Table of contents
The expression class used in iLand was originally developed for the Picus model and aims at a high performance for repeated executions. Expressions are parsed only once and converted to an internal "program" which can be executed with very little overhead. When using more complex mathematical functions (e.g. exp()), the overall performance is comparable to hard coded C++.
The meaning of a variable name within an expression depends on the context. There are three main cases:
- Expressions bound to "Objects" e.g. to Trees. In that case, the variable names are defined by the respective object. If an expression is bound to trees, one can use context sensitive variables like "dbh" or "volume" for trees. Depending on the context, this can be tree variables, resource unit variables, or sapling variables.
- named variables: the names of variables are fixed programmatically. E.g: a expression with two named parameters ("x", "n") to calculate a weight could be "x/n" or more sophisticated "x*x/n*n"
- unbound variables: In that case the name of the variable can be chosen freely. The value for the variable is set during the model execution and depends on the context. Example: the expressions for the calculation of biomass compartments (allometric equations) has one parameter (dbh). Valid expression are, e.g.: "0.1*dbh^2" or "0.1*d^2" or "0.1*fish^2".
Basically, expressions can consist of basic arithemtic operators, variables and functions. The basic operators +,-,* and / work as expected (incl. precedence rules). Additionally, the caret "^" can be used for power functions (e.g. x*x can be written as x^2). Parentheses work as expected (e.g. (a+b)*c is different from a+(b*c)).
Floating point numbers must use the dot (".") for constants (e.g. 0.001). The comma "," is used to separate arguments in function calls.
Expressions can be used to evaluate logical expressions, i.e. expressions with a result value of either "true" or "false". Operators for logcal expressions are "and" and "or". Logical expressions are typically used to filter or select from a set of objects based on a criterion. E.g.: using the expression "dbh>30 and (stress>0.5 or leafarea<1)" as a filter, would result in a list of large, but stressed trees.
Logical and "mathematical" operators can be used together: every non-zero value is evaluated as "true", zero (0) as "false". 'true' and 'false' are also available and internally converted to 1 and 0, respectively. Hence, a logical "not" can be expressed as ex
Expressions provide a basic support for constants. A constant is a system-defined name with a fixed value that can be used instead of the numerical value. Currently, the species names / IDs are available and linked to the numerical index of the species for the current run:
Therefore, you can use, e.g., 'species=Tsme' in a management-filter expression (having 'Tsme' as the ID of one tree species). Note, that no apostrophs are required. 'Tsme' is replaced with the numerical value (e.g. '1') during parsing time of the expression.
The general form of function is:
functionname(list of arguments)
Generally, mathematical functions are executed without checking for the validity of the arguments (e.g. division by 0, or tan(pi/2)).
|sin(x)||the sin of x, x as radians.||sin(x)|
|cos(x)||the cosine of x, x as radians||cos(x)|
|tan(x)||the tangens of x, x as radians||tan(x)|
|exp(x)||exponential function, e(1)=2.7182...||exp(-k*LAI)|
|ln(x)||the logarithm of base e, ln(2.7182...)=1||ln(x)|
|sqrt(x)||the square root of x (equivalent to x^0.5)||sqrt(x)|
|mod(x,y)||return the modulo (remainder) of x/y. e.g. mod(13,10)=3||if(mod(id,2), 1, 0)|
|round(x)||Returns the integral value that is nearest to x, with halfway cases rounded away from zero.||round(x)|
|min(x1,x2,...,xn)||returns the minimum value of the arguments. Argument count must be >1 and <10||min(x,0)|
|max(x1,x2,...,xn)||returns the maximum value of the arguments. Argument count must be >1 and <10||max(min(x,1),0)|
|if(condition, true, false)||logical if-then-else construct. if "condition" is true, the "true" is returned, "false" otherwise. E.g.: a abs()-function: if(x<0;-x;x). Note that both clauses are calculated in every case!||if(x<0;-x;x)|
|in(value, arg1, arg2, ... , argn)||returns true if value is in the list or arguments, false otherwise.||in(year,100,200,300)|
|incsum(fn)||when used in SQL like expressions (e.g., management, the function incsum cumulates its value over several calls. See the management functions mean and sum functions.||incsum(basalArea)<40|
|polygon(value, x1,y1, x2,y2, x3,y3, ..., xn,yn)||return is: y1 if value<x1, yn if value>xn, or the lineraly interpolated numeric y-value.||polygon(x, 0,0, 1,0.5)|
|sigmoid(x, type, param1, param2)||The value of "sigmoid" curve at x. The type of curve is designated by type with the two parameters param1 and param2. 0: logistic, 1: Hill-function, 2: 1-logistic, 3: 1-hill||sigmoid(x, 0, 10, 100)|
|rnd(from, to)||returns a uniformly distributed random function between from and to.||rnd(0,1)|
|rndg(mean, stddev)||returns a random number drawn from a Gaussian normal distribution with 'mean' as the mean value 'stddev' as the standard deviation.||rndg(0,1)|
- create an output only every 10 years:
To improve the calculation performance in certain situations, expressions can use precalculated function results to interpolate a function numerically. This works best for expressions with one variable and a defined range for the input, but also an implementation for expressions with two variables is available.
