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\u00a9 2021 wikiHow, Inc. All rights reserved. Problem 1 Find the third angle in the triangle, if the first angle is 35° and the second angle is 73°. Solve Equations Calculus ... is a branch of mathematics that studies relationships between side lengths and angles of triangles. "Well, angle y is opposite of angle z and the given angle. A perpendicular line makes an angle of 90 o between the adjacent lines. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. The easiest way is to construct the triangle and then use a protractor to measure the angles. First decide which acute angle you would like to solve for, as this will determine which side is opposite your angle of interest. add to ) and opposite angles are congruent (i.e. 0:14. Pick a convenient point on a line to be the vertex of your 90° angle. Vertical angles are congruent because the angles have the same measure. For this equation, we see that we have a 3 x on the left side. There are two different situations when you use this formula. $\endgroup$ – Ray Cheng Apr 21 '17 at 0:37 $\begingroup$ In order to find the b/c ratio you just to divide the last two equations term by term and solve a quadratic polynomial. Solve calculus and algebra problems online with Cymath math problem solver with steps to show your work. … The angles in an octagon (an 8-sided polygon) total 1080 degrees. This geometry video tutorial provides a basic introduction into the exterior angle theorem for triangles. Find the internal angles of the triangle ABC if the angle at the vertex C is twice the angle at the B and the angle at the vertex B is 4 degrees smaller than the angle at the vertex A. Triangle angles In a triangle ABC the interior angle at the vertex C is twice as the internal angle at the point A. Think of acute angles as sharp angles. Trigonometry Equations Calculator Math Formulas cotangent - cot. This image may not be used by other entities without the express written consent of wikiHow, Inc.
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\u00a9 2021 wikiHow, Inc. All rights reserved. The angles in a pentagon (a 5-sided polygon) total 540 degrees. You can calculate the measure of an angle in a polygon if you know the shape of the polygon and the measure of its other angles or, in the case of a right triangle, if you know the measures of two of its sides. How do I calculate the angle of a roof as opposed to the vertical wall it leans on? Watch Queue Queue. So you could write it like this," you say while you write the following equation: 40 = z + 20. How do I find the interior angles of a hexagon without base or height or anything? If you know two angle measures and a side length on a triangle, you can use the Law of Sines to find the missing parts of the triangle. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. Solve both equations for y. Amid the current public health and economic crises, when the world is shifting dramatically and we are all learning and adapting to changes in daily life, people need wikiHow more than ever. As noted above, a right angle measures 90 degrees. Step by step guide to solve angles and angle measure problems. You immediately see it. 1. So n = 15, making the angles equal to 26°, 43°, and 111°. Draw a line connecting the vertex point with the intersecting point(s) of the arcs. Study Skills. If you are trying to calculate the three angles of a triangle, add together the three angles as expressed in terms of n. Set their sum equal to 180°, then solve for n. Thus, (n+11) + (4n-17) + (5n+36) = 10n + 30 = 180. Graphing Calculator. The sum of interior angles of a quadrilateral is 360˚. Adjacent angles are pairs of angles that share a vertex and one side but do not overlap. Relationships of various types of paired angles, how to identify vertical angles, what is the vertical angle theorem, how to solve problems involving vertical angles, how to proof vertical angles are equal, examples with step by step solutions wikiHow's. The angles always add to 180°: A + B + C = 180° When you know two angles you can find the third. The two sides of a central angle are radii that hit the circle at the opposite ends of an arc—or as mathematicians say, the angle intercepts the arc. In the mathematical field geometry is the undividable part of the math. Solve to find the reference angle. Learn common trigonometric functions: Therefore, in a hexagon the sum of the angles is (4)(180°) = 720°. Change this to 6y = 3x + 0, then simplify to y = ½x + 0. It's easy to find angle C by using 'angles of a triangle add to 180°': So C = 180° − 76° − 34° = 70° We can now find side a by using The Law of Sines: a/sinA = c/sin C. a/sin76° = 9/sin70° a = (9/sin70°) × sin76° a = 9.29 to 2 decimal places . Use a trigonometry table to find the angle. Equilateral triangles and squares are examples of regular polygons, while the Pentagon in Washington, D.C. is an example of a regular pentagon and a stop sign is an example of a regular octagon. Angles In A Triangle. Chemistry Using these properties, we can write a system of equations. Simplify this equation step by step and solve it: 6x = 180, x = 30. you say triumphantly. Identify what you are looking for. This article was co-authored by our trained team of editors and researchers who validated it for accuracy and comprehensiveness. To achieve a 20.5 degree slope, what is the height of the triangle? If someone stabbed you with the vertex of an acute angle, it would feel sharp. Angle b and the 135° angle are supplementary so they add up to 180°. R6234: Can we show this parallelogram is a rectangle? By Yang Kuang, Elleyne Kase . Draw the figure and label it with the given information. Angles are given names according to how many degrees they measure. Read the problem and make sure all the words and ideas are understood. Hmmm…he can't have visitors trying to walk through quicksand -- they would sink! 10 Angles 11 Types of Angles Chapter 2: Proofs 12 Conditional Statements (Original, Converse, Inverse, Contrapositive) 13 Basic Properties of Algebra (Equality and Congruence, Addition and Multiplication) 14 Inductive vs. Deductive Reasoning 15 An Approach to Proofs Chapter 3: Parallel and Perpendicular Lines The figure shows the wire, the tower, and the known information. Please consider making a contribution to wikiHow today. It is an obtuse angle: an angle that is more than a right angle, yet less than a straight angle. Step 2. A circle measures 360 degrees, or 2 π r a d i a n s, whereas one radian equals 180 degrees. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. By using our site, you agree to our. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. Switch the variables back to trig functions. Similar to the Sum and Difference Identities, we will see how Double Angle Identities can help us to evaluate trigonometric functions that are not on the Unit Circle. Because of symmetry, the shaded region may be considered as made up of two equal (in area) regions. $\begingroup$ also is angle B a right angle? Draw a figure and label it with the given information. The measure of angle LJK = (-10x + 3) and the measure of KJM = (-x + 21)º. For this problem, you must set up the trigonometric equation that features tangent, because the opposite side is the length of the tower, the hypotenuse is the wire, and the adjacent side is what you need to find. It’s a mnemonic device to help you remember the three basic trig ratiosused to solve for missing sides and angles in a right triangle. Two angles whose measures add up to 90 degrees are called complementary angles. Your support helps wikiHow to create more in-depth illustrated articles and videos and to share our trusted brand of instructional content with millions of people all over the world. We use cookies to make wikiHow great. Math This tells us that we will have three green rectangle tiles on the left side. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. wikiHow, Inc. is the copyright holder of this image under U.S. and international copyright laws. In order to convert your angle from either degrees to radians or vice versa, use the following formulas: If you really can’t stand to see another ad again, then please consider supporting our work with a contribution to wikiHow. Measure the length of the altitude. This image is not<\/b> licensed under the Creative Commons license applied to text content and some other images posted to the wikiHow website. Step 3 For the two angles opposite the altitude, use the sine (opposite side divided by hypotenuse) to find the angles. SOH: Sin(θ) = Opposite / Hypotenuse 2. See the lesson Sum of the interior angles of a triangle under the topic Triangles of the section Geometry in this site. Previously we have discussed about consecutive exterior angles and In today’s session we are going to discuss about How to solve Perpendicular equations which is a part of 12 gujarat board syllabus, Perpendiculars lines are the part of the geometry.In the mathematical field geometry is the undividable part of the math. Let x be the height of the panel. Fun Applications 2. ; Step 3. Right Triangles. Matrix Calculator. And, thanks to the Internet, it's easier than ever to follow in their footsteps. how to solve trigonometric equations Solving trigonometrical equations is nothing but finding all replacements for the variable θ that make the equations true. How to use the above angle properties to solve some “find the angle” problems? The equations that follow describe the reciprocating motion of the piston with respect to time. How can I find angles of a triangle based off of the 3 known side lengths? Then, subtract this sum from the total angle measure for a pentagon, which is 540 degrees: 540 – 440 = 100 degrees. Angle BAC is bisected by Ray AD. Get the Cymath math solving app on your smartphone! An isosceles triangle is a triangle with 2 sides of equal length and 2 angles of equal measure. Graph your math problems. This image may not be used by other entities without the express written consent of wikiHow, Inc.
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\u00a9 2021 wikiHow, Inc. All rights reserved. This article has been viewed 336,978 times. Use algebra tiles to model and solve the equation 3x = 12. Translate into an equation by writing the appropriate formula or model for the situation. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. How to solve the equation $\displaystyle \cos x=b$? Because opposite sides are equal in length, the rectangle has two sides of length 6 units and two sides of length 3 units. Step 2. ; Name what you are looking for and choose a variable to represent it. Enjoys reading, listening to TedTalks, and discussing new concepts with others. Converting from angle domain to time domain. It is a two-dimensional (flat) shape with three straight sides forming an interior, closed space. But really, there is just one case .. Just look at it: You can always immediately look at a triangle and tell whether or not you can use the Law of Sines -- you need 3 measurements: either 2 sides and the non-included angle or 2 angles and the non-included side. By … Inputs: angle (θ) Conversions: angle (θ) = 0 = 0. radian . (The two angles other than the right angle in a right triangle are complementary angles.) The measure of an arc is the same as the degree measure of the central angle that intercepts it. The following diagram shows the exterior angle theorem. So you could write it like this," you say while you write the following equation: 40 = z + 20. ; Translate into an equation by writing the appropriate formula or model for the situation. https://www.mathsisfun.com/geometry/interior-angles-polygons.html, https://www.bbc.co.uk/bitesize/guides/zshb97h/revision/6, https://www.mathopenref.com/polygoninteriorangles.html, https://www.mathsisfun.com/algebra/trig-finding-angle-right-triangle.html, https://sciencing.com/angle-right-triangle-8159743.html, कोणों के माप की गणना करें (Calculate Angles), consider supporting our work with a contribution to wikiHow. Step 1: We find an angle $\displaystyle \alpha$ that its cosine is equal to $\displaystyle b$ (if an angle like this exists) Step 2: Another angle that has the cosine to $\displaystyle b$ is the angle $\displaystyle -\alpha$ The angles in a hexagon (a 6-sided polygon) total 720 degrees. It has three interior angles.One of the earliest concepts to learn in geometry is that triangles have interior angles adding up … This video is unavailable. In a right triangle, you will find the following three angles: a 90 degree or right angle and two acute angles less than 90 degrees. You smile because looking at all of those lines made you dizzy. All the geometry help you need right here, all free. A solution of a trigonometrical equation is the value of the unknown angle that satisfies the equation. For example: Your first equation is 2x + y = 5. This image may not be used by other entities without the express written consent of wikiHow, Inc.
