
Q29. The third term of a G.P. is 4. The product of the first five terms is
Sol: Let a and r the first term and common ratio, respectively.
True/False Type Questions
Q30. Two sequences cannot be in both A.P. and G.P. together.
Consider the sequence 3,3,3; which is A.P. and G.P. both.
Q31. Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessarily be true.
Sol: True –
Consider the progression a, a + d, a + 2d, … and sequence of prime number 2, 3, 5, 7, 11,…
Clearly, progression is a sequence but sequence is not progression because it does not follow a specific pattern.
Q32. Any term of an A.P. (except first) is equal to half the sum of terms which are equidistant from it.
Q33. The sum or difference of two G.P.s, is again a G.P.
Q34. If the sum of n terms of a sequence is quadratic expression, then it always represents an A.P.
Match the questions given under Column I with their appropriate answers given under the column II.


On 20.07.2017 15:00, Aleksandar Trifunovic wrote:
> I was looking for a way to prevent adding of parallel edges in the graph
> i.e. I would like to create a graph that is not a multigraph. Now I do that
> manually but that is quite prone to bugs.
> Is this possible to do this with graph tool in any way?
You can check for the existence of an edge with Graph.edge(), and you can
use Graph.edge(s, t, add_missing=True) to fetch edges or add missing ones if
they do not yet exist.
Alternatively, you can remove parallel edges a posteriori with the function


Blog post by Tina Winterlik © 2012
Whooohooo! This guy is AWESOME!!
He makes soooooooooooooo MUCH sense. I have math story. Want to hear it?
I had a math teacher in Grade 8 that was not very good at his job. I fell behind and never could catch up. I managed to get thru Grade 10 Math but HATED it! I never wanted to take Math again. And so I didn't.
I took ART! I LOVED ART! I had great Art teachers in 8, 9, and 10...but 11 & 12, different story. And that's another story. But part of that story is because of the issues with Art I changed to photography. Anyways, I couldn't wait to graduate. I never wanted to go to school again.
When I graduated I went to work in labour. I worked on big production farms and in factories. Fast forward 10 years and I get tendonitis really bad working in a repetitive labour intensive job. I go back to school. I want to take Professional Photography, BUT I need MATH and not only that PHYSICS.
My Math is so old now I have to re-take grade 10 Math so I can take the grade 11 Math. Thankfully I had an awesome teacher, BUT I also had a boyfriend I was living with and everytime I got stuck on a problem he got me thru it and I could finish my homework. I got an A. But many of the other kids/adults did not. Some even failed.
Then I had to take the Grade 11 Math. I was so incredibly stressed, I needed this stupid course with quadratic equations so I could take my Photography course, it was what was standing in the way of my dream.
I remember watching the teacher mark my paper and she was going over my mistakes and tears were falling and some landed on the page, and I couldn't help it and it was just awful.
Anyways I managed a B somehow, and while accomplishing this helped boost my ego a lot when I took the Business Math in the Photography program I sucked and barely passed. I am still not sure how I did it, but 2 other kids didn't. Mainly because they didn't go every single day, which I did.
The Physics was another battle and that I did thru correspondence. Yikes, Yuck!
Thankfully that's all behind me now. And you know what. I don't use any of it.
And I am a very good photographer and artist.
My girl is learning Math now. She had to go to public school recently because they were renovating our building and the noise was so bad that I had to send her for our sanity. She came home with Math headaches. It's CRAZY!! Why are we doing this to kids? God didn't invent Math. You don't see bears and deers in the forest calculating and using math problems to figure things out.
This speaker, John Bennett, is so RIGHT ON!! Stop teaching Math after age 10. Let kids learn cool brain games, there are lots on the internet and in boardgames and other places.
Analytical thinking makes sense. Having common sense and figuring out consequences that makes sense. Math makes no sense. Even many mathematicians say that algebra is all imaginary.
It's unhealthy to stress child this way and at the same time stress there parents and everyone around them.
It's unhealthy to test children period, I believe. Anyways, I love this guy and I hope people wake up to this and stop teaching useless Math in school.
Instead teach them gardening. Teach them to grow food in raised bed gardens. How to compost and grow seeds and plants in popbottle greenhouses. Teach them about GMO and seedbanks. Teach them how to harvest the food, how to weigh it and measure it and how to cook with it.
Teach them how to pick it and pack it and sell it in a market. These are the skillls they need. This is the reality today. You could also teach them how to build cob houses. That would be a super useful skill. A pratical skill. These are the skills and tools need today.
If a kid really wants Math fine. But don't force anyone to learn what they don't want and which they see no point in learning. Help them learn the tools they need to truly succeed and be happy in life.


Reasoning Mind (RM) is a low-cost, nationally scalable solution to the three most pressing problems in K-12 math education: human capital, curriculum quality, and student engagement. RM partners with schools where student use the RM system during the school day, in the time set aside for mathematics instruction. Each student learns mathematical theory and solves problems at his or her own computer – and consequently, at his or her own rate. Gifted children aren't held back, and struggling children aren't left behind. The teacher is freed to work with students one-on-one or in small groups.
RM was founded in 2000 and while they are classified as a 501(c)3 nonprofit, they operate like a for-profit startup social enterprise and are working towards 100% self-sufficiency. Today, RM is serving about 70,000 students.
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Make a fabulously fun hidden numbers game for preschoolers to practise learning their numbers!
I love hands on learning activities that have a hint of magic and real fun about them.
This activity is both VERY easy and VERY easily adaptable to suit a range of different learning objectives. So hopefully you will find a way to use it for your own kids!
How to make a hidden numbers game for your kids
You will need:
- transparent serving or oven dish
- piece of card
- marker pen
- rainbow rice (click here to find out how to make this easily!)
- paint brush
Simply find a piece of paper to match the size of the underside of the transparent baking dish. For us, A4 worked perfectly.
Onto this paper write all of the numbers that your child is currently earning, or moving onto learning next so that there is an element of challenge. My 4 year old loves “big” numbers and can now read two digit numerals with some ease. He is really interested in hundreds numbers but not yet able to read them with confidence, so this is the next step.
You could choose to write their name, letters that they are learning, sight words or even calculations to solve. Draw shapes, stick animal/ vehicle / dinosaur stickers to find- anything is possible! Start with your child and take it from there.
Now slide this sheet under the dish ready to go.
Into the dish pour a thin layer of rainbow rice (or choose dyed salt or another sensory base filler material.)
Then simply set out a paintbrush next to it and ask your child to find the hidden surprises!
This is such a fun and rewarding little game that is sure to delight any child as they just LOVE secret surprises!
Watch the simple how-to video and snapshots of play here too:
I hope you loved this super simple hidden numbers game for preschoolers! Tag me if you make a version, I would love to see it and re-share on Instagram!
Find lots more MATHS GAMES and DIY RESOURCES here in the archives too.


Last week I had the chance to speak to 6-8th graders at Prince of Peace in Hoover about how I use math in my job. One of my great friends is the math teacher, and has had several speakers over the course of the semester. She was particularly interested in showing the kids that even creative careers involve math on a daily basis.