For complex mathematical expressions the linearized version is up to 10 times faster than the regular expression engine.
The linearization is globally enabled/disabled by the switch system.settings.expressionLinearizationEnabled in the project file. Note, that this feature must be explicitly enabled for each expression used in the source code.


It is a fortunate coincidence that Saturday is not just Pi Day but also Albert Einstein’s birthday. Museums and destinations across the country have created terrific ways for families to celebrate with pi parades, Albert Einstein look-a-like contests, pie eating contests, and interactive numbers games.
I think Mr. Einstein would smile at the assorted events created to honor the most memorable number sequence on the planet. If you are not following me, pi usually recognized by the Greek letter “π” is the symbol used in math to represent a constant — the ratio of the circumference of a circle to its diameter which is 3.14159265358…. and on it goes. The following events prove math can be fun for everyone and who doesn’t love pie.
Princeton, New Jersey is hosting a string of crazy events including an Einstein look-a-like contest and a Dinky Train ride with reenactors playing Einstein, his mother and their good friends.
Naturally MoMath, the National Museum of Math in Manhattan will begin the day’s festivities begin at 3/14/15 at 9:26 with a scavenger hunt through the exhibits and oodles of pi puzzles to be solved.
Any day at the Franklin Institute in Philadelphia offers exceptional rewards for families but the Museum’s National Pi Day celebration will add extra geekiness with pi reciting contests and a pie launch competition
Adler Planetarium’s daytime Pi Day activities include a pie eating contest, a parachuting pies challenge, and pie-in-the-sky solar observations.
At San Francisco’s Exploratorium the day begins at 9:26:53 and includes pi processional, light circles and pizza pie dough tossing.
If none of these work for you there’s always the excuse to bake a pie and do your own calculations per serving.
For the latest family travel news follow the Family Travel Files on Twitter (@FamTravelFiles)
Nancy Nelson-Duac, Curator of the Good Stuff


Book Title: Discovering Mathematics: Common Core, Workbook 7B
Publisher: Star Publishing
Discovering Mathematics Common Core Workbooks are designed for middle school students. Developed in collaboration between Star Publishing Pte. Ltd. and Singapore Math Inc,. these workbooks follow the Singapore Framework and also cover the topics in the Common Core State Standards.
Each workbook is written as a supplement to the textbook, Discovering Mathematics Common Core, to give students more practice in applying the concepts learned. Students may refer to the summary of the important concepts in each chapter of the textbook for a quick review before attempting the questions in the workbook. After completing the exercises, students will not only polish their own analytical skills, but also develop a stronger foundation in mathematics.
The questions in each workbook chapter are categorized into 4 pars according to the level of difficulty and the thinking skills involved:
Basic Practice: simple questions that drill comprehension of concepts
Further Practice: harder questions that involve direct applications
Challenging Practice: questions that require synthesis ability
Enrichment: questions that demand higher order thinking
These questions encourage students to think analytically, reason logically, use appropriate connections between mathematical ideas, and apply problem-solving skills in daily life situations.