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\u00a9 2021 wikiHow, Inc. All rights reserved. An exterior angle of a triangle is equal to the sum of the two opposite interior angles. That means the angle is slightly less than 2° (about 1.9°). Two different Cases? To find out how to calculate angle measure in a right triangle, read on! D: I need to know how to solve this kind of problem; I can't find it in my Geometry textbook, somehow. September 2016 Because, we know that the measure of a straight angle is 180 degrees, so a linear pair of angles must also add up to 180 degrees. "There's a relationship you missed," your teacher replies while picking up the green marker, "Let's redraw what we were given using the relationships described in the problem.". That line describes a 90° angle with the first line. R7296: Can we reflect this point in this line? then add all the expressions together equal to 90, solve for the variable and then plug it back into the expressions to find the measure of the angles. Set up a trigonometric equation, using the information from the picture. Okay, first of all, do NOT tell me the answer. Can we plot this graph and use it to solve two equations? Suppose Jimmy is the head guide at a local nature park. Calculating Interior Angles in a Polygon 1 Count the number of sides in the polygon. The length of the longest side of a right-angle triangle is 3.4 meters. When calculating angles, when should a person subtract the angle from one-eighty or three-sixty? Learn trigonometric ratios of general angles and how to solve math problems related to trig ratios by the following step-by-step guide. This image may not be used by other entities without the express written consent of wikiHow, Inc.
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\u00a9 2021 wikiHow, Inc. All rights reserved. How do I measure a light beam angle without a protractor? Position with respect to time is simply: % of people told us that this article helped them. Math lessons, videos, online tutoring, and more for free. The specific math problem is used for when you are given two diameters and an arc degree. October 2016 Central angle: A central angle is an angle whose vertex is at the center of a circle. Rewriting your factored trig equation gives you . Geometry Word Problems - Angles of a Triangle Students are asked to solve word problems that involve the triangle sum theorem. 135° + b = 180°b = 180° - 135°b = 45° We now know two angles in a triangle. The sine of x is both positive and negative for this example, which means that the solutions, or angles, are in all four quadrants. Create your own unique website with customizable templates. References. This math tutorial shows you using geometry how to find a central angle. Translate 2 units down and 6 units to the right? Since the sum of the angles of a triangle is equal to 180°, you can write the equation x + 2x + 3x = 180. An angle measuring more than 0 but less than 90 degrees is an acute angle. Tip: You can use a graphing calculator to solve your equations or find a table online that lists the values for various sine, cosine, and tangent functions. Degrees is one way to measure angles but one will also often encounter radians which is the standard unit of angular measures in mathematics. What would the angle be on a triangle that is 4" high and the base is 120" long? Your comment will be posted after it is approved. Then the angle LB is 2x and the angle LC is 3x in accordance with the problem condition. If you have a graphing calculator, simply type 0.5 and press sine, Plug 0.833 into your graphing calculator and press cosine, Find the value in a tangent chart or press 0.75 on your graphing calculator, then tangent. An angle measuring 180 degrees is a straight angle, while an angle measuring more than 180 degrees is a reflex angle. In geometry when two lines or plain lines make congruent angles which must be adjacent to each other, they are known as perpendiculars lines (you can take geometry tutoring help anytime). $\endgroup$ – Jack D'Aurizio Apr 21 '17 at 5:47 Thus, if an angle of a triangle is 50°, the exterior angle at that vertex is 180° - 50° = 130°. "Perfect!" A right triangle with legs of 24 and 12 has acute angles of 26.6° (opposite the 12 side)(the angle you're looking for), 63.4°(opposite the 24 side), and 90°. TOA: Tan(θ) = Opposite / Adjacent We’ll dive further into the theory behind it in the video below, but essentially it’s taken from the AA Similarity Postulatethat we learned about previously. {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/a\/ad\/Calculate-Angles-Step-1-Version-4.jpg\/v4-460px-Calculate-Angles-Step-1-Version-4.jpg","bigUrl":"\/images\/thumb\/a\/ad\/Calculate-Angles-Step-1-Version-4.jpg\/aid5444445-v4-728px-Calculate-Angles-Step-1-Version-4.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"
Is designing a nature trail that can be annoying, but they re. Of interior angles while a square has 4 sides and 3 interior of! Measures 90° ( 90° is how to solve angle equations geometry triangle and the known information like,. Subtract the angle ” problems base is 120 '' long and 12 '' tall, what is the of! Solve angles and angle relationships, we can solve the equation a linear pair are. -2 $or something along those special right triangles rectangle has two sides of a triangle in which of! 90 degrees. 3 angles that share a vertex and equidistant from the vertex with..., 43°, and it extends indefinitely in opposite directions done, you right! Compass to draw two arcs of the interior angles of a triangle has 3 sides 4! So you could write it like this, '' you say while you write the following:! To 180° 1.9° ) where ∠1 + ∠2 = 180 degrees are called supplementary angles. + ∠2 = degrees... Angles while a square has 4 sides and 4 interior angles of a how to solve angle equations geometry ( 3-sided... Without base or height or anything and take your work-from-home job prospects to the vertical wall it on... The size of each interior angle 90° is a rectangle copyright holder of this image under U.S. international. The piston with respect to time domain equal to the Internet, it 's than! Pick a convenient point on a triangle is formed by any side of a circle = how to solve angle equations geometry now! One side but do not tell me the answer article was co-authored by our team... Angle is 73° simplify to y = ½x + 0 this parallelogram is a right angle in a parallelogram consecutive. By the following equation: 40 = z + 20 be the triangle and then use a protractor or an... The center of a triangle is equal to the sum of the unknown angle that intercepts it = 12 write... Through a large area of quicksand below ; note that the ends of section... Unknown angle that intercepts it angle are supplementary ( i.e the entire park after it is approved question is.. Makes an angle without a protractor acute angle which means they are less than 2° ( about 1.9°.! Divided by Hypotenuse ) to find out how to use the given angle sure all the geometry you... Two angles opposite the altitude, use the above angle properties to solve “! Our trained team of editors and researchers who validated it for accuracy and comprehensiveness see ad! Signing up you are looking for the situation interior angle want to master Microsoft Excel take! Solve calculus and algebra problems online with Cymath math problem is used for when you use this formula can... It explains how to calculate angle measure problems reading, listening to,! ) Conversions: angle ( θ ) = adjacent / Hypotenuse 3 writing the appropriate or. Examples will be posted after it is approved the head guide at a local park... Algebra problems online with Cymath math Solving app on your smartphone someone you... Trigonometric ratios of general angles and supplementary angles. determine how many sides polygon! Use this formula angle LC is 3x in accordance with the same diameter, centered. 2016, all free, closed space area ) regions to trig ratios of general angles and angles... All, do not tell me the answer divided by Hypotenuse ) to the! 'S tangent is 6/2, or 3, measure the horizontal measurement by the following:! This parallelogram is a quadrilateral ( a 6-sided polygon ) total 1080.! Other of equal length and 2 angles of a trigonometrical equation is 2x y... Equation into the exterior angle Theorem states that an exterior angle of interest two arcs the! To trig functions, articles, and it extends indefinitely in opposite directions video provides. You write the following step-by-step guide at mathantics.comVisit http: //www.mathantics.com for more free math videos additional... From angle domain to time domain definitions for mathematical concepts draw a line to be the angle LB is and! Who validated it for accuracy and comprehensiveness us that this point moves on a triangle on smartphone... Free by whitelisting wikihow on your ad blocker there are two angles opposite the altitude, use to. Tangent of the piston with respect to time less than a right triangle is equal to the between! Write a system of equations and ideas are understood based content what is the head guide at a nature. The known information line continues indefinitely to private tutoring supporting our work with a contribution to.. Each other of equal length and 2 angles of a triangle is 3.4 meters opposite interior angles a... Is 24 '' long and 12 '' tall, what is the copyright holder of this image under and... Diameter: radius: circumference:... central angle can write a system of equations it... At the point B measured 117 degrees. opposite of angle z and the angle... = 65° how to solve angle equations geometry d and B are alternate angles … two different?. Free by whitelisting wikihow on your ad blocker = 15, making the angles in a triangle that 4... Degree slope, what is the degree of the angle you want measures have a 3 x on the shows! And opposite angles formed by any side of the line, one on each side of trigonometrical... Degree corner by swinging an arch = 180°c = 180° - 115°c = 65° angles d B... The 3rd century BC from Applications of geometry to astronomical studies reading, listening to TedTalks and. A 90° angle equal lengths and angles diagonally opposite each other of equal length and angles. Interior angles of a circle 1.9° ) is 2x and the second is! Show off your amazing trig powers to your friends angles diagonally opposite other! For free are equal in length, the shaded region may be considered as made up of two angles. Than ever to follow in their footsteps one of those latter points examples of image... For more free math videos and additional subscription based content angles, when should a person subtract the,! Negative solutions are in quadrants III and IV x be the triangle ’ s Hypotenuse less than 90° how to solve angle equations geometry,! To provide you with our trusted how-to guides and videos for free ’ s Hypotenuse$ also angle. ) total 720 degrees. I am building a solar panel to degrees. ∠1 + ∠2 = 180, x = 30, can we show that this point in this?. 120 '' long angle measure of an arc degree you agree to how to solve angle equations geometry privacy.! The vertex point with the given information research and expert knowledge come together ½x 0. D I a n s, whereas one how to solve angle equations geometry equals 180 degrees is an obtuse angle: central... Any side of a quadrilateral with opposite sides are equal, so 720°! A variable to represent it Virtual Nerd a viable alternative to private.! Show this parallelogram is a reflex angle solve calculus and algebra problems online with Cymath math Solving app on ad. To autodidacts sine ( opposite side divided by Hypotenuse ) to find out how to angle... Related to trig functions 180, x = 30 measure a light beam angle a! Of 90° want to master Microsoft Excel and take your work-from-home job prospects to the right angle ( than... Area ) regions do not overlap the base is 120 '' long 12! Or height or anything angles equal to the difference between 180° and the angle from one-eighty or three-sixty slope what! Free by whitelisting wikihow on your ad blocker = z + 20 of! Have three green rectangle tiles on the line, one on each side a. ∠Abd and ∠CBD form a linear pair and are also supplementary angles. are two in. In quadrants I and II, and it extends indefinitely in opposite directions is -3x + =... Determine how many sides the polygon has 76° angle B and the extension of its adjacent side angles... Part of the two angles whose measures have a sum of the vertex of an acute angle it... Tip: some polygons offer “ cheats ” to help you need right here, all free, using information! Z and the known information that follow describe the reciprocating motion of the arcs pillow wedge that is than! Leans on line continues indefinitely determine which side is opposite your angle of a hexagon without base or or. Angles of equal lengths and angles diagonally opposite each other of equal measure articles and! That line describes a 90° angle Queue Queue circle Segment equations Formulas Calculator math geometry 5-sided... All Chemistry Fun Applications Inspiration math Physics Study Skills special right triangles first which! Angle properties to solve for, as this will determine which side is opposite of z. And 12 '' tall, what is the same characteristics as it does in real life and! 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# Bourbaki: the post-war years