So, inspired by Dick Hardt’s Identity 2.0 presentation I put together this little group of slides. If you haven’t seen the video of that presentation it is very much worth a watch. At any rate, the final product has little in common with Mr. Hardt’s presentation, but it was fun to throw together at midnight the night before and the kids got a kick out of it.
(Does anybody actually use picas? I threw it in for the sake of the math problem, but have never really needed to use a pica for anything.)
(And also, I blatantly ripped images from the web for this little teaching moment. Only a few images are actually my own, and if you see one of yours comment and I’ll credit you here.)


Mrs E Law
Miss S Hoare
- Monday, Tuesday and Wednesday morning - Mrs Law
- Wednesday afternoon, Thursday and Friday - Miss Hoare
Reading - Children should read at home for 30 minutes daily as part of their homework.
Maths fluency- Children have a weekly maths fluency focus which is rehearsed each day in school and quizzed on a Friday. Details of our weekly fluency focus is found on Eschools, under 'me', then 'My groups'. Children can also practice maths fluency on Numbots and TTRockstars.
Spelling - children have a list of spellings to learn at home each week. We rehearse these in school each day and quiz on them on Fridays. Details of our weekly spelling focus is found on Eschools, under 'me', then 'My groups'.


One pillar of the modern world was the project to transform science from a discipline for contemplating nature into a tool for mastering it. Queen among the new sciences was mathematical physics, made possible by a corresponding transformation of mathematics. Ancient mathematicians, said René Descartes, had misunderstood their subject. They offered a procession of dazzling spectacles, but mathematics properly understood is not the presentation of beautiful chance discoveries. It must instead provide a systematic method for solving problems.
True and oft-heard though this story is, it is also easily taken for granted. Harvey Flaumenhaft’s new book Insights and Manipulations seeks to remedy that with a hands-on guide to this momentous change. To understand that change, he says, and to understand it as progress, “we need to know what it was a step from as well as what it was a step toward.” He presents it as a movement from the Conics of Apollonius of Perga, a brilliant and difficult triumph of ancient mathematics written in the third century b.c., to Descartes’s 1637 Geometry, a decisive stride into modernity. For Apollonius, mathematics was a way of gaining insights into the nature of geometrical forms by envisioning them in the mind; Descartes made mathematics into an activity of manipulation — hence the title of Flaumenhaft’s book.
But appreciating scientific progress should not become a license to forget the past:
We often take for granted the terms, the premises, and the methods that prevail in our time and place. We take for granted, as the starting points for our own thinking, the outcomes of a process of thinking by our predecessors.
What happens is something like this: Questions are asked, and answers are given. These answers in turn provoke new questions, with their own answers. The new questions are built from the answers that were given to the old questions, but the old questions are now no longer asked. Foundations get covered over by what is built upon them.
Progress thus can lead to a kind of forgetfulness, making us less thoughtful in some ways than the people whom we go beyond. We can become more thoughtful, though, by attending to the originating thinking that while out of sight is still at work in the achievements it has generated.
Insights and Manipulations is thus a guidebook helping us to retrace the steps of discovery from ancient mathematics to Descartes, who conceived the world anew. For “only by actively taking part in scientific discovery — only through engaging in re-discovery ourselves — can we avoid both blind reaction against the scientific enterprise and blind submission to it.” What’s at stake for Flaumenhaft is more than being thoughtful about the world we inhabit; it is whether we are to understand the foundations of modern science well enough to sustain it.
Classic works of science and mathematics make up a substantial part of the liberal arts curriculum at St. John’s College in Annapolis, Maryland, where Flaumenhaft has taught for more than fifty years. Students there struggle through Apollonius and Descartes not as an antiquarian exercise but as a way to see the modern world come into being.
Insights and Manipulations also offers a model of how to read a scientific classic: Presume that it has been written with care and treat it, like any other work of art, as a unified whole whose very form helps to tell its story. (Flaumenhaft provides, at times, more about the detailed structure of these works than many readers may want to know, but that’s a quibble.)
The first three-quarters of this hefty book is devoted to Apollonius, and is a considerable pedagogical achievement in itself. Apollonius studies conic sections, the curves obtained by slicing straight through a cone — circle, ellipse, parabola, hyperbola. The Conics is, to say the least, austere, a sequence of definitions, propositions, and proofs with little motivating explanation. It also assumes knowledge of sophisticated parts of Euclid’s Elements, a classical masterwork of geometry that preceded Apollonius by roughly a century.
Flaumenhaft, supplying background from Euclid as needed, carefully explains what each definition and proposition says (and, equally important, what it doesn’t). Before offering a formal proof, he prepares readers by informally explaining why each proposition ought to be true. He supplies abundant illustrations, not just geometric diagrams but also flowcharts that offer an overview of how every part of an argument fits together. The diagrams are great improvements on those typically provided in other presentations of Euclid or Apollonius. Instead of a single, sometimes massively complex figure, he accompanies each proof with a series of simpler pictures, each of which isolates only what is needed to explicate a single logical step. He also provides views from different angles (bird’s eye, side view), allowing the reader to imagine walking around a 3-D model. These illustrations are hand-drawn, which lends them a certain charm, as if one is following a teacher at a blackboard.
The story frequently steps back from the local questions of understanding individual definitions and arguments to the global question of why the results have been organized and presented as they are. Flaumenhaft tends, at times, to over-explain — as Marian Adams said of Henry James, to chew a bit more than he bit off — but that’s the defect of a virtue.
Conic sections still appear in high school mathematics today, but a contemporary student will encounter nothing like what we find in Apollonius. Apollonius starts with a geometrical image, a slice through a cone. Everything that follows is expressed in terms of geometrical constructions built on the cone, the curve, and the plane making the slice; there are no coordinates, formulas, equations, or even numbers. By contrast, a modern textbook will, pro forma, provide a picture of a cone, but will have no real use for it. The curves to be studied are defined by formulas, and the terminology of “conic sections” is a vestige of forgotten technology, as when we speak of dialing a phone. The modern student is really doing algebra.
Apollonius thinks very differently, guided by a set of clear logical distinctions that made it impossible to mix geometry with arithmetic (and thus with algebra). One must appreciate these distinctions to understand the radical step Descartes took in breaking them down, the power he unleashed in doing so, and the intellect that could do that.
After the extended treatment of Apollonius, and brief excursions into works by Diophantus of Alexandria (200s a.d.) and François Viète (1500s), Insights and Manipulations concludes with the 1637 Geometry. Here Descartes accuses the ancients of withholding explanations of how they discovered their results in order to make themselves look good. Further, the ancients insisted on a fundamental logical distinction between arithmetic and geometry — to Descartes, a pesky scruple that prevented them from finding the proper mathematical method. By merging the two fields, Descartes unveiled a royal road to discovery — inaugurating, as Flaumenhaft puts it, a decisive shift from seeing “with the mind’s eye” to performing “mental manipulation.”
Descartes famously distinguished mind, whose essence is to think, from matter, whose essence is “extension” — roughly, the ability to occupy space. The physics of the material world must therefore be explained as matter moving and rearranging in space. So a systematic method for solving geometrical problems, for mentally manipulating space, must have seemed the key to a comprehensive explanation of the material world.