We hope that these comprehensive workbooks will give students the tools and the confidence to handle mathematical questions and apply mathematical concepts to real-life situations. By achieving this, students will find learning mathematic an interesting and exciting experience.


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


A friend's Christmas present to himself (and indirectly his family and friends) was a MakerBot Replicator 3-D printer. Down at Bristol Hackspace we have several repraps in various stages of completion, but none yet ready for novice use. Marc's printer seems to have worked out of the box, so I was able to get my first test piece send via Dropbox to London, printed appropriately enough, since it involved hearts, in a pretty pink plastic and posted to Bristol in a couple of days.
The key to my personal progress was the discovery of OpenSCAD. I've looked at using GUI-based 3-D tools like Blender but OpenSCAD is a programming environment, just my kind of 3-D tool. I soon found a weath of freely avaliable tutorials, reference material and scripts:
OpenSCAD uses two modes of construction: Constructive Solid Geometry (CSG) using the Computational Geometry Algorithms Library (CGAL); and extrusion(two modes) of a 2-D shape.
Some lessons learnt:
The challenge for the budding designer is to learn how to compose a desired shape by the application of the operators to the primitives. In the physical world, we typically construct objects in a 'carpentry' mode, glueing non-intersectiing objects (union) and removing waste (difference). Also our range of primitives is vast and I generally have no idea how, say, a nut is formed. CSG allows us to union() overlapping objects, get the intersection of objects and use the less-known operators minkowski sum and hull. To see what experienced openSCAD programmers use, I analysed the collection of 22 library files for basic shapes, gears and threads etc. There were a total of 10000 lines including comments. I used grep to do the counting of patterns such as 'rotate\s*(' to get counts of each construct in the language.
|union||576||Severe underestimate since union() is the default for a sequence of objects|
|intersection_for||2|| replacement for loop within intersection which doesnt work
This data seems to indicate that designers are mainly using operators which correspond to the physical operations (carpentry mode) and are generally not using the advanced operators. This is not surprising because even intersection can yield surprising results. An example in the wikibook shows the construction of a dodecahedron from the intersection of 6 boxes:
but I havent been able to trace the source of this construction. It's tempting to look for ways of constructing the other regular solids using intersection and indeed playing with this code, I found I could make an octahedron:
What other solids could I make this way? One feature of OpenSCAD is very useful here - the ability to display a parameterized sequence of objects as an animation. The system variable $t changes from 0 to 1 in increments determined by the total number frames entered in the animation panel, so to see what solids are generated as the dihedral angle is changed :
and its fun to watch the shapes morph, becoming dodecahedrons at several angles and curious asymmetric faceted shapes in between. Moreover you can edit the script as it runs to change other parameters such as the number of intersecting boxes. You can also create a video. Each frame is saved as a .png file. I used ffmpeg installed on Ubuntu:
> ffmpeg -f image2 -i frame%05d.png -r 12 dodec.avi
to create this little video


Elementary Apps for the Classroom
Check out these great apps for your students.
Arithmetic Invaders Express: Grade K-2 Math Facts - Express addition, subtraction, and multiplication games to defend the solar system and master K-2 arithmetic. Also, learn some basic facts about the Solar System.
Cloud Math Free - Excellent app for practicing and learning math." The difficulty level is easily controlled by a choice of operations, number ranges, numbers of answers to choose from, the number of problems in a set, and a Fail-safe mode. There are problems with addition and subtraction operations on numbers up to 50.
Geoboard, by the Math Learning Center - Geoboard is a tool for exploring a variety of mathematical topics introduced in the elementary and middle grades. Learners stretch bands around the pegs to form line segments and polygons and make discoveries about perimeter, area, angles, congruence, fractions, and more.
Math Puppy - Your younger students will be able to enjoy an interactive, learning environment of fun and adventure as they master basic math skills. Perfect for early childhood students.