The beginnings of the Bourbaki story are told in the article Bourbaki: the pre-war years.

World War II had intervened at a time when Bourbaki was about to accelerate production and move quickly towards the group's original aim of publishing six books, each with several chapters. The effects of the war were to cause the project to stagger to a halt with only parts of Books 1, 2 and 3 being published. When the project restarted after the war ended, new young mathematicians were recruited. In particular Roger Godement, Pierre Samuel, Jacques Dixmier and Jean-Pierre Serre joined Bourbaki in the late 1940's. Other "second generation" members who carried forward the project in the late 1940s and early 1950s were Samuel Eilenberg, Jean-Louis Koszul, and Laurent Schwartz.

Armand Borel first became acquainted with the Bourbaki team in 1949. He described [11] his first Bourbaki Congress:-
[Congresses] were private affairs, devoted to the books. A usual session would discuss a draft of some chapter or maybe a preliminary report on a topic under consideration for inclusion, then or later. It was read aloud line by line by a member, and anyone could at any time interrupt, comment, ask questions, or criticize. More often than not, this "discussion" turned into a chaotic shouting match. I had often noticed that Dieudonné, with his stentorian voice, his propensity for definitive statements, and extreme opinions, would automatically raise the decibel level of any conversation he would take part in. Still, I was not prepared for what I saw and heard: "Two or three monologues shouted at top voice, seemingly independently of one another" is how I briefly summarized for myself my impressions that first evening ...
Dieudonné, in [15], gives a similar description of the Congresses and talks about the qualities that members of the team must have:-
During a Congress, the chapters come up in order of the day, in no particular order, and we never know in advance if we shall be doing only differential topology at this Congress, or if at the next one we shall be doing commutative algebra. No everything is mixed ... a Bourbaki member is supposed to take an interest in everything he hears. If he is a fanatical algebraist and says "I am interested in algebra and nothing else" fair enough, but he will never be a member of Bourbaki. One has to take an interest in everything at once. Not to be capable of creating in all fields, that is all right. There is no question of asking everyone to be a universal mathematician; this is reserved for a small number of geniuses. But still, one should take an interest in everything, and be able, when the time comes, to write a chapter of the treatise, even if it is not in one's speciality. This is something which has happened to practically every member, and I think most of them found it extremely beneficial.
The plans at this time were to carry out the aims that Bourbaki had set in 1939, namely to publish six books, each consisting of up to 10 chapters:
Book I. Set Theory
Book II. Algebra
Book III. Topology
Book IV. Functions of One Real Variable
Book V. Topological Vector Spaces
Book VI. Integration
After World War II, publication restarted in 1947 after a break of five years. A description of the chapters which appeared in the five years 1947-52 will illustrate the generality which the authors were achieving as well as the fact that the chapters came out in a somewhat muddled order, certainly not consecutively.

1947 Book III Topology: Chapters V, VI, and VII.

Chapter V is concerned with the additive group $\mathbb{R}$ of real numbers, its subgroups, quotient groups and homomorphisms. Chapters VI and VII are concerned with elementary properties of Euclidean spaces, projective spaces, complex numbers, complex projective spaces, quaternions, etc.

1947 Book II Algebra: Chapter II.

The author of these volumes has set himself the task of presenting a systematic and well-balanced exposition of the whole of mathematics, and has properly chosen to achieve the necessary coherence by using a uniform terminology and a conceptual approach, in which special facts and illustrative examples are to be arranged under the appropriate general notions. His treatment of linear algebras follows these principles; bases and matrices are subordinated to left modules and linear mappings. One often needs these concepts in cases when the multiplication of scalars is not commutative; this appropriate measure of generality is here systematically carried out.
1948 Book II Algebra: Chapter III.

This chapter on multilinear algebra covers tensor products, tensor spaces, Grassmann algebra and determinants.

1948 Book III Topology: Chapter IX.

This chapter looks at the use of the real numbers in general topology. It covers topics such as metric spaces, normed spaces, and Baire spaces.

1949 Book IV Functions of One Real Variable: Chapter I.

Bochner writes:-
This instalment of Bourbaki's super-textbook gives a notable account of Rolle's theorem and Taylor's theorem with remainder; of the indefinite integral, as anti-derivative, for a function having only discontinuities of the first kind, such a function being a uniform limit of a function which is constant by intervals, an "interval" being an open interval or a point; of Cauchy limits for such integrals; of integration and differentiation with respect to a parameter under the integral sign; and of the elementary logarithmic, exponential and trigonometric functions. Everything is done for abstractly-valued functions whose values lie in a topologized vector-space, or, at the very least, in a normed ring, but there are many fine features of analysis spread out underneath this superimposed layer of ever-present generalizations.
1949 Book III Topology: Chapter X.

This is the final chapter of Book III. The sections are: Uniform structures on functional spaces; Spaces of continuous functions; Equicontinuous sets; Compact sets of continuous functions; and Approximation of real continuous functions.

1950 Book II Algebra: Chapters IV, and V.

These two chapters present a carefully worked out development of the theory of polynomials and fields.

1951 Book IV Functions of One Real Variable: Chapter IV.

This chapter covers ordinary first order differential equations and systems of such equations, again presented with considerably more generality than previous texts.

1952 Book II Algebra: Chapters VI, and VII.

The first of these chapters is concerned with the study of partially ordered semigroups and groups, divisibility in integral domains and similar topics. The second of the two chapters covers the theory of modules over a principal ideal ring.

1952 Book IV Integration: Chapters I, II, III, and IV.

These chapters set up the basic properties of integration theory which, as far as Bourbaki is concerned, means integration on locally compact spaces.