Geometry was published in a volume comprising four short works. The first is the famous Discourse on the Method of Rightly Conducting One’s Reason and Seeking Truth in the Sciences — the source of “I think, therefore I am.” The other three offer examples of applying his method, which we can’t retrace here, to problems in different fields of science. Optics works out a (correct) law for the refraction of light. Meteorology explains (again correctly) the cause of a rainbow. Those two achievements, he says, are persuasive evidence for the value of his method.
And Descartes presents Geometry, the concluding essay, as definitive proof. He first describes a procedure for solving any problem in classical, Euclid-style geometry. A problem asks for a construction, such as: Given a rectangle, construct a square with the same area. I’ll use this very simple problem as a running example. Descartes doesn’t deign to illustrate his procedure by applying it to a specific example. Readers will learn more, he says, by working things out for themselves.
Instead, he turns to a problem that had been open for more than a millennium — actually to a family of problems called “locus” problems. They were formulated by Pappus of Alexandria, the last of the great, ancient Greek mathematicians, in the fourth century a.d. A locus problem asks a question of the form: What is the curve containing all points that satisfy some constraint? To take a simple example, if the constraint is “All points that lie at a given distance from a fixed center point,” the answer is a circle. Pappus formulated his problem in a very general way by establishing a systematic, stylized way to express the constraints. (See Supplement 1 below for an illustration.) While the answers to some relatively simple locus problems are the conic sections of Apollonius, the general problem posed by Pappus — an infinite family of increasingly complex problems — remained open.
Descartes is interested not merely in providing the answer to this or that previously unsolved problem, the stuff of which reputations in mathematics had always been made. What matters is the discovery of a systematic method for problem-solving — a way to attack any of Pappus’s locus problems, however elaborate the constraints. By doing so, he says, “I think I have entirely satisfied what Pappus tells us had been sought in this by the ancients,” and he adds, cavalierly, “I shall try to put the demonstration of it in a few words, for I am already bored from writing about it so much.”
The conceptual tools available for performing constructions in classical geometry are the straight edge and the compass. Although we may use physical versions of them on a blackboard, we should think of them as theoretical devices, imaginative ways to express two of Euclid’s postulates: that we can construct a straight line connecting any two points and extend it as far as we like (straight edge); and that we can construct a circle with any given center and radius (compass). Descartes’s methods allow him to see how far it’s possible to go using only straight edge and compass, and to identify the fancier tools required to solve more complex problems. He introduces what’s come to be called a “geometrical compass,” an indefinitely extendable device that allows additional moving straight-edges that act like levers:
As one adds more levers, one can draw more complex curves, with each added lever corresponding to added complexity in the algebraic expressions needed to formulate a problem.
When Descartes made his new beginning … , he said that the ancients were handicapped by their having a scruple against using the terms of arithmetic in geometry…. Before modern readers can appreciate why Descartes wanted to overcome the scruple, and what he saw that enabled him to do it, they must be clear about just what the scruple was.
We may think about the ancient scruple in the following way. The science we now have is what we might call “quantitative” and “numerical.” Quantities admit comparisons of less and greater. To us it seems natural to identify quantitative with numerical — to represent any quantity as a numerical value, for example a decimal number or a location on what we now call a “number line.” For that notion I’ll use the ungainly term number-in-our-sense.
To our great ancestors this conception of quantity, far from being natural, would have seemed incoherent. They recognized a fundamental difference between quantities arising in geometry (lengths, areas, volumes) and quantities arising in arithmetic (the numbers we count with). Our physical science required that Descartes break down that barrier by introducing arithmetical methods into geometry.
What was the problem? First, as noted, quantities fall naturally into two radically different categories: multitudes (collections of distinct individuals that can be counted, like cows) and magnitudes (which can be made continuously smaller or greater, like lines or areas or volumes). Ordinary speech reflects the difference. Of multitudes we ask “How many?” but of magnitudes “How much?” A herd of cows is a multitude; we ask how many cows are in a field, not how much cows there are. We get the answer by counting them. The water in a pond is a magnitude. We ask how much of it there is, not how many.
To define a multitude, we have to specify which individuals it consists of. One can’t point to a field and ask “How many?” without specifying how many of what: horses, cows, hooves. Euclid’s Elements, which lays the foundation for Apollonius, says that “a unit is that by virtue of which each of the things that exist is called one.” In this sense, “cow” is a unit but “water” is not; we can count cows but not water because there’s such a thing as “a cow” but no such thing as “a water.” There is an unfortunate collision between Euclid’s meaning of “unit” and our use of “unit” to denote some arbitrarily chosen magnitude — inch, pound, gallon — employed as a reference for measurements. It seems we might be able to paper over the differences between multitudes and magnitudes by using such reference values: We can’t ask how many water are in a pond, but we can ask how many gallons of water are in it. As we will see, that simple strategy — changing the question so as to think of magnitudes as multitudes — can’t be made to work. Something much deeper will be required.
Euclid goes on to say, “A number is a multitude composed of units.” Euclid’s units are, by definition, indivisible, so multitudes are discrete: There is no multitude consisting of more than three cows but fewer than four. Multitudes defined by different units can be compared: It makes sense to ask whether there are more chickens in the barn than eggs. From now on, I’ll use the word number, unqualified, to mean a number in this ancient sense but will sometimes, for emphasis, say “counting number,” and we will see how this is different from number-in-our-sense, which allows us to represent any quantity as a numerical value.
Magnitudes, by contrast, are not discrete; a line, unlike a herd of cows, can repeatedly be divided into smaller pieces as often as you like. (For Euclid, “line” always means what today we would prefer to call a “line segment”: limited in length, having two end points.) Equally important, magnitudes are of different kinds, and those of different kinds are incomparable; we can’t, for example, ask whether (the length of) a line is greater or less than (the area of) a two-dimensional figure such as a square.
The preceding sentence hid the references to length and area in parentheses because they suggest a way of thought according to which there is, on the one hand, a line (or a figure) and, on the other hand, some sort of number-in-our-sense that is its length (or its area). But the possibility or intelligibility of such numbers-in-our-sense is precisely what is at issue. When, for example, Euclid shows that a certain square equals a certain triangle, that amounts not to some numerical comparison but to showing that the figures can be cut into identical collections of pieces — or, if you like, cut up and then rearranged to make the same figures.
Consider a further distinction between multitudes (“how many”) and magnitudes (“how much”): Different things can be done with each. For example, two multitudes can be multiplied but two magnitudes cannot. The result of multiplying four cows by three is (to use a modern notation that is in this case not misleading) 4 + 4 + 4 cows. The multitudinous-ness of three is what tells us how many times to add groups of four cows together. By contrast, although we can multiply a magnitude by a multitude — say, doubling a line — we cannot multiply two magnitudes, even if they are of the same kind. We can’t multiply a line by a line or one figure by another, because lines and figures provide no natural answer to the question “How many times?”