Counting Bills & Coins - Counting Bills & Coins lets you practice identifying and solving math problems with money. Count, match, and make change with coins up to quarters and bills up to $20. Practice money skills in five unique activities: counting money, show me the money, making change, matching amounts, and show values.
Math Ninja - Use your math skills to defend your tree house against a hungry tomato and his robotic army in this fun action packed game! Choose between ninja stars, smoke bombs, or ninja magic - and choose your upgrades wisely!
Interactive Telling Time Lite - Interactive Telling Time is great for kids from ages 3 to 12 and comes in 3 difficulty levels so that it helps kids master telling time progressively (statistics are provided to help keep track of progress). Learn to set the time via interactive clocks with movable hour and minute hands, how to read a clock/to tell time, the conversion between analog clocks and digital clocks, as well as concepts such as ‘o’clock‘, ‘midnight’, ‘half past’, ‘quarter past’, ‘quarter to’, ‘past’, ‘to’, etc.
Dominoes Easy Match - Dominoes Easy Match skills focus on: concepts of numbers, identification of numbers 1-9, identification of simple arrays, rote counting, and one-to-one correspondence. This is a great app for beginning math learners.
MathTappers: Multiples - MathTappers: Multiples is a simple game designed first to help learners to make sense of multiplication and division with whole numbers, and then to support them in developing fluency while maintaining accuracy.
Teaching Number Lines: Little Monkey Apps Number Lines helps to introduce the concept of number lines through a variety of of modules such as: counters, drawings, and number lines in order to explain and physically model problems. Little Monkey Apps Number Lines aims to visualize numbers for rote counting and ordering and see the physical position of a number linking patterns and relationships. Great for student in K through 2.
K12 Equivalence Tiles - K12 Equivalence Tiles lets you compare the values of fractions, decimals, and percents up to 1 by using draggable tiles of varying values. K12 is not a calculator; it is a tool for students who have trouble understanding rational number equivalence (for example, 0.5, 1/2, 50% are all equal). This tool also allows students to visually perform simple operations like addition, subtraction, comparison, and ordering. This is a great app for upper elementary grades.
Rounding - Rounding is a great way to test your rounding skills and have fun at the same time. Select the number of problems and other options in order to correlate with your skill level.
Interactive Math Glossary -The Interactive Math Glossary App is provided by the Texas Education Agency to help teachers explore and understand mathematics vocabulary used in the grades K–8 Texas Essential Knowledge and Skills. Each glossary word is displayed in a four quadrant Frayer Model.
Science 360 - This app will allow your students to scroll the 3D sphere that is made of thumbnails of various science photos and videos. Tap on any one of those thumbnails to bring up the recent updates from National Science Foundation (NSF) or institutions funded by NSF. All contents are multimedia and are updated weekly, so you always get the most recent development of science and technology.
NASA - The NASA app carries a wealth of NASA information right on your iPad. The app collects, customizes and delivers an extensive selection of dynamically updated mission information, images, videos. It has information about the planets, live streaming of NASA TV, and on demand NASA video.
WWF Together- Experience the world's most amazing and endangered animals in one app – together. This interactive experience brings you closer to the stories of elephants, whales, rhinos and other fascinating species. Discover their lives and the work of WWF in a way you’ve never seen before. Try out “tiger vision,” stay as still as the polar bear during a hunt, and chop the panda’s bamboo. Explore the animals’ stories, then fold them up and share them with the world.
Robots for IPADS - Robots for iPad is the best, most complete guide to the world of robotics. This fun, highly interactive app lets you explore over 150 real-world robots, with hundreds of animations, photos, videos, and articles
Kids Can Match - An interactive, adaptive and fun memory game for children of all ages. This app contains amazing, one of a kind, collection of over 70 authentic animal images and sounds. The perfect way to engage children in a fun game while they learn about a broad range of animals from all over the world. Tailor made and carefully designed by experienced educators to fit the needs of small children. This educational game is a wonderful asset in any animal loving family.
Talkboard- Talkboard is the canvas for collaborative learning. Work with your students, wherever they are, on a shared whiteboard to bridge the gap in distance between you. Then harness your creativity to bring color and clarity to your lessons. Talkboard’s simple yet elegant drawing tools and integrated audio allow you to engage students in a whole new way.