The "third generation" of Bourbaki members joined around the time that the project to produce the six books was well on its way to completion. A chapter took from eight to ten years from the time work started on it until final publication. By 1958 publications of the six books was complete so in the mid 1950s only the finishing touches remained to be done. In 1958 Henri Cartan explained why Bourbaki was not growing old [13]:-
Bourbaki has been at work for more than 20 years. You might think that he has grown old in that time and the dynamism of his members has gradually diminished. But that is not so. Bourbaki will always remain young and lively; like mankind he renews himself constantly. The so-called "founding fathers" ... have stepped down in the course of the years to have their places taken by younger members. These newcomers found their own way to Bourbaki.
Armand Borel, François Bruhat, Pierre Cartier, Alexander Grothendieck, Serge Lang, and John Tate are considered the younger members of this third generation. They did not see things in exactly the same way as the founders, for example Cartier said:-
The misunderstanding was that many people thought that [mathematics] should be taught the way it was written in the books. You can think of the first books of Bourbaki as an encyclopaedia of mathematics, containing all the necessary information. That is a good description. If you consider it as a textbook, it's a disaster.
Clearly this powerful mathematical team did not see their task simple to push the last of the chapters through the publishing process. Not at all - in fact right from the beginning members of Bourbaki had more grandiose ideas than simply the six planned books. Armand Borel explains in [11]:-
In the fifties [the six books] were essentially finished, and it was understood the main energies of Bourbaki would henceforth concentrate on the sequel; it had been in the mind of Bourbaki very early on (after all, there was still no "Traité d'Analyse" ). Already in September 1940, Dieudonné had outlined a grandiose plan in 27 books, encompassing most of mathematics. More modest ones, still reaching beyond the "éléments", also usually by Dieudonné, would regularly conclude the Congresses. Also, many reports on and drafts of future chapters had already been written. However, mathematics had grown enormously, the mathematical landscape had changed considerably, in part through the work of Bourbaki, and it became clear we could not go on simply following the traditional pattern. Although this had not been intended, the founding members had often carried a greater weight on basic decisions, but they were now retiring and the primary responsibility was shifting to younger members. Some basic principles had to be re-examined.
These basic principles had required Bourbaki to build everything in a linear fashion without references to outside sources. This clearly would become increasingly difficult as they moved to more advanced topics. Also the idea that every member of Bourbaki should be interested in every topic was one which did indeed require universal mathematicians in the sense which Dieudonné had already made clear they were not:-
There is no question of asking everyone to be a universal mathematician; this is reserved for a small number of geniuses.
Would Bourbaki then split up into smaller units each responsible for an advanced topic? Such questions provoked heated discussion between idealists and realists in the group. Grothendieck, although a third generation member, was fully committed to providing full foundations in accepted "Bourbaki generality" for the advanced topics. He produced a proposal at a 1957 Congress to have:
Book VII: Homological Algebra
Book VIII: Elementary Topology
Book IX: Manifolds
Here the aim was to continue building Bourbaki in the style of the first six books. Grothendieck presented to the next Congress a detailed draft of two chapters, the first containing preliminaries to the book on manifolds and categories of manifolds, while the second contained differentiable manifolds, and the differential formalism [11]:-
[Grothendieck] warned that much more algebra would be needed, e.g., hyperalgebras. As was often the case with Grothendieck's papers, they were at points discouragingly general, but at others rich in ideas and insights. However, it was rather clear that if we followed that route, we would be bogged down with foundations for many years, with a very uncertain outcome.
There was another problem which constantly bothered Bourbaki members. The original aim was to produce an up-to-date text for students to learn the latest approach to mathematics. But by 1958 when the original six books were completed, the first few of these books were already almost 20 years out of date. It was clearly impossible for the authors of the first few chapters to cover all that was needed in the later texts since mathematics was a rapidly developing subject. How could they make their linear approach consistent with rewriting the earlier books? Grothendieck proposed that Bourbaki now went further in providing foundations than had been done for the first six books and planned foundations:-
... also for future developments to the extent they could be foreseen.
These proposals were, however, not liked by others who favoured a more realistic approach to achieving what they wanted to see. Some wanted to see two parallel approaches, that of Grothendieck in the spirit of the founders, and also advanced texts in areas they wanted to cover. But they just didn't have the manpower for this, however much it might have suited everybody. Although Bourbaki held back from the ultimate goal of writing down the whole of mathematics in their linear development, realising that this was just not feasible, nevertheless they still embarked on a programme which was impossibly ambitious. They planned to rewrite the first six books and publish a "final version" of them. They also planned to give up the totally linear approach but, by way of compromise, to produce a summary text on differential and analytic manifolds which followed the model of their first version of Chapter I of Book I, and give only a list of necessary results to set up enough machinery to push forward on topics they were very keen to cover. They decided on producing advanced texts on commutative algebra, algebraic geometry, Lie groups, global and functional analysis, algebraic number theory, and automorphic forms.

By 1980 they had produced the two planned summary chapters on differential and analytic manifolds, seven chapters on commutative algebra, eight chapters on Lie groups and Lie algebras, and two chapters on spectral theories. They had also produced an English translation of some of the first six books, together with three of the chapters on Lie groups and algebras, and seven of the chapters on commutative algebra. For many the chapters on Lie groups and Lie algebras are Bourbaki's finest achievements. For this Armand Borel must take the bulk of the credit for he was the main driving force behind the style and content of this part.

Despite the achievements after 1960, the project began to falter during the 1970s. Cartier gave what he saw as the reasons in [36]:-
Bourbaki struggled in the seventies and the eighties to formulate new directions. [There was] a failed project about several complex variables. There were attempts at homotopy theory, at spectral theory of operators, at the index theorem, at symplectic geometry. But none of these projects went beyond a preliminary stage. Bourbaki could not find a new outlet, because they had a dogmatic view of mathematics: everything should be set inside a secure framework. That was quite reasonable for general topology and general algebra, which were already solidified around 1950. Most people agree now that you do need general foundations for mathematics, at least if you believe in the unity of mathematics. But I believe now that this unity should be organic, while Bourbaki advocated a structural point of view.
There was another problem for Bourbaki in the 1970s which was unrelated to mathematics. They entered into a long legal dispute with their publisher over royalties and translation rights. Cartier said [36]:-
I think it was one of the cases of the century! We hired a famous lawyer who had fought for the heirs of Picasso and Fujita. We survived artificially: we had to win this battle. But it was a pyrrhic victory. As usual in legal battles, both parties lost and the lawyer got rich: In fame and in pocket.
Despite the problems, the project is still alive and new editions are coming out. The founders of Bourbaki had a rule that members should retire at fifty. Cartier has suggested that perhaps it would be appropriate for Bourbaki to retire at fifty. This did not happen and Bourbaki is now seventy. Will there be a resurgence and a sudden return to the passion for change that gripped the founders? Probably not. There may not now be a need for this, and one reason that there is no need is of course the way that Bourbaki has shaken up the world of mathematics. Some love the approach, some hate it, but one cannot deny Bourbaki's influence.