A modern text that formulates the Pythagorean Theorem by saying “a right triangle with sides a, b, and hypotenuse c satisfies a2 + b2 = c2” is therefore saying something radically different from what Euclid or Apollonius or Pythagoras said. It assigns to the sides the lengths a, b, and c, which are numbers-in-our-sense; it performs arithmetical operations on them (squaring, adding) to compute other numbers-in-our-sense; and then it asserts that the results are numerically equal. Euclid puts this very differently. He says that if you’re given a right triangle and construct squares on all its sides, then the squares on the two sides containing the right angle taken together are equal to the square on the other one. Numbers don’t enter the picture, and the proof is done geometrically.
We are so used to thinking in terms of numbers-in-our-sense that the difference can be hard to think about, but it is critical for understanding what ancient mathematicians were doing, and what marked the divide between geometry and arithmetic that Descartes broke down. But before we get there, we must get a taste for the challenges the ancients faced — and the insights they gained — in doing geometry without arithmetic.
Earlier, I dismissed a maneuver that seems as if it could bridge the two domains — namely, to treat a magnitude as a kind of multitude, an exact multiple of some magnitude used as the reference for a measurement. Thus we might be able to regard some line as a multitude of feet or meters or cubits. If a right triangle has sides that are 3 feet, 4 feet, and 5 feet, the Pythagorean Theorem might be expressed as a fact about calculations with those numbers: 32 + 42 = 52. (Ignore for now that the theorem would no longer describe visualizable relationships among geometric figures.)
Why does this trick fail? It can be applied only if all the lines in the problem are exact multiples of the same reference line — if, in Euclid’s terminology, all the lines have a “common measure.” Unfortunately, one of the most famous proofs in ancient mathematics shows that that’s not always possible. A simple example: the side of a square and its diagonal have no common measure. This example also shows that the trick can’t be used to formulate the Pythagorean Theorem. It can’t be applied to the right triangles that result from slicing a square in half along its diagonal.
Ancient mathematicians devised an elegant way around this by appealing to the notion of ratio. Quantities of the same kind — two lines, for example — will have a ratio even if they lack a common measure, and ratios can be compared. We can say, for example, that the ratio of 2 to 5 is the same as the ratio of 4 to 10, and is greater than the ratio of 2 to 10. Crucially, we can compare any ratios — we can, for example, compare the ratio of two counting numbers to that of two areas, or the ratio of two lengths to the ratio of two volumes.
Modern mathematics is indifferent to any distinction between ratios and numbers-in-our-sense; it is happy to identify the ratio of 2 to 5 with the number-in-our-sense 2/5, or the ratio between a square’s diagonal and its side with the number-in-our-sense called “the square root of two.” For us, the proportion stating that “the ratio of 2 to 5 is the same as the ratio of 4 to 10” can be expressed as the equation “2/5 = 4/10.”
Euclid and Apollonius have no words to denote “the square root of two” but do have a powerful theory of ratios that makes it possible to establish connections among quantities of all kinds. That theory gets a rigorous basis in the most technically sophisticated book of Euclid’s Elements.
Consider a simple example, expressing the relation between the area of a rectangle and the lengths of its sides. In modern mathematics we represent the sides with numbers-in-our-sense — say, 2.6 and 3.5 — and define the area to be their numerical product: 2.6 * 3.5 = 9.1. In a practical application, the numbers we use will depend on whether we choose to express the lengths in inches, meters, furlongs, or whatever.
Euclid expresses that relationship more elegantly with ratios, needing no resort to an arbitrarily chosen standard of measurement. Look first at a simple case. If, given a rectangle, we leave its height unchanged, and double its width, we will have doubled its area. Similarly, to use modern terms, if we change the width by a factor of 50 percent, or by a factor of , we will change the area by the same factor. That is, if we write “:” to abbreviate “the ratio” between two quantities, we can in this simple case express the relation between the sides and area of a rectangle as a proportion:
[old rectangle : new rectangle] is the same as [old width : new width]
If we now change both the height and width, then [old rectangle : new rectangle] will be a ratio that depends on both [old width : new width] and [old height : new height]. It will be what Euclid calls the ratio compounded of those two; given such a pair, Euclid shows how to construct a pair of lines whose ratio is their compound.
These are the terms in which Apollonius thinks, and he deploys them with great virtuosity.
Flaumenhaft wants to help us teach ourselves to think like Apollonius, who proves theorems using proportions, and to compare that to the Cartesian practice of solving problems using equations. The root of the Greek word for “theorem” means “to look at” or “behold.” And “proportion” has to do with recognizing a similarity between different things. This reflects what Apollonius does: behold certain similarities in things. A problem, by contrast, presents us with a task: to find a solution. The terms in an equation are symbolic. Their purpose is not to represent something that we see by looking at or through it, but to be material for manipulation.
For Descartes, Flaumenhaft writes, “beholding is subordinate to mastery,” and “wonder should give way to problem-solving. What Descartes brought to this modern project which took mastery of nature as its end was an emphasis on mathematics as means to the end.”
We can further contrast the two approaches by considering the difference between synthesis and analysis. Euclid and Apollonius present their results by synthesis. A synthesis is a proof that begins from postulates and things already proven, then deduces a sequence of further results until it manages to establish the desired conclusion. Those deductions often concern not only the lines and figures given in the statement of a theorem, but additional lines and figures introduced, often with great ingenuity, in order to establish in some roundabout way relationships among the elements of the original material. How the proposition or its proof were discovered is not manifest.
Descartes, by contrast, provides a method of analysis. Starting with a problem, he will transform it into a sequence of other problems, all of which are equivalent, until arriving at one he knows how to solve. Since the problems are equivalent, a solution to any of them gives a solution to all of them. The procedure is transparent because the analysis itself shows how the result was obtained.
Analysis is what we do in a high school algebra class. We’re given the problem of finding an x such that 3x – 2 = 7. Adding 2 on each side of the equation transforms it to 3x = 9. Dividing both sides by 3 transforms that to x = 3. Maneuvers of this kind were hardly unknown before Descartes. Flaumenhaft walks us through the way Diophantus applied them to problems about numbers and Viète applied them to what he called “species” — entities that share some characteristics with numbers and some with geometrical magnitudes. Species, however, are still encumbered by something like the distinctions among kinds of quantities that Descartes eliminates.
Descartes claims that any geometric problem can be expressed as the task of finding the lengths of certain lines. To do that by analysis, he first restates the geometric problem using equations that relate the unknowns (lines to be found) to the knowns (lines given in the statement of the problem).
Consider again our familiar example: Given a rectangle, construct a square of the same area. The knowns are the two sides of the rectangle; call them A and B. There’s one unknown, the side of the square we’re looking for (since all sides of the square are the same); call it x.
The equation that relates the areas of the two figures may look misleadingly, comfortably familiar:
x * x = A * B
We’ll have more to say about that later, but, to avoid jumping to unjustified conclusions, note that Descartes is doing geometry: A, B, and x are straight lines, not numbers.
Algebraic manipulations then transform these equations into a solution — that is, a collection of equations in which each unknown is set equal to an expression involving only known quantities.