BY PROF JACOB
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics c. 1900 BC), the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC) and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction". Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics. Chinese mathematics made early contributions, including a place value system. The Hindu-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.
From ancient times through the Middle Ages, bursts of mathematical creativity were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day.
 Prehistoric mathematicsThe origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. The idea of the "number" concept evolving gradually over time is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.
The oldest known possibly mathematical object is the Lebombo bone, discovered in the Lebombo mountains of Swaziland and dated to approximately 35,000 BC. It consists of 29 distinct notches cut into a baboon's fibula. Also prehistoric artifacts discovered in Africa and France, dated between 35,000 and 20,000 years old, suggest early attempts to quantify time.
The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be as much as 20,000 years old and consists of a series of tally marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six month lunar calendar. In the book How Mathematics Happened: The First 50,000 Years, Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."
Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design.
All of the above are disputed however, and the currently oldest undisputed mathematical usage is in Babylonian and dynastic Egyptian sources. Thus it took human beings at least 45,000 years from the attainment of behavioral modernity and language (generally thought to be a long time before that) to develop mathematics as such.
 Babylonian mathematicsBabylonian mathematics refers to any mathematics of the people of Mesopotamia (modern Iraq) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. It is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics.
In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.
The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.
The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of regular reciprocal pairs. The tablets also include multiplication tables and methods for solving linear and quadratic equations. The Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.
Babylonian mathematics were written using a sexagesimal (base-60) numeral system. From this derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors. Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the decimal system. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.
 Egyptian mathematicsEgyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars.
The most extensive Egyptian mathematical text is the Rhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000-1800 BC. It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge, including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory (namely, that of the number 6). It also shows how to solve first order linear equations as well as arithmetic and geometric series.
Another significant Egyptian mathematical text is the Moscow papyrus, also from the Middle Kingdom period, dated to c. 1890 BC. It consists of what are today called word problems or story problems, which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum: "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. You are to take 28 twice, result 56. See, it is 56. You will find it right."
Finally, the Berlin papyrus (c. 1300 BC) shows that ancient Egyptians could solve a second-order algebraic equation.
 Greek mathematicsGreek language from the time of Thales of Miletus (~600 BC) to the closure of the Academy of Athens in 529 AD. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics.
Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them.
Greek mathematics is thought to have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.
geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. Pythagoras established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers.
Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others. His Platonic Academy, in Athens, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus, came from. Plato also discussed the foundations of mathematics, clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions. The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.
Eudoxus (408–c.355 BC) developed the method of exhaustion, a precursor of modern integration and a theory of ratios that avoided the problem of incommensurable magnitudes. The former allowed the calculations of areas and volumes of curvilinear figures, while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, Aristotle (384—c.322 BC) contributed significantly to the development of mathematics by laying the foundations of logic.
In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria. It was there that Euclid (c. 300 BC) taught, and wrote the Elements, widely considered the most successful and influential textbook of all time. The Elements introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. In addition to the familiar theorems of Euclidean geometry, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry, including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.
Archimedes (c.287–212 BC) of Syracuse, widely considered the greatest mathematician of antiquity, used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, 310⁄71 < π < 310⁄70. He also studied the spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid), and an ingenious system for expressing very large numbers. While he is also known for his contributions to physics and several advanced mechanical devices, Archimedes himself placed far greater value on the products of his thought and general mathematical principles. He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume a cylinder circumscribing the sphere.
Apollonius of Perga (c. 262-190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone. He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond"). His work Conics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton. While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.
Around the same time, Eratosthenes of Cyrene of Cyrene (c. 276-194 BC) devised the Sieve of Eratosthenes for finding prime numbers. The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline. Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers. Hipparchus of Nicaea (c. 190-120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle. Heron of Alexandria (c. 10–70 AD) is credited with Heron's formula for finding the area of a scalene triangle and with being the first to recognize the possibility of negative numbers possessing square roots. Menelaus of Alexandria (c. 100 AD) pioneered spherical trigonometry through Menelaus' theorem. The most complete and influential trigonometric work of antiquity is the Almagest of Ptolemy (c. AD 90-168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years. Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.
Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics. During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis, which is also known as "Diophantine analysis". The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations. The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares). Diophantus also made significant advances in notation, the Arithmetica being the first instance of algebraic symbolism and syncopation.


Maths and Science Tutor wanted immediately: APPLY NOW
Master Maths Morningside is currently looking for mathematics and/or physical science tutor.
Master Maths Morningside is part of a well-established franchise concern, which has been providing extra tuition for over 40 years, to learners from junior level to adults. Our successful candidate must have exceptional customer service skills, over the phone and in person. Must be computer savvy.
The closing date for applications is 1 December 2019.
Please only submit your CV if the following applies to you:
You achieved 70% or more for maths or physical science in the final matric exam, or you have a relevant tertiary qualification.
You will be required to tutor up to grade 12 level for maths and physical science.
70% or more for maths and/or physical science in the final matric exam.
Speak fluent English.
South African passport or valid work visa.
Motivated by success in your job.
A people’s person and you think you might enjoy tutoring.
Able to interact successfully with young children.
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Requires an ability to explain mathematics or physical science pleasantly and with an enthusiastic personality.
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Math with Bad Drawings: Illuminating the Ideas That Shape Our Reality (Paperback)
In Math with Bad Drawings, Ben Orlin reveals what math is all about. His tools are unorthodox: jokes, cartoons, strange-but-true stories, and beneath it all, the empathy of a veteran teacher who believes that math should belong to everyone. Orlin helps us to think like mathematicians by teaching a brand-new game of tic-tac-toe, profiling the ten people you meet in line for the lottery, and documenting the headaches that ensue when the Evil Empire attempts to build a spherical Death Star. Math with Bad Drawings will change the way you see the subject—and the world.
Ben Orlin is the author of Math with Bad Drawings (as well as the blog of the same name), Math Games with Bad Drawings and the companion game kit The Ultimate Game Collection, and Change Is the Only Constant. His writing on math and education has appeared in The Atlantic, the Chicago Tribune, the Los Angeles Times, Slate, Vox, and Popular Science. He has taught middle and high school mathematics and has spoken about math and education at colleges and universities across the United States. He lives with his family in St. Paul, Minnesota.
"Ben Orlin is terribly bad at drawing. Luckily he's also fantastically clever and charming. His talents have added up to the most glorious, warm, and witty illustrated guide to the irresistible appeal of mathematics."—Hannah Fry, mathematician, University College London and BBC presenter
"Brilliant, wide ranging, and irreverent, Math with Bad Drawings adds ha ha to aha. It'll make you smile - plus it might just make you smarter and wiser."—Steven Strogatz, Professor of Mathematics, Cornell University, author of The Joy of x
"MATH WITH BAD DRAWINGS is a gloriously goofy word-number-and-cartoon fest that drags math out of the classroom and into the sunlight where it belongs. Great for your friend who thinks they hate math - actually, great for everyone!"—Jordan Ellenberg, author of How Not To Be Wrong
"Ben Orlin has hit the seemingly unattainable sweet spot. He has written a book that is funny and serious, that is entertaining and informative, and that would interest a reader with or without a background in mathematics. Math with Bad Drawings would be a wonderful book for people who love math, used to love math, want to love math, want to know what math is good for, or just want to know what math really is."—Math Horizons
"Orlin's ability to masterfully convey interesting and complex mathematical ideas through the whimsy of drawings (that, contrary to the suggestion of the title, are actually not that bad) is unparalleled. This is a great work showing the beauty of mathematics as it relates to our world. This is a must read for anyone who ever thought math isn't fun, or doesn't apply to the world we live in!"—John Urschel, mathematician named to Forbes® "30 Under 30" list of outstanding young scientists and former NFL player
"Illuminating, inspiring, and hilarious, Math with Bad Drawings is everything you wanted to learn in class but never thought to ask. A joyful romp through mathematics and all its wisdom."—Bianca Bosker, author of the New York Times-bestselling Cork Dork


Dr Mahinthakumar is interested in large scale modeling of subsurface flow and transport, parallel and distributed computing, optimization and inverse problems, water distribution system analysis
Dr. Mahinthakumar's (referred to as Dr. Kumar by his students) long term goal is to develop efficient algorithms and tools to solve large scale civil and environmental engineering problems.