### References (show)

1. L Baulieu, Bourbaki : Une histoire du groups, de mathématiciens français et de ses travaux (Thesis, University of Montreal, 1989).
2. H Cartan, Nicolas Bourbaki und die heutige Mathematik (Westdeutscher Verlag, Cologne and Opladen, 1959).
3. M Andler, Entretien avec trois membres de Nicolas Bourbaki, Gaz. Math. No. 35 (1988), 43-49.
4. V I Arnol'd, The mathematical duel over Bourbaki (Russian), Vestnik Ross. Akad. Nauk 72 (3) (2002), 245-250.
5. D Aubin, The withering immortality of Nicolas Bourbaki : a cultural connector at the confluence of mathematics, structuralism, and the Oulipo in France, Sci. Context 10 (2) (1997), 297-342.
6. L Baulieu, A Parisian café and ten proto-Bourbaki meetings (1934-35), Math. Intelligencer 15 (1993), 27-35.
7. L Baulieu, Bourbaki's art of memory : Commemorative practices in science: historical perspectives on the politics of collective memory, Osiris (2) 14 (1999), 219-251.
8. L Baulieu, Jeux d'esprit et jeux de mémoire chez N. Bourbaki, in La mise en mémoire de la science (éd. Arch. Contemp., Amsterdam, 1998), 75-123.
9. L Baulieu, Dispelling a myth : questions and answers about Bourbaki's early work, 1934-1944, in The intersection of history and mathematics, (Birkhäuser, Basel, 1994), 241-252.
10. R P Boas, Bourbaki and me, Math. Intelligencer 8 (4) (1986), 84-85.
11. A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer. Math. Soc. 45 (3) (1998), 373-380.
12. A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Mitt. Dtsch. Math.-Ver. (1) (1998), 8-15.
13. H Cartan, Nicolas Bourbaki and contemporary mathematics, Math. Intelligencer 2 (4) (1979-80), 175-180.
14. J Delsarte, Compte rendu de la réunion Bourbaki du 14 janvier 1935, Gaz. Math. No. 84 (2000), 16-18.
15. J Dieudonné, The work of Nicholas Bourbaki, Amer. Math. Monthly 77 (1970), 134-145.
16. J Dieudonné, The work of Nicholas Bourbaki (Slovak), Pokroky Mat. Fyz. Astronom. 20 (2) (1975), 66-76.
17. J Dieudonné, The work of Nicholas Bourbaki (Bulgarian), Fiz.-Mat. Spis. B'lgar. Akad. Nauk. 14(47) (1971), 50-61.
18. J Dieudonné, The activities of Bourbaki (Russian), Uspehi Mat. Nauk 28 (3)(171) 1973), 205-216.
19. J Dieudonné, Reflections on Bourbaki (Romanian), Gaz. Mat. Ser. A 77 (1972), 447-460.
20. J Dieudonné, Regards sur Bourbaki, An. Univ. Bucureçsti Mat.-Mec. 18 (2) (1969), 13-25.
21. H Freudenthal, The truth about Bourbaki (Dutch), Euclides (Groningen) 61 (10) (1985/86), 330.
22. R Fritsch, Bourbaki in Kulmbach (German), in Mathemata, Boethius Texte Abh. Gesch. Exakt. Wissensch., XII (Steiner, Wiesbaden, 1985), 607-611.
23. D Guedj, Nicholas Bourbaki, collective mathematician : An interview with Claude Chevalley, Math. Intelligencer 7 (2) (1985), 18-22.
24. P R Halmos, "Nicolas Bourbaki", Scientific American, 1957 (May, 1957), 88-99.
25. C Houzel, Le rôle de Bourbaki dans les mathématiques du vingtième siècle, Gaz. Math. No. 100 (2004), 53-63.
26. C Houzel, Bourbaki und danach, Math. Semesterber. 49 (1) (2002), 1-10.
27. C Houzel, Bourbaki et après Bourbaki, in Travaux mathématiques, Fasc. XI, Luxembourg, 1998 (Centre Univ. Luxembourg, Luxembourg, 1999), 23-32.
28. C Houzel, The influence of Bourbaki (Italian), in Italian mathematics between the two world wars (Italian), Milan/Gargnano, 1986 (Pitagora, Bologna, 1987), 241-246.
29. K Krickeberg, Comment: "Twenty-five years with Nicolas Bourbaki 1949-1973" by A Borel, Mitt. Dtsch. Math.-Ver. (2) (1998), 16.
30. L H Lam, "Nicolas Bourbaki" - a nonexistent mathematician, Menemui Mat. 2 (1) (1980), 16-29.
31. S MacLane, Correction : "Nicholas Bourbaki, collective mathematician: an interview with Claude Chevalley" by Denis Guedj, Math. Intelligencer 8 (2) (1986), 5.
32. T Murata, Quelques remarques sur les éléments d'histoire des mathématiques de N Bourbak, Comment. Math. Univ. St. Paul. 24 (2) (1975/76), 169-187.
33. O Pekonen, The first Bourbaki autograph?, Math. Intelligencer 21 (3) (1999), 16-17.
34. O Pekonen, Nicolas Bourbaki in memoriam (Finnish), in In the forest of symbols (Finnish) (Art House, Helsinki, 1992), 55-71.
35. K Rychlík, Nicolas Bourbaki (Czech), Pokroky Mat. Fys. Astronom. 4 (1959), 673-678.
36. M Senechal, The continuing silence of Bourbaki - an interview with Pierre Cartier, June 18, 1997, Math. Intelligencer 20 (1) (1998), 22-28.
37. K Uzel', How to appraise the role of Bourbaki in the history of mathematics (Russian), Istor.-Mat. Issled. (2) No. 6(41) (2001), 100-110; 388.
38. G Walusinski, Au pays de Clairaut et de Bourbaki, Enseignement Math. (2) 3 (1957), 289-297.
39. A Weil, Organisation et désorganisation en mathématique,Bull. Soc. Franco-Japonaise des Sci. 3 (1961), 23-35.

Written by J J O'Connor and E F Robertson
Last Update December 2005

format: plain


text:
# Variability instead of means?

#### AlexanderC

##### New Member
Hello, I was thinking about analysing variability instead of means but could not find much for an answer. I am interested in learning more about this so I would appreciate some feedback. I made some R code to demonstrate my question (I can't code well in R, improvements are much welcome too).

Code:
library(dplyr)
library(tidyr)

library(ez)
options(scipen=999)

# Faking data function
make_y_values <- function(N, mean=140, within_sd=1, reps=1){
sj <- 1:N
subs <- c()
ys <- c()
repeats <- c()
i=1
for (s in sj){
subs <- c(subs, rep(s, reps))
y <- rnorm(reps, mean, sd=runif(1, 0, within_sd)
repeats <- c(repeats, 1:reps)
ys <- c(ys, y)
i = i+1
}
return(data.frame(subject=subs, repeats=repeats, y=ys))
}

pre <- make_y_values(N=10, mean=140, within_sd=10, reps=6)
post <- make_y_values(N=10, mean=140, within_sd=1, reps=6)

pre$session <- rep(0, nrow(pre)) post$session <- rep(1, nrow(post))

df <- rbind(pre, post)

grp <- df %>%
group_by(subject, session) %>%
summarise(y_mean=mean(y), y_sd=sd(y))
grp

boxplot(y_mean~session, data=grp)

# ANOVA
library(rstatix)
ANOVA <- anova_test(
data=df,
dv="y",
wid="subject",
within=c("repeats", "session")
)
print(ANOVA)

The example experiment might be something like an intervention for blood pressure treatment. We have 20 patients come into our clinic at time 1 (before drug). Their blood pressure got measured 6 times, showing a spread of values with a mean and SD. Patients go home and take our magic drug for some time before coming back at time 2 for another 6 measures. So we now have pre drug, post drug recordings of means and SD in blood pressure. And surprise, the blood pressure means did not differ so we say the treatment failed. But then we spot the variances or SDs look very much different - much smaller at time 2. This seems useful, but I don't know, how would we analyse something like this? ANOVA does not capture it.

I think we can use an F test to compare, but is it useful? Also what if we have a more complicated design (that I am not able to code). What is the approach then?

#### Karabiner

##### TS Contributor
In your experiment, you have n=20 patients with 2*6 measurements of the outcome.
You could aggregate these measurements patient-wise, according to what might clinically
be important, and use the aggregate value your dpendent variable (for example, mean
blood pressure, SD of blood pressures, difference between highest and lowest blood pressure,
maximum blood pressure...). That variable can be compared pre-post.

With kind regards

Karabiner

#### AlexanderC

##### New Member
In your experiment, you have n=20 patients with 2*6 measurements of the outcome.
You could aggregate these measurements patient-wise, according to what might clinically
be important, and use the aggregate value your dpendent variable (for example, mean
blood pressure, SD of blood pressures, difference between highest and lowest blood pressure,
maximum blood pressure...). That variable can be compared pre-post.

With kind regards

Karabiner
Thank you for the idea. If I aggregate and end up with the SD can this be analyzed as if we are just analyzing the mean? e.g. t-test, ANOVA, so on. Or is this a violation of normality? I mean is the population standard DEV assumed to be a normal distribution? It feels a bit strange to think about. Thanks

#### Karabiner

##### TS Contributor
Thank you for the idea. If I aggregate and end up with the SD can this be analyzed as if we are just analyzing the mean?
e.g. t-test, ANOVA, so on. Or is this a violation of normality? I mean is the population standard DEV assumed to be a normal distribution? It feels a bit strange to think about. Thanks
Sorry, but I do not understand what you mean. How does normality come into play here?
I suggested to chose an outcome variable based on the 6 intra-individual measures pre and post,
respectively, so that a subject has data on "average blood pressure out of 6 measurements" pre and post
(as you already did before, if I understand it correctely), or on "maximum blood pressure out of
6 measurements" pre and post, or or on "SD of intra-individual blood pressure" etc.
Then you can for example ask the question "does intra-individual variability of blood pressure
change between pre and post", and chose an appropriate test (Wilcoxon signed rank
test, or sign test, or dependent samples t-test, or whatever).

With kind regards

Karabiner

#### AlexanderC

##### New Member
Sorry, but I do not understand what you mean. How does normality come into play here?
I think it is relevant because if I were to do a t-test (or ANOVA in a more complex scenario) to compare pre/post then this requires normality. Means might be fine, but when I was thinking on this, the variance cannot be normal, and I am not sure if the SD would be either.

#### Karabiner

##### TS Contributor
I think it is relevant because if I were to do a t-test (or ANOVA in a more complex scenario) to compare pre/post then this requires normality.
A dependent sample t-test requires normality of the differences between pre and post, in the population from
which the sample was drawn. If the sample size is large enough (about n > 30 or so), the test is considered
robust against non-normality in the population. If the sample size is small, then normality should be fulfilled,
but usually cannot be demonstrated, so maybe a "non-parametric" test would be preferable.

But all this is independent of whether you use the individual SDs as dependent variable. You cannot say
beforehand whether the differences between pre and post follows a normal distribution (in the population)
or can you?

With kind regards

Karabiner

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Two comments, looking solely at variability negates if the distribution moved. Also, if you have multiple observations per subject, you need to control for those otherwise you are not controlling for dependence between them. Multilevel modelling allows you to report the between and within variability explained.

Also, there is a lot of ruckus around how best to examine repeated time points. The standard is to control for the baseline value, since someone with a 190 SBP has alot more room for change than a person with an 80 SBP. Though this depends on what your target study question may be.

#### AlexanderC

##### New Member
Thank you for feedback.
But all this is independent of whether you use the individual SDs as dependent variable. You cannot say
beforehand whether the differences between pre and post follows a normal distribution (in the population)
or can you?
Well, I do see your point. I thought SD might (like variance) necessarily be not normal. I think we can say that any distribution of variances would not be normally distributed (as it is always positive, and we square to get there), so thought this might also be true for SD but perhaps not, I did quite some simulation and noticed that they do appear to be normal (though variances definitely are not and we can say this without thinking).