In this case, the solution (also misleadingly familiar) is:
Now comes a piece of magic. Descartes does not find x by calculating some value for the expression ; instead, as we will see below,“” is interpreted as a set of instructions for performing a geometrical construction that gives the desired answer.
He is doing geometry, but the most important steps take place in a symbolic world where we (literally) lose sight of the original subject matter. We can, however, extract a geometrical meaning from the symbolic solution that emerges.
Before looking at how Descartes solves the problem, we must consider how he explains arithmetical manipulation of lines. The terms in Descartes’s equations use the operations of addition, subtraction, multiplication, division, and the taking of roots (square root, cube root, and so forth). As Descartes defines these operations, both their inputs and outputs are lines. Thus, all the manipulations act on a single kind of thing — on a single “type,” to use the terminology of modern logic and computer science. Operating on a single type, regardless of what it represents, is a crucial simplification.
Addition and subtraction of lines are explained in the obvious way: A + B is a line that results from extending A by attaching B at one of its endpoints. A – B is a line that results from lopping B off the end of A, and is not defined if B is greater than A. Descartes defines multiplication, division, and the taking of roots by analogy. Take multiplication as an example. He first finds a recipe that characterizes multiplication for counting numbers in terms of ratios and proportions — the domain in which multitudes and magnitudes participate on something like an equal footing. He then uses that recipe as the definition of multiplication for lines.
Here’s how that works. For counting numbers, we can describe the result of multiplying 2 by 3 as a proportion by saying that
(2 * 3) : 2 is the same ratio as 3 : 1
In other words, because 3 is what you get by tripling 1, the value of 2 * 3 is what you get by tripling 2. Descartes simply copies that: If A and B are lines and we arbitrarily choose any other line u, then — in terms of that choice of u — we define A * B to be a line such that
(A * B) : A is the same ratio as B : u
That is, whatever “multiple” B is of u, A * B is that same multiple of A. (Supplement 2 below provides a geometrical construction for finding A * B.) Descartes calls u a “unit,” using the word in essentially the modern sense — unlike Euclid’s usage, in which a unit is a concept that specifies the kind of thing (a cow) assembled in a multitude (a herd).
It’s far from obvious what those definitions buy us, since the line we obtain by calculating the value of some complex expression will depend entirely on the arbitrarily chosen unit (u). But Descartes doesn’t want to calculate with these expressions; he only wants to manipulate them. His strategy works, because, for any fixed choice of a unit, all the usual rules for algebraic manipulation are valid. For example, consider the expressions (A * B * C) / A and B * C. Whatever line we choose as the unit u, the value of the two expressions calculated in terms of u will be lines of the same length. Descartes can therefore say that (A * B * C) / A = B * C, and justify the algebraic maneuver of “canceling the A’s.”
Here are the steps by which Descartes’s method would solve our simple sample problem: Given a rectangle, construct an equal square.
We choose names for the knowns, the sides of the rectangle: call them A and B. We choose a name, x, for the side of the square we’re seeking.
We next apply certain rules for translating the geometry into symbols. One of them tells us to represent (the area of) that rectangle by the expression A * B. By the same rule, x * x represents the square we want to find. (For x * x, we’ll use the modern notation x2, which Descartes introduced.) We thus reformulate the problem as a request to solve x2 = A * B. A symbolic solution requires just two steps:
|taking the square root of both sides|
Descartes provides another set of rules for translating the solution thus found into a geometric construction. Relegating the explanation to Supplement 3 below, I’ll simply assert that the solution given above corresponds to a geometric construction solving the problem:
Find an x so that B : x is the same as x : A.
All together, this amounts to a geometrical argument in which all the work is done by routine symbolic manipulation.
The rules for translating geometric problems into equations, and for then translating their algebraic solutions back into geometric constructions, are based on plausible analogies. Further, because the rules for algebraic manipulation are valid regardless of the choice of unit, it’s plausible that those manipulations won’t corrupt the geometrical meaning that we ultimately wish to extract. Those deep insights don’t truly amount to proof that his method works, although using modern logical techniques such a proof would be straightforward.
Descartes asserts, leaving it to the reader to persuade himself, that the method he describes suffices to solve any problem solvable by the constructions available in Euclid. But that’s only a warmup.
The definitive proof for the value of Descartes’s method is that it has enabled him to discover how to systematically classify and solve an unlimited number of problems of arbitrary complexity. Locus problems with more and more elaborate constraints will result in equations containing terms of higher and higher power, something that might in the past have seemed problematic. We can understand X2 — indeed, visualize it — as representing the area of a square, and we can understand X3 as the volume of a cube. But what about X4, X5, X6, … ? What could be the sense of those? Descartes has made that problem disappear. There is a method. One applies it.
He thus helped set in motion a chain of developments that led mathematics to become formal. Euclid and Apollonius understand themselves to be talking about lines, circles, figures, and solids. They single out certain obvious truths as postulates and prove others, including some that are not at all obvious, from those postulates. Those proofs allow us to see that the less-obvious things are true, but don’t make them true; rather, their truth follows from the nature of lines, circles, figures, and solids.
A modern treatment of Euclidean geometry, by contrast, might begin by saying something like this: Suppose we have two collections of things, which we choose to call points and lines, and suppose that they satisfy such-and-such a list of postulates. It doesn’t matter, and we don’t care, what the things called “lines” and “points” are in relation to space or to anything in the physical world; the question doesn’t arise. Such meaning as “lines” and “points” have is acquired from the role they play in the system. Bertrand Russell, with his talent for aphorism, expressed this by saying that “mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”
Practicing mathematicians do of course form mental pictures of what they talk about. Few have carefully thought-through positions on the philosophical status of those entities, but, professionally speaking, all play the formal game because it has immense power — and is the only way to make their results acceptable to the mathematical community at large.
Insights and Manipulations is written, the introduction says, “for serious amateurs by a serious amateur,” and even its smallest details reflect the author’s long career as a teacher. Its meticulous layout, for example, arranges that an illustration and any references to it will lie on the same page or on a facing page, so that the reader won’t be compelled to flip back and forth. The original texts, including Flaumenhaft’s new translation of Apollonius, are presented in gray boxes and arranged to help the eye take them in, with lengthy and complex sentences often displayed so that each line on the page consists essentially of a logical unit. The publisher deserves compliments for agreeing to design decisions that make a big book even bigger.
A serious amateur could use this book for self-study with a good prospect of success. Small seminars, I’m told, have worked through the entirety of early drafts with university students. A profitable course could be made from selections that focus, like this essay, on the beginning and end of the story — Apollonius and Descartes.
Insights and Manipulations is part of a long-term pedagogical project, of the author and others, to reintroduce classic scientific works into liberal education. Intellectually serious students are owed that experience. Great scientific works are of enduring interest for their daring and imagination, and as chronicles of heroic adventures of the mind. “To be thoughtful human beings,” Flaumenhaft says, “to be thoughtful about what it is that makes us human, we need to read the record of the thinking that has shaped the world around us, and continues still to shape our minds.”