Dr. Kumar is currently focused on 1) real time optimization and inverse modeling for water distribution systems analysis, 2) large scale modeling and characterization of groundwater flow and transport systems, 3) parallel and distributed computing algorithms and tools for environmental applications.
In CCEE, Dr. Kumar collaborates with Dr. Arumugam, Dr. Brill, Dr. DeCarolis and Dr. Ranjithan.
At the graduate level, Dr. Kumar teaches Introduction to Numerical Methods for Civil Engineers (CE 536), High Performance Computing for Civil Engineers (CE 791A), and Inverse Modeling for Civil Engineers (CE 791B). In Numerical Methods, he teaches application of common numerical methods to civil engineering problem solving. The high performance computing course is a project based course focused on parallel and distributed computing algorithms for large scale civil engineering applications. The inverse modeling course focuses on heuristic and gradient based search techniques as well as Markov Chain Monte Carlo methods for the solution of civil engineering parameter estimation and system identification problems. The graduate students who work with Dr. Kumar enjoy computer programming and are interested in developing innovative methods and tools for large scale civil engineering problem solving.
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|Golden Toy Robot|
Golden Toy Robot is a decoration object that can be found through Treasure Hunting. It is found in the Safes that can be dug up. The Golden Toy Robot is the rare variant of the regular Toy Robot and is much rarer. When looting a Safe, there's a 2.3% chance of it being any robot in the first place. After becoming a robot, the game decides whether it be a regular Toy Robot (94.8%) or a Golden Toy Robot (5.2%).
Following paragraph describes the calculation of the chance of getting a Golden Toy Robot. To get the Golden Toy Robot, the player must first find a Safe. Every treasure that is dug up has a 20% chance of being a Safe. This translates to 1/5. Getting a robot in the first place is 2.3%, which is roughly 1/50, and it being golden is another 5.2%, which is roughly 1/20. Getting this specific combination of rolls is therefore a 1/5000 or 0.02%, making the Golden Toy Robot the rarest item in the game. Note that this equation is rolled for all items within the container, meaning that if there are five items in the Safe, the player rolls for the 1/1000 chance five times. This means that on average, when looting a Safe, the player has roughly 0.5% chance of getting a single Golden Toy Robot.
- Used to decorate the raft.
|Update 12||Golden Toy Robot added to the game.|


This riveted me; I love mental maths. To cube root a number (assuming it’s a perfect cube), here are the steps, using 250,047 as our protagonist cube:
- Ignoring the last 3 digits of the cube, find the first cube which is less than the remaining digits. 250 (ignoring 047) is the number in question, and 216 (63) is the first cube less than 250 — giving us 6.
- Find the number from 1-10 that, when cubed, ends in the last digit of your cube. 250,047 ends in 7, which corresponds to the last digit of 27 (33). This number is the last digit of the cube root.
- Voila! Those are the 2 digits of the cube root — 6 and 3 — giving us 63 as the cube root of 250,047.


Subtraction with Addition: Two's Complement
Subtracting numbers on paper is easy, but how does a computer do it? Most computer operations can be reduced to additions of binary numbers… even when the operation is subtraction. To understand how this works, let’s first think about how an odometer works.
Let’s say you’ve driven 7631 miles and really want to celebrate your 9999th mile and see the odometer return to 0000. You would have to drive 2368 miles to reach 9999 miles. When you’ve hit 9999 miles, the odometer returns to 0000 and the cycle continues.
And so the total amount of miles you had to drive was 2368 to hit 9999 and then 1 mile to hit 10000.
The formula to calculate the amount of miles you have to drive to hit 9999 miles is:
7631 + x = 9999 x = 9999 - 7631 x = 2368 2368 + 1 = 2369 miles driven in total for odometer reset.