Two comments, looking solely at variability negates if the distribution moved. Also, if you have multiple observations per subject, you need to control for those otherwise you are not controlling for dependence between them. Multilevel modelling allows you to report the between and within variability explained.

Also, there is a lot of ruckus around how best to examine repeated time points. The standard is to control for the baseline value, since someone with a 190 SBP has alot more room for change than a person with an 80 SBP. Though this depends on what your target study question may be.
Thank you for your feedback hlsmith, you are correct it might be nice also to see if the distribution moved. But I do not understand what you mean by controlling for subjects - isn't this done by repeated measures t-test or ANOVA? Could you maybe explain a little bit more about what you mean by controlling for multiple observations per subject? If you mean about the 6 observations in each time point (in my example) then as Karabiner suggests I would be aggregating.

Maybe a normalization subject-by-subject would help to overcome the problem with baseline values, or not? I am not very familiar with the concept of controlling for baseline value beyond throwing all my data onto a repeated measures test.


text:
# Given: segment WX is perpendicular to segment UV; segment WX perpendicular to segment YZ; segment WU is congruent to segment

###### Question:

Given: segment WX is perpendicular to segment UV; segment WX perpendicular to segment YZ; segment WU is congruent to segment WV
Prove: whatever you can about angles 1, 2, 3, and 4.

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text:
## Average Acceleration Question

1. The problem statement, all variables and given/known data
"A bullet is fired through a 5.0 cm thick board of some material. The bullet strikes the board with a speed of 200 m/s, and emerges out the other end at 100 m/s.

2. Relevant equations
a= (change in velocity)/(change in time)
average velocity = .5(initial v + final v)

3. The attempt at a solution

a= (-100 m/s)/(.000333 seconds) = -300300 m/s^2
I found .000333s as the time it took for the bullet to pass through the board, using the average velocity of 150 m/s.

Is this correct?

SOLVED.

 PhysOrg.com science news on PhysOrg.com >> 'Whodunnit' of Irish potato famine solved>> The mammoth's lament: Study shows how cosmic impact sparked devastating climate change>> Curiosity Mars rover drills second rock target
 Recognitions: Homework Help It helps, if you are unsure of your reasoning, to relate what you did with some other representation... eg. from your equations: ave speed = distance over change in time:$$\frac{d}{T} = \frac{1}{2}(v + u) \Rightarrow 2d = (v + u)T$$... using distace d, change in time T, final velocity v, initial velocity u. acceleration is change in speed over change in time:$$a = \frac{v - u}{T} \Rightarrow v = aT + u$$ Soooo... combining them:$$2d = \big ( (aT + u) + u\big )T = aT^2 +2uT \Rightarrow d = uT + \frac{1}{2}aT^2$$... which you will recognize as a kinematic equation. ... so what you've done is basically the same as assuming a constant acceleration, and is consistent with other physics you know.

 Tags 5.0 cm, acceleration, board, bullet, velocity


text:
# Watts and Kelvin

• July 18th 2008, 11:13 AM
Skewed
Watts and Kelvin
Hi,

I have a question regarding light, specifically artificial light. Please forgive me I am not a physics guru by any means and my terminology may be completely wrong, but hopefully someone will be able understand what I am trying to ask.

First, the only way I know how to ask is in basic terms, I know that there is much more to it than that. So, I am asking in terms that I feel that I understand.

Ok here it goes.

Lets assume you have two lights, one light is 5000k and the other is 2700k. Considering that both lights are 23 watts each. Which light has the energy to penetrate...lets say a plant canopy better? The blue or red?

Or another another way to ask could be..
A 23watt 5000k bulb could produce light that would travel 2 miles. A 23watt 2700k bulb produces a light that would travel 1 mile. In order to to get the red light to reach 2 miles I would have to increase the 2700k bulb to 40 watts????

Here is what I think. I feel that the blue light will penetrate deeper than the red given that they are the same wattage. Now, if I am correct in my thinking would it be fair to say that if I should want the red light to penetrate to the level of the blue light at 23watts I would need to increase the wattage of the red bulb to lets say 40 watts? I know the numbers may not be correct, this is all generally speaking of course. I am just trying to keep it on a basic understanding level at this moment. TIA.
• July 18th 2008, 11:36 AM
CaptainBlack
[quote=Skewed;168217]

Quote:

Or another another way to ask could be..
A 23watt 5000k bulb could produce light that would travel 2 miles. A 23watt 2700k bulb produces a light that would travel 1 mile. In order to to get the red light to reach 2 miles I would have to increase the 2700k bulb to 40 watts????
.
They both have infinite range other things being equal. Also without absorption the intesities are equal at equal ranges.

RonL
• July 18th 2008, 11:37 AM
CaptainBlack
Quote:

Originally Posted by Skewed
Lets assume you have two lights, one light is 5000k and the other is 2700k. Considering that both lights are 23 watts each. Which light has the energy to penetrate...lets say a plant canopy better? The blue or red.

I think you had better tell us why you are asking, so that we can answer the real question.

RonL
• July 18th 2008, 11:52 AM
Skewed
Quote:

Originally Posted by CaptainBlack
I think you had better tell us why you are asking, so that we can answer the real question.

RonL

Fair enough.

Where my question derives from is this. When I first started indoor gardening I was using the 23watt 5000k bulbs and 2700k bulbs. The 23 watt 5000k bulbs did an excellent job at maturing the plants growth during the vegetative stage. But during the fruit/flowering stage where the 2700k lights are used, the outcome was less than desired. However, when I switched to the 40watt 2700k bulbs the outcome was excellent. From my observations of the two scenarios, the increase in power on the red light was able to penetrate the plants and do a better job at producing the fruit. What aspect am I missing and not understanding as to what the increase in power, intensity, or whatever the correct term is called.
• July 18th 2008, 01:54 PM
CaptainBlack
Quote:

Originally Posted by Skewed
Fair enough.

Where my question derives from is this. When I first started indoor gardening I was using the 23watt 5000k bulbs and 2700k bulbs. The 23 watt 5000k bulbs did an excellent job at maturing the plants growth during the vegetative stage. But during the fruit/flowering stage where the 2700k lights are used, the outcome was less than desired. However, when I switched to the 40watt 2700k bulbs the outcome was excellent. From my observations of the two scenarios, the increase in power on the red light was able to penetrate the plants and do a better job at producing the fruit. What aspect am I missing and not understanding as to what the increase in power, intensity, or whatever the correct term is called.

Plants have evolved to use sunlight as an energy source in photosysntesis, which means they are (to a hand waving approximation)optimised to use light of with a spectrum similar to that of the daylight. This is close to that of a black body at the same equivalent temprature as the Sun, and corresponds to 5000-6000K.

So the plants are making better use of the energy from the 5000K bulbs than they are from the 2700K ones. To get the same energy into the plants acceptance band with "cooler" bulbs requires greater brightness and so a higher powered bulb.

However, ripening may require warmth rather than light for photosynthesis, and the 2700K bulbs will be radiating more energy in the infra-red which may be keeping the ripening fruit warmer.

RonL
• July 21st 2008, 11:23 AM
Matt Westwood
I don't know the full technical details (biology is sketchy with me) but the wavelength may be more crucial than that.

I presume the 5000k bulb produces a light with a shorter wavelength (and hence by $E = h \nu$) a higher energy than the 2700k one.

It may not be the number of photons that arrives on the plant that makes the difference, but the energy of the individual photons that stimulates the chlorophyll into doing what it's supposed to do.

Check out the photoelectric effect, in which you need light above a certain frequency threshold to kick the electrons free of the atoms. If you need blue light to do it, then it doesn't matter how bright the red light you shine on it, you're not going to see the effect happen.
• July 21st 2008, 11:37 AM
CaptainBlack
Quote:

Originally Posted by Matt Westwood
I presume the 5000k bulb produces a light with a shorter wavelength (and hence by $E = h \nu$) a higher energy than the 2700k one..

Black body radiation (that is thermal radiation) like that from the Sun and/ot incandesant light has a continuous spectra, that is all wavelengths are present over a band wider than the optical.

No one said it was the crude number of photons that was important, but the numbers in the bands that plants were using. Increasing the brightness while maintaining the same colour temorature increases the numbers of photons in all bands.

RonL


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## Intermediate Algebra (6th Edition)

$\frac{9}{10}\times\frac{x}{2}$
The commutative property of multiplication holds that $a\times b=b\times a$. In this example, we can set $a\times b=\frac{x}{2}\times\frac{9}{10}$. So, $a=\frac{x}{2}$ and $b=\frac{9}{10}$. So an equivalent expression would be $b\times a=\frac{9}{10}\times\frac{x}{2}$.