A simple three-line locus problem is illustrated below:
We’re given the “reference” lines A, B, and C. Here is the constraint
determining whether a point P lies on the locus:
Draw perpendiculars from P to each of the reference lines, calling
those perpendiculars a, b, and c. P satisfies the constraint if
you can draw a rectangle with sides a and b that has the same
area as a square with side c.
In this diagram, c is larger than both a and b. Thus we would expect the square with side c to be larger than the rectangle with sides a and b, and so P does not lie on the locus.
Pappus introduced a very general way to formulate constraints: There can be any number of reference lines, not just three. Instead of insisting that the line drawn from P meets each reference line at a right angle, one can specify the angle for each of them — the line from P to A must make this angle; the line from P to B must make that one; and so forth. Instead of requiring that the areas of the rectangle and square be equal, one can specify that the areas must have a certain ratio. (In, say, a six-line problem, the way to formulate the comparison of ratios based on the six lines drawn from P to the reference lines is more complex.)
Given A, B, and u, lay them out as in the left-hand side of the figure
A and B meet at their end points to form an angle (it doesn’t matter what size). Lying on one of those sides is u (it doesn’t matter which side; in the figure I’ve chosen A), with one of its end points at the angle’s vertex. Neither does it matter whether u is greater than, less than, or equal to the side on which it lies.
The right-hand side of the figure then shows how to construct the side that will be defined as the length A * B with respect to the unit u: Draw a line (shown with heavy dashes) from the end of u to the end of B. Then draw a line (shown with lighter dashes) parallel to that line at the end of A, extending that line, and adding an extension to B, until they meet.
The desired result is the line marked A * B. Because the two dashed lines are parallel, the two triangles in the figure are similar, and corresponding sides will therefore have the same ratio. So, as desired, (A * B) : A is the same as B : u.
Let’s first explain why this equation corresponds to the proportion:
B : x is the same as x : A
This can be seen by backing up to the preceding equation, x2 = A * B, and rewriting it as follows:
|x * x = A * B||by the definition of x2|
|x = (A * B) / x||dividing both sides by x|
|x / A = (A * B) / (x * A)||dividing both sides by A|
|x / A = B / x||“canceling” the A’s|
This last equation corresponds to the proportion
B : x is the same as x : A
The figure below shows the construction of x from B and A. Draw a semicircle whose diameter is A + B. The line x is the perpendicular drawn from the common point of A and B to the circle. The ability to draw circles (a compass) makes it possible to construct square roots. Successive add-ons to Descartes’s improved compass make it possible to extract roots of higher powers.
Math and Modernity


Brook Green Centre for Learning’s aim is to develop successful learners who are numerate, creative and enquiring and able to solve problems.
Our highly dedicated Mathematics team possess an outstanding range of skills and experience in education and the wider world including uniformed services and engineering. The department is headed by a Mathematics specialist, working alongside a specialist Scientist, our ICT Lead teacher, specialist Technology teacher (with a strong interest in electronics and engineering) and our Humanities Lead teacher; all are Secondary trained. We also have a specialist Higher Level Teaching Assistant for the maths department who uses intervention strategies to offer further support to pupils on a one to one or small group basis. The diversity we offer supports and enhances our strong Mathematics scheme of work and helps our students become more aware of the challenges facing them as they leave school.
We seek to create a supportive learning environment. We make great use of a variety of software, books, web-sites, games and real-life materials to engage students and help them learn in the way that best suits their needs. We are constantly looking for innovative ways and resources to help our students to not just ‘do’ mathematics but to understand ‘why’ and ‘how’ as well.
We have comprehensive schemes of work available as electronic resources on our school server allowing teachers to better consider and reflect on their teaching and improve creativity and effectiveness in the classroom.
Numeracy is developed across the school as well as cross-curricular provision. Targeted Intervention provision is delivered by our secondary trained Intervention Lead Teacher and is used extensively to help students who encounter specific difficulties in an area of Mathematics that result in barriers to learning.
The department has a very strong progress tracking and analysis system which proves invaluable in identifying strengths and weaknesses both in teaching and learning, enabling us to always find opportunities for improvement.
Years 7 & 8
Year 7 & 8 children are set by ability. Year 7 setting is achieved by analysing their KS2 levels prior to entry and clarified by our base-line assessments. The Year 7 & Year 8 analyses are compared and both year groups are set by ability into two sets.
This system allows us to deliver the best teaching possible by matching teachers with certain skill-sets to appropriate ability ranges.
Students follow the National Curriculum Renewed Framework for Mathematics. We adapt the Numeracy Strategy to meet our children's needs.
Pupils make good progress in all aspects of mathematics, including number, algebra, shape and measure and data handling.
Activities for our less able children's needs include counting, measuring, comparisons, pattern making and problem solving. The work is often practical and may include songs, rhymes and play activities (pretend shopping, etc.)
All pupils will be expected to complete one mathematics piece of homework each week.
Years 9, 10, 11
All of our children have the opportunity to start working towards an Entry Level Pathways qualification. The most able children will successfully complete the level three course in year 9 and the majority will complete the level three course in year 10. For those that find this too difficult they will be entered for the level two course. We expect all students to gain at least WJEC Entry Level 2 Mathematics.
The year 10 and 11 pupils will follow the WJEC GCSE mathematics syllabus at foundation level. This will give them a solid grounding in all areas of mathematics, increasing their confidence and abilities in the subject. Most of our pupils will be entered for the WJEC GCSE Mathematics exam.
For less able children in Year 10 & 11, there is more emphasis on real life skills, such as time and money whilst working on their relevant course. These skills are further enhanced by the many extra-curricular visits and courses, including College courses and our fantastic Citizenship programme.


What Is the Mode of Math – What Is It?
The fantastic thing about Motion Alarm Clock is it forces users to continue to keep your iPhone in motion for a couple seconds to block the ringing alarm. Not just that, but even after solving the issue, the app asks users to talk out the answer loud enough to block the alarm. So, in the event the native alarm app of your iPhone failed to wake you up in the early hours, then it’s possible to consider a number of the alternate alarm apps listed inside this write-up.
All About What Is the Mode of Math
Line plots are another means to organize data as it’s being collected. There are definition posters for each one of the data landmark conditions. resume writer Line graphs are utilized to demonstrate how data changes as time passes.
You will likely still hit a vital site. It is by far the most typical member of the group. In statistics, there are 3 common techniques to get the average of a group of information.
The Awful Side of What Is the Mode of Math
Even though the mean is easily the most frequent kind of average, the median may also be utilized to express the average of a group. The you started studying statistics and all a sudden the average is currently known as the mean. Median is another type of average.
Mean is the average of all of the values. It is also the number that is halfway into the set. Thus, it is the value in the center position.
When you pay a trip to the A version, you’re likely to be in a place to observe any other versions which exist. Robots could also be beneficial in assisting in surgery, manufacturing, etc. When you’ve gathered all of the vital tools, you’re now prepared to commence the job of fixing your own vehicle.
What You Need to Do About What Is the Mode of Math Before You Miss Your Chance
There https://seras.uib.es/ciencia/ are 3 distinct numbers that could represent the idea of a regular price, and it’s important to comprehend which one is used, and whether that’s appropriate. The remedy to an addition problem is called the sum. The mean is going to be displayed in the event the calculation is successful.