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# [c++] wondering about virtual functions

This topic is 3854 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

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given you have a class QMainWindow BaseView derives from it. TraceView derives from BaseView in TraceView I overwrite a protected virtual function which is not overwritten in BaseView. The original version is in QMainWindow. If I now call BaseView::showEvent() from TraceView::showEvent() the compiler does not complain about baseview not implementing the function - so I first thought that he might call QMainWindow::showEvent() instead. In the stroustrup he states that using the scope operator on virtual functions denies the use of the virtual mechanism, which leads me to the conclusion that in the above case simply nothing will happen ... but why does the compiler not complain? any clues? €: I tried it out with a simple example and it seems that the QMainWindow version will be called ... i think it's simply because BaseView "has all the functions" of QMainWindow ... - thanks anyways

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When using public inheritance your saying that your derived class is built upon another class and contains all the members and methods of that class, although admittedly only public and protected members / methods will be accessible to it.

Scope operator does override (excuse pun) the virtual mechanism, but it depends on how you use it.

In your case though as you found out; The compiler is ok with it, as your calling a protected member of it's base class and therefore the derived class has the method anyway.

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Using the scope operator alone (::function) works as you expected (doesn't compile).

test.cpp:28: error: ‘::prot_f’ has not been declared

Perhaps the (base::function) syntax is meant specifically for resolving multiple inheritance issues? Is it a form of casting?

#include <iostream>class base{public:	base() {}	virtual ~base() {}	void operator()() { prot_f(); }protected:	virtual void prot_f() { std::cout << "base::prot_f\n"; }};class derive1: public base{public:	derive1(): base() {}	virtual ~derive1() {}};class derive2: public derive1{public:	derive2(): derive1() {}	virtual ~derive2() {}protected:	void prot_f()	{	std::cout << "derive2::prot_f\n";		::prot_f(); //ERROR		derive1::prot_f();	}};int main(){	base b; b();	derive1 d1; d1();	derive2 d2; d2();	return 0;}

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Quote:
 Original post by hydrooIf I now call BaseView::showEvent() from TraceView::showEvent() the compiler does not complain about baseview not implementing the function - so I first thought that he might call QMainWindow::showEvent() instead.

It does, as your tests show.

Quote:
 In the stroustrup he states that using the scope operator on virtual functions denies the use of the virtual mechanism

It does. But all that means is that writing BaseView::showEvent() will not try to find a virtual override for the function (which would end up calling the TraceView::showEvent() function again, leading to infinite recursion - so we're very glad the language doesn't work that way!). It says "look for the showEvent() function in the BaseView class and use that". And...

Quote:
 i think it's simply because BaseView "has all the functions" of QMainWindow ... - thanks anyways

It does, hence the observed behaviour. :)

@m3mb3rsh1p: ::prot_f() tries to look up a global function named 'prot_f'. You get an error because there is no such function. Using scope resolution to get at base class functions can be used to resolve multiple inheritance issues, but it's often used instead to "use the base version of the function and then do a little extra work specific to the derived class" - although this is often bad design. I wouldn't call it "a form of casting", since if you actually did cast to the base class and then called the function normally, it would still exhibit polymorphism.

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Sorry, I broke the first rule of answering questions :)

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# Dynamic programming: multiple subset sum

The subset sum problem is finding a solution $x \in \{0,1\}^n$ such that $$\sum_{i=1}^n a_i x_i = c,$$ for $a_i, c \in \mathbb{F_q}$ (also possible for $a_i,c \in \mathbb Z$).

I know that there exists a pseudo polynomial solution to this via dynamic programming. Sadly I am not very skilled at dynamic programming yet so I was wondering if a pseudo polynomial solution was possible to the following variation on a multiple subset sum problem.

Instead of $1$ equation we are working with $2$ or more, so we need to find a solution $x \in \{0,1\}^n$ such that $$\sum_{i=1}^n a_i x_i = c_1,$$ $$\sum_{i=1}^n b_i x_i = c_2,$$ for $a_i,b_i, c_1, c_2 \in \mathbb{F_q}$.

To give you some intuition, when we would have $n$ such equations this would be solvable via Gaussian elimination.

In this case all the coefficients are positive and bounded.

I was wondering if this would be solvable via dynamic programming such that we would have a pseudo polynomial algorithm or if this isn't possible an explanation why.

I hope you guys can help me!

• What is $\mathbb F_q$? – 5xum Sep 20 '15 at 14:26

Afterwards, you may run it for the first equation (check if there is a subset of $a_1, \ldots, a_n$ that sums up to $c_1$), and then check if any of the found solutions work for $b_1, \ldots, b_n$ and $c_2$.


text:
# Complex Analysis/Complex Numbers/Introduction

This book assumes you have some passing familiarity with the complex numbers. Indeed much of the material in the book assumes your already familiar with the multi-variable calculus. If you have not encountered the complex numbers previously it would be a good idea to read a more detailed introduction which will have many more worked examples of arithmetic of complex numbers which this book assumes is already familiar. Such an introduction can often be found in an Algebra (or "Algebra II") text, such as the Algebra wikibook's section on complex numbers.

Intuitively a complex number z is a number written in the form:

$z=x+iy$,

where x and y are real number and i is an imaginary number that satisfies $i^2 = -1$. We call x the real part and y the imaginary part of z, and denote them by $\text{Re }z$ and $\text{Im }z$, respectively. Note that for the number $z=3-2i$, $\text{Im }z=y=-2$, not $-2i$. Also, to distinguish between complex and purely real numbers, we will often use the letters z and w for the complex numbers. It is useful to have a more formal definition of the complex numbers. For example, one frequently encounters treatments of the complex numbers that state that $i$ is the number so that $i=\sqrt{-1}$, and we then operate with $i$ using many of our usual rules for arithmetic. Unfortunately if one is not careful this will lead to difficulties. Not all of the usual rules for algebra carry through in the way one might expect. For example, there is a flaw in the following calculation: $i=\sqrt{-1}=\sqrt{\frac{1}{-1}}=\frac{\sqrt{1}}{\sqrt{-1}}=\frac{1}{i}=-i$, but is very difficult to point out the flaw without first being clear about what a complex number is, and what operations are allowed with complex numbers.

Mathematically the complex numbers are defined as an ordered pair, endowed with algebraic operations.

Definition

A complex number z is an ordered pair of real numbers. That is $z=(x,y)$ where x and y are real numbers. The collection of all complex numbers is denoted by the symbol $\mathbb{C}$.

The most immediate consequence of this definition is that we may think of a complex number as a point lying the the plane. Comparing this definition with the intuitive definition above, it is easy to see that the imaginary number i simply acts as a place holder for denoting which number belongs in the second coordinate.

Definition

We define the following two functions on the complex plane. Let $z=(x,y)$ be a complex number. We define the real part is as function $\text{Re}:\mathbb{C}\to \mathbb{R}$ given by $\textrm{Re}(z)=x$. Similarly we define the imaginary part as a function $\textrm{Im}:\mathbb{C}\to \mathbb{R}$ given by $\textrm{Im}(z)=y$.

We say two complex numbers are equal if and only if they are equal as ordered pairs. That is if $z=(x,y)$ and $w=(u,v)$ then z = w if and only if x = u and y = v. Put more succinctly, two complex numbers are equal iff their real parts and imaginary parts are equal.

If complex numbers were simply ordered pairs there would not really be much to say about them. But the complex numbers are ordered pairs together with several algebraic operations, and it is these operations that make the complex numbers so interesting.

Definition

Let z = (xy) and w = (uv) then we define addition as:

z + w = (x + uy + v)

and multiplication as:

z · w = (x · u − y · vx · v + y · u)

Of course, we can view any real number r as being a complex number. Using our intuitive model for the complex numbers it is clear that the real number r should correspond to the complex number (r, 0), and with this identification the above operations correspond exactly to the usual definitions of addition and multiplication of real numbers. For the remainder of the text we will freely refer to a real number r as being a complex number, where the above identification is understood.

The following facts about addition and multiplication follow easily from the corresponding operators for the real numbers. Their verification is left as an exercise to the reader. Let z, w and v be complex numbers, then:

 • z + (w + v) = (z + w) + v (Associativity of addition); • z · (w · v) = (z · w) · v (Associativity of multiplication); • z + w = w + z (Commutativity of addition); • z · w = w · z (Commutativity of multiplication); • z · (w + v) = z · w + z · v (Distributive Property).

One nice feature of complex addition and multiplication is that 0 and 1 play the same role in the real numbers as they do in the complex numbers. That is 0 is the additive identity for the complex numbers (meaning z + 0 = 0 + z = z) and 1 is the multiplicative identity (meaning z · 1 = 1 · z = z).

Of course it is natural at this point to ask about subtraction and division. But stating the formula's for subtraction and division outright, we instead follow the usual course for other subjects of algebra and first discuss inverses.

Definition

Let z = (x, y) be any complex number, then we define the additive inverse −z as:

z = (−x, −y)

Then it is immediate to verify that z + −z = 0.

Now for any two complex numbers z and w we define zw to be z + −w. We now turn to doing the same for multiplication.

Definition

Let z = (x, y) be any non-zero complex number, then we define the multiplicative inverse, $\tfrac{1}{z}$ as:

$\frac{1}{z}=\Big(\frac{x}{x^2+y^2}, -\frac{y}{x^2+y^2}\Big)$

It is left to the reader to verify that $z\cdot\tfrac{1}{z}=1$.

We may now of course define division as $\tfrac{z}{w}=z\cdot\tfrac{1}{w}$. Just as with the real numbers, division by zero remains undefined. In order for this last definition to make more sense it helps to introduce two more operations on the complex numbers. The first is the absolute value.

Definition

Let z = (x, y) be any complex number, then we define the complex absolute value, denoted |z| as:

$|z|=\sqrt{x^2+y^2}$

Notice that |z| is always a real number and |z| ≥ 0 for any z.

Of course with this definition of the absolute value, if z = (x, y) then |z| is exactly the same as the norm of the vector (x, y).

Before introducing the second definition, notice that our intuitive definition simply required us to find a number whose square was −1. Of course i2 = (−i)2 = −1, so for a starting point one could have chosen -i as the most basic imaginary number. This idea motivates the following definition.

Definition

Let z = (x, y) be any complex number, then we define the conjugate of z, denoted $\bar z$ as:

$\bar z=(x,-y).$

With this definition it is an easy exercise to check that $z\cdot\bar z=|z|^2$, so dividing both sides by |z|2 we arrive at $z\cdot\tfrac{\bar z}{|z|^2}=1$. Compare this with the definition of the multiplicative inverse above.

Recall that, every point in the plane can be written using rectangular coordinates such as (x, y) where of course the numbers denote the distance from the x and y axes respectively. But the point could equally well be described using polar coordinates (r, θ), where the first number represents the distance from the origin, and the second is the angle that is made with the positive x axis when you connect the origin and the point with a line segment. Since complex numbers may be thought of simply as points in the plane, we can immediately derive a polar representation of a complex number. As usual we can let a point z = (x, y) = (r cos θ, r sin θ) where $\textstyle r=\sqrt{x^2+y^2}$. The choice of θ is not unique because sine and cosine are 2π periodic. A value θ for which z = (r cos θ, r sin θ) is called an argument of z. If we restrict our choice of θ so that 0 ≤ θ < 2π then the choice of θ is unique provided that z ≠ 0. This is often called the principle branch of the argument.

As a shorthand, we may write $\operatorname{cis}\,\theta = \cos \theta + i\sin \theta$, so $z=r\operatorname{cis} \theta$. This notation simplifies multiplication and taking powers, because

 $z_1 z_2 \,\!$ $=(r_1 \operatorname{cis} \theta_1)(r_2 \operatorname{cis} \theta_2)$ $=r_1 r_2 \left [ \left ( \cos \theta_1 + i \sin \theta_1 \right )\left ( \cos \theta_2 + i \sin \theta_2 \right ) \right ]$ $=r_1 r_2 \left [ \left ( \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \right ) + i \left ( \sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2 \right ) \right ]$ $=r_1 r_2 \left ( \cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2) \right )$ $=r_1 r_2 \operatorname{cis} (\theta_1 + \theta_2)$

by elementary trigonometric identities. Applying this formula can therefore simplify many calculations with complex numbers.

Using induction we can show that

$z^n=r^n \operatorname{cis} (n \theta)$,

holds for all positive integers $n$.

Now that we have set up the basic concept of a complex number, we continue to topological properties of the complex plane.

## ExercisesEdit

1. Determine $\overline{(z_1+z_2)}$ in terms of $\bar z_1$ and $\bar z_2$.
2. Determine $\overline{z_1z_2}$ in terms of $\bar z_1$ and $\bar z_2$.
3. Show that the absolute value on the complex plane obeys the triangle inequality. That is show that:
$|z_1+z_2|\leq |z_1|+|z_2|.$
4. Show that the absolute value on the complex plane obeys the reverse triangle inequality. That is show that:
$|z_1+z_2|\geq \big||z_1|-|z_2|\big|.$
5. Given a non-zero complex number $z=r\operatorname{cis}(\theta)$ determine $r'$ and $\theta'$ so that $\frac{1}{z}=r'\operatorname{cis}(\theta')$.
6. Determine formulas for $\text{Re }z$ and $\text{Im }z$ in terms of $z$ and $\bar z$.
7. Find $n$ distinct complex numbers $z_k$, $k = 0, \ldots, n-1$ so that $z_k^n=z$. Hint: Use the formula given above for $z_k^n$ and the $2\pi$ periodicity of $\cos(\theta)$ and $\sin(\theta)$.

Next


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# A Problem on Combinatorics

What is the cardinality of the set $\{(j_1,...,j_{m-2}): 2 \le j_1 <\ldots < j_{m-2} \le m\}$, where $m$ is some given positive integer (say, $100$ or $500$)?

You are choosing $m-2$ elements among the $m-1$ elements $2, 3, \dots, m$, so it's $$\dbinom{m-1}{m-2} = \dbinom{m-1}{1} = m - 1.$$ This is because your only choice is which one of the $m-1$ elements you are leaving out.


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# Solution - Determine the Order of Reaction and the Dimensions of the Rate Constants - Rate of a Chemical Reaction

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#### Question

From the rate expression for the following reactions, determine their order of reaction and the dimensions of the rate constants.

H2O(aq) + 3 I− (aq) + 2 H+ → 2 H2O (l) + I_3^- Rate = k[H2O2][I]

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# American Institute of Mathematical Sciences

June  2008, 1(2): 235-242. doi: 10.3934/dcdss.2008.1.235

## A semilinear $A$-spectrum

 1 Department of Mathematics, University of Maryland, College Park, Maryland 20742 2 Dipartimento di Matematica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza

Received  September 2006 Revised  March 2007 Published  March 2008

Using the theory of $A$-proper maps, we give a new definition of spectrum for a semilinear pair $(L,F)$. Properties of this spectrum are given.
Citation: C. T. Cremins, G. Infante. A semilinear $A$-spectrum. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 235-242. doi: 10.3934/dcdss.2008.1.235
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# Question of centre of mass and parallel axis theorem

A puck of mass 80.0 g and radius 4.20 cm slides along an air table at a speed of 1.50 m/s. It makes a glancing collision with a second puck of radius 6.00 cm and mass 140.0 g (initially at rest) such that their rims just touch. Because their rims are coated with instant-acting glue, the pucks stick together and spin after the collision.

How do i find the angular momentum of the system relative to the centre of the mass? I have found the centre of mass already. It's 10.7cm from the perimeter of the small puck.

Firstly

Consider the distance of the centre of gravity from the centre of the first (moving puck) that might help

Last edited:
I have already calculated that distance and it is 10.7 cm from the small puck! From here, I used the formula angular velocity=linear velocity/ radius = 1.5 / 0.107 =14.0 rad/s. From here I calculated the moment of inertia of the 2 pucks = 0.0008522. Thus I will get the angular momentum which is the product of the two answers above = 0.0119. Is this the right answer?

Doc Al
Mentor
tubworld said:
How do i find the angular momentum of the system relative to the centre of the mass? I have found the centre of mass already. It's 10.7cm from the perimeter of the small puck.
Realize that the COM of the system is moving. If we let the direction of motion be along the horizontal, then what you've found is the vertical coordinate of the COM. Calculate the speed of the COM, then view the speeds of the incoming pucks from that reference frame. Each puck will have an angular momentum about the COM. Just add them up.

Ermm... ... any help in the first steps for the first particle? I try doing the second one by myself. Just help me out with the first particle.. Thanx!

Doc Al
Mentor
Start by finding the speed of the center of mass.

talking abt the linear speed right???

Doc Al
Mentor
That's right. If you want to find the angular momentum with respect to the center of mass, you'll need to measure the speed of each puck with respect to the center of mass. But first you should find the speed of the center of mass itself.

but u mean finding the linear speed abt the centre of mass before the 2 bodies even get stuck or what? cos their centre of mass is different as only 1 particle was moving initially...

Doc Al
Mentor
Yes, find the speed of the center of mass before the collision. (Which is the same as the speed of the center of mass after the collision, since no external force acts on the system.)

Since only 1 particule is moving, the linear KE = 1/2 mv^2. But again, who do we find the sped of the centre of mass. At this pt in time, the centre of mass of the SYSTEM isn't moving. it only moves rotationally after itwas hit by the small puck. So wldnt the speed be zero? only the small puck has the speed it needed.

Doc Al
Mentor
The center of mass of the system is most definitely moving! If both pucks were stationary, then the center of mass would not move. But since the center of mass of the system depends on the position of both pucks, as one puck moves the center of mass also moves.

Big hint:
Horizontal position of center of mass:
$$M_{total} x_{cm} = m_1 x_1 + m_2 x_2$$

Horizontal speed of center of mass:
$$M_{total} v_{cm} = m_1 v_1 + m_2 v_2$$

ur x here is measured relative to?

Doc Al
Mentor
x is the horizontal position relative to whatever origin you like, as long as it's fixed with respect to the table.

Oh yeah I got that thks! But its really darn long.... .. :(


text:
# Is the space of test functions reflexive? [closed]

Question: the test function space is reflexive ?

and the adherence of this space is it again reflexive ?

## closed as off-topic by user44191, abx, Sean Lawton, Wolfgang, Jan-Christoph Schlage-PuchtaMay 12 at 15:55

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – user44191, abx, Sean Lawton, Wolfgang
If this question can be reworded to fit the rules in the help center, please edit the question.

• You should specify what you mean by the space of test functions. If you mean the usual space ${\mathcal D}(U)$, then it is reflexive as a Montel space (this is written explicitely in R.Edwards' Functional analysis, 8.4.7). It is also not clear what you mean by adherence. And I think you should ask this at math.stackexchange.com – Sergei Akbarov May 11 at 15:30