All ACT statistics questions are just variations on the exact theme, so knowing your foundations is vital. Just make sure you keep your mind sharp and your work organized, and you’ll be in a position to tackle SAT statistics problems instantaneously. If you’re attempting to rush, it may get second nature to glance with an m-word and begin in immediately on solving the issue.
What Is the Mode of Math – What Is It?
Later, you can want to read more about the way to include more packages and the way you can create packages of your own. Next a meeting date will be set. While mean and median supply you with a tremendous picture idea, the mode provides you a notion of what number you’re likely to encounter.
An overlay with similar picture resembles an S-curve. It’s usually faster than fully sorting the array, if you don’t want to compute it for at least a few orders. Unfortunately, these types of twists on simple concepts can arrive in many different distinct forms.
What Is the Mode of Math Secrets
10 minutes a day is all it can take to find a substantial improvement in your children math abilities. An excellent case of this is figuring out what the normal score for all of the students who took a test in a course. Most games are appropriate for elementary and middle grades.
The Ugly Side of What Is the Mode of Math
If it is a positive number, you want to decrease 1. You should not presume that your mean is going to be one of your initial numbers. These videos will explain to you how to locate the median.
Whatever They Told You About What Is the Mode of Math Is Dead Wrong…And Here’s Why
Appropriate comprehension of given situations and contexts can often offer a person who has the tools essential to ascertain what statistically relevant procedure to use. Median in math is a term that’s used when you’re trying to find the mid point. You use various averages in a number of scenarios, dependent on what you would like to communicate with your sums.
The Fight Against What Is the Mode of Math
Pause the video here to see whether you can discover the mean of this data collection. A comprehensive training pass over the entire data set such that each and every example was seen once. Employing the list feature is simply as simple as entering the data onto the home screen, and it contains the extra advantage that it is possible to use the data for different purposes after you’ve calculated your mean, mode and median ( for instance, you may want to create a TI 83 histogram).
The mode has become the most frequent value in a set of information. It is not often used in describing data, but it can be useful in certain circumstances. You also need to be mindful that there can be more than one mode.
Just because you’re able to use mean, median and mode in the actual world doesn’t signify that every measure applies to any scenario. Display mode equations must show up on their very own line. And finally, mode questions very rarely appear on the ACT.
The table indicates the results. If you select a marble from the bag without looking, you can select a red marble or you are able to select a blue marble. It isn’t possible you will decide on a blue marble.
Because these kinds of questions will appear 1 to 2 times on the test, you will notice them in a selection of distinct forms. Sets can be categorized into many types. Certain pathological distributions ( for instance, the Cantor distribution) have no defined mode whatsoever.


- "Number and Operations" forms the basic math knowledge of a students quantitative abilities. It tests a student's abilities in basic arithmetic in areas such as properties of integers, elementary number theory, fractions/decimals, percentages as well as ratios and proportions. Other thinking and arithmetic skills include the ability to compute and use number sequences, understanding unit conversions as well as logical reasoning skills.
The SSAT requires some knowledge of Algebra. Students should know what Algebra is, its uses as well as how to answer questions involving Algebra. This includes simplifying algebraic expressions, solving simple equations as well as working with word problems involving algebra. Substitution or replacing of variables with numbers and evaluating these expressions are also an integral part of Algebra for the purposes of the SSAT.
- Geometry skills required for the SSAT include the ability to work with points, lines and angles in a 2-dimensional plane. Students need to demonstrate the ability to work with simple shapes and figures such as triangles, quadrilaterals as well as circles and to calculate the area and perimeter of these shapes. Geometry also includes coordinate geometry where students need to understand and be able to apply linear equations, determine slopes and read coordinates. Visualization of solid figures will also be tested in Geometry.


Alright, finally found it
" Take your Actual AFR and subtract it from your Desired AFR. Now take that number you get and divide it by your Desired AFR. Take that number and times it by 100. That will give you the percentage you need to add or subtract.
So example. My AFR is 10.5 but I want 11.7. So 10.5 -11.7 = -1.2. Then -1.2/11.7 = -0.103. Now -0.103 X 100 = -10.3. Round that to the nearest 0 or 5 so you would subtract 10.5% from the DSP under the wot fuel modifier. "
I started by removing 5.5 % at 3-4, 5-6, 6-7 and 4.5% 1-2, 2-3 thousand rpm's and I'll update the datalog and see where I'm sitting at. I'm shooting for 11.7-11.5 And I'll backoff the timing when I get my fuel where I want it to try and combat knock.
I know the pid's are missing alot but using the intune + dataviewer to log my wideband is a pain in the ass. Works 1/2 the time, other times all the data in the log is insanely off.
I wish they would fix dataviewer with the intune, but they already have my $$$ .......so I'll probably have to live with it working 1/2 the time and saving a successful log 1/4 of the time.


Boise State University has been awarded a $2 million National Science Foundation grant to increase evidence-based best teaching and learning practices in foundational science, technology, engineering and math (STEM) courses.
The three-year grant aims to engage faculty members across college and departmental boundaries to put into place practices that research shows promote student success in STEM subjects.
Applying an intentional change process is a unique element of this project and puts Boise State in the national spotlight for education reform, said Susan Shadle, director of the Center for Teaching and Learning and principal investigator for the project.
Courses targeted during the three-year project include general chemistry, calculus and pre-calculus, general physics, lower-division engineering and upper-division geoscience courses. These courses were chosen because many students take them as gateways to upper-division study in STEM majors. Students who major in fields other than STEM also take many of these courses.
Goals of the project include: Æ Continuous improvement in the teaching of STEM subjects Æ Increasing the number of STEM majors Æ Attracting more women and underrepresented groups to STEM majors Æ Retaining STEM students from the first to the second year Tony Marker, associate professor of organizational performance and workplace learning, will use a business model for facilitating change to guide implementation of the project, Shadle said.
Leaders in the project include Tony Roark, philosophy professor and dean of the College of Arts and Sciences, Amy Moll, dean of the College of Engineering, Eric Landrum, psychology professor, and Doug Bullock, mathematics associate professor.


If ever you will be needing assistance with algebra and in particular with vertex form online calculator or calculus come visit us at Algebra-equation.com. We offer a ton of good reference materials on topics varying from quadratic function to rational functions.
Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. The vertex can be found from an equation representing a quadratic function.. The domain of a quadratic function is all real numbers. The range varies with the function.Polynomial equations in factored form All equations are composed of polynomials. Earlier we've only shown you how to solve equations containing polynomials of the first degree, but it is of course possible to solve equations of a higher degree.However, some polynomials of higher degree can be written in quadratic form, and the techniques used to factor quadratic functions can be utilized. Changing Form Factoring a polynomial, such as x.
Identify the vertex and general forms of a quadratic function. Find the vertex of a quadratic function. Identify the vertex as the minimum or maximum point on the graph of a quadratic function. Write a quadratic function given in vertex form, in general form instead. Write a quadratic function given in general form, in vertex form instead.
Find Equation In Vertex Form Calculator Five Facts About. Vertex From Standard Form Calculator Five Things You Most. What Is Vertex Form Example Get Education. Completing The Square To Write A Quadratic In Vertex Form. Standard Form To Factored Calculator Math Write Polynomial. Factored Form To Standard Calculator Math Vertex Definition.
Vertex Form Of An Equation Calculator Five Precautions You. What Is Vertex Form Example Get Education. Vertex From Standard Form Calculator Five Things You Most. Write In Vertex Form Math Converting From Factored To. Standard Form To Factored Calculator Math Write Polynomial. Factored Form To Standard Calculator Math Vertex Definition.
So i am being told to find the vertex form of a cubic. Further i'd like to generalize and call the two vertex points (M, S), (L, G). I understand how i'd get the proper x-coordinates for the vertices in the final function.
I am taking an online how to write equation in vertex form course. For me it’s a bit difficult to study this course all by myself. Is there some one studying online? I really need some guidance.
Explanation:. This graph has zeros at 3, -2, and -4.5. This means that, , and .That last root is easier to work with if we consider it as and simplify it to .Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.
Polymathlove.com contains both interesting and useful material on writing a standard equation in vertex form, functions and final review and other math subjects. Just in case you have to have advice on inverse as well as factoring, Polymathlove.com is certainly the right place to have a look at!
The vertex form of a quadratic equation can help you quickly identify the vertex of that quadratic. Follow along with this tutorial to see how to use the completing the square method to change a quadratic equation from standard form to vertex form!
Factoring-polynomials.com makes available insightful info on standard form calculator, logarithmic functions and trinomials and other algebra topics. In the event that you need to have advice on practice or even math, Factoring-polynomials.com is the ideal site to take a look at!
Vertex: Axis of symmetry: Domain: (All real numbers, just like any other polynomial function) Range: The lead coefficient is negative, therefore the parabola opens downward and the vertex represents a maximum value of the function. That maximum value is the -coordinate of the vertex, namely. The range is unconstrained less than the maximum, so.
How to write vertex form, right triangle trig calculator, graphing rational functions with asymptotes online, 2nd grade printouts, inequality on a number line using a calculator. Factor machine polynomials, solving inequalities calculator free, Grade 11 free maths textbook, focus of a circle?, quadratic formula ti 83 plus, adding, subtracting multiplying decimals worksheet, bond ks2 past.
When giving a final answer, you must write the polynomial in standard form. Standard form means that you write the terms by descending degree. That may sound confusing, but it's actually quite simple. Here's what to do: 1) Write the term with the highest exponent first 2) Write the terms with lower exponents in descending order.
This Quadratics: Polynomial Form Lesson Plan is suitable for 9th - 12th Grade. Throught this subscription-based sight, learners explore different aspects of the parabola by changing equations from standard to vertex form. Next, find the general form of the vextex based on the values of a, b, and c, and investigate the minimum and maximum points of a real-world example.


Welcome to our introduction about blackjack
If you want to fully master the game of blackjack, you must really obtain a deep understanding of blackjack probabilities and odds. It is crucial to know why the casino gains its edge over all players in the long run and how you can minimize this edge by playing an optimal blackjack strategy. Note that blackjack is very different from poker, as there is no bluffing at all.
A casino game like blackjack demands a considerable element of skill, substantial knowledge of the gameís know how and a constant dose of practice. Rather than simple and sheer luck, blackjack is one card game whose probabilities of success depend more on good play.
However it is as important to accept that it is next to impossible to beat the game consistently. Losses are an equal part of blackjack probabilities. By clearly understanding the table of numbers that represent blackjack probabilities, it will immensely help players to understand the game and increase their chances of winning.
The blackjack probability table also indicates the chances of winnings. The blackjack probability table also indicated the chances of beating the dealer by advising bets tactics in different gaming situations. Blackjack probabilities are not nearly as complicated as you might perceive.
In fact you can substantially increase your odds when playing Blackjack by simply developing a clear and thorough understanding of the game. Since blackjack, contradictory to other card games, is a game of skill and expertise, it demands a familiarity of the mathematical underpinnings in order to beat the house.
The gambling odds in a lot of casino blackjack games are 0.8% that arises from the fact that the dealer plays after all the players have crossed the possibility of busting. It is extremely important to see when you should hit of not depending upon the hand combination you have. If you hand count is 21, blackjack probability to hit is 100%. Gradually, as you hand count falls lower than 21 to 20, 19, and 18 and so on, the winning odds also simultaneously dip. If you hand count is 11 or less, then the blackjack probability to win nil or 0%.
A very important thing you need to remember is that all your hands that are 16 or below are of approximately the same value, since your only chance of winning in that case is only if the dealer busts. In a 2 card game, the chances of being dealt a natural 21 are only 4.8%. A value of 17-30 has a probability percentage of 30% and being dealt a value of 1-16 is 38.7%. And blackjack probability of being dealt a No Bust is 26.5%.
A mental mastery of these numbers will enable you to make the best decision since being familiar with the percentages is quite beneficial when playing. The correct application and implementation of the basic blackjack strategy lowers the casinos odds to 0.5%. The dealerís probability of busting depends on the face up card. The croupier or the dealer has the highest probability to exceed 21 with a 5 face up card and the player has the least blackjack probability when the croupier has a 10 card.
Good luck to you at the online casino tables.
Start playing online blackjack today at absolute poker.


One of our favorite math tools my kids and I created were the mini-offices using math printables and display boards that we cut in half horizontally. These were pulled out during math time to use for easy reminders of math concepts, formulas and more. Above is what ours looked like.
There are all kinds of ways to create a mini-office in addition to the way we chose to do ours. Some folks use file folders while others keep notebooks. It is all a matter of preference but for us the display board method worked great.
For our mini-office, I printed out things I found on the internet or made copies from other resources we had such as textbooks and workbooks. Ours were made when my kids were in middle and high school so what we included helped them with some of what they were working on at the time. The things we included on our board were (the ones with the * are included in our free Math Mini-Office Printables PDF we created for you to get you started on creating your own math-mini office):
*A Hundred's Chart (we colored all the prime numbers)
*A multiplication chart (we colored the square numbers)
*English vs Metric Measuring System chart
* A mini time chart
*A chart that explained mean, mode, median, and range
*A chart to explain chance, probability, odds
* Unit Equivalents Tables
* Divisibility Rules
* Place Value Chart
* Properties Charts
* Time Equivalent Chart
* Triangle classification chart
*Classification of quadrilaterals
*Regular and irregular polygons
*Types of angels
*Geometric Solids chart
* A How to Do Math Functions Tab Booklet
Ruler measurements; US and metric
A fraction/decimal equivalent chart
A booklet about angles, lines, rays, etc…
Here are a couple of websites that share their mini-office versions along with some printables you can use.
Need a starting point of printables for your office? Check out these sites:
And don't forget to grab our free Math Mini-Office Printables PDF.
Our math offices got a lot of use over the years and proved to be a wonderful tool for helping my kids be independent in their math studies and have information at the ready if the need arose.