The mind-blowing part of this is that 2369 is the negative equivalent of 7631. How?
Let’s pick another random 4-digit number and subtract 7631 from it:
9473 - 7631 = 1842
That's also equivalent to adding negative 7631:
9473 + (-7631) = 1842
Let’s substitute -7631 for its positive equivalent:
9473 + 2369 = 11842
11842 looks familiar doesn’t it? Since we’re operating in a 4-digit system, we drop the first digit and we have 1842, the correct result of the subtraction.
We can also use this negative representation of the number in other equations:
2354 - 7631 = -5277
So -5277 is the correct answer.
Let’s convert that answer back to the positive equivalent:
5277 - 1 = 5276 9999 - 5276 = 4723
Now… let’s see what happens when we only use addition with the positive equivalent number:
2354 + (-7631) 2354 + 2369 = 4723
🎉 Voila! Obviously, converting 4723 to its negative equivalent will result in -5277!
However! Instead of reversing the calculation like we just did, we can also covert a negative to a positive using the exact same procedure without reversing it:
9999 - 5277 = 4722 4722 - 1 = 4723
🎉 And the result is back to 4723.
This method is called the two's complement and is exactly how computers implement signed binary numbers! We just looked at base-10 numbers, but computers work in binary, base-2.
The same idea applies in binary. Take the binary number 0110.
The negative number of 0110 can be calculated in the same way, except the odometer resets at 1111 and not 9999.
1111 - 0110 = 1001 1001 + 0001 = 1010
With that, the negative number for 0110 is 1010. We can then do similar calculations, let’s say 7 minus 6.
7 - 6 = 1 (in decimal) 0111 - 0110 (in binary) 0111 + (- 0110) 0111 + 1010 = 10001, drop the first digit, resulting in 1!
Typically the computer uses 16-bit inputs, following this example means instead of 4-digit, computers use 16-digit binary numbers.
The system can code a total of 2n signed numbers, where 2n-1 - 1 and -2n-1 represent the number of positive values and negative values respectively. This means in a 4-bit system, the total number of numbers (haha!) represented is 24 = 16, where we can go up to positive 7 and as low as -8.
The two's complement is implemented in the hardware of the Arithmetic Logical Unit chip that makes up the CPU.


The Early Life of Ada Lovelace
Ada Lovelace, born Augusta Ada Byron, is widely regarded as the world's first computer programmer. She was born on December 10, 1815, in London, England. Ada had a fascinating upbringing, being the daughter of Lord Byron, the famous poet, and Anne Isabella Milbanke, a mathematician.
Despite her parents' separation when she was just a few weeks old, Ada's mother ensured that she received a well-rounded education. At an early age, Ada developed a keen interest in mathematics and science, showing exceptional aptitude in these subjects.
Ada's passion for numbers and calculations grew as she delved deeper into her studies. She was tutored by some of the best mathematicians and scientists of her time, including Augustus De Morgan and Mary Somerville. These mentors recognized Ada's brilliance and nurtured her love for mathematics and logical reasoning.
Ada's Introduction to Charles Babbage
Ada Lovelace's interest in mathematics and science was further nurtured by her mother, who introduced her to prominent mathematicians and scientists of the time. At the age of 17, Ada met Charles Babbage, a renowned mathematician and inventor. Babbage's invention, the Analytical Engine, is considered the precursor to modern computers. This encounter would prove to be a turning point in Ada's life.
Charles Babbage recognized Ada's exceptional mathematical abilities and intellect. He became her mentor and close collaborator, fostering her interest in the world of computing and laying the foundation for her groundbreaking work in computer programming.
The Collaboration Between Ada and Charles Babbage
Ada Lovelace's collaboration with Charles Babbage was instrumental in shaping her legacy as the pioneer of computer programming. She translated and annotated an article written by Italian engineer Luigi Menabrea about Babbage's Analytical Engine. Ada's annotations, which were three times longer than the original article, included a method for calculating Bernoulli numbers using the Analytical Engine.
What set Ada's annotations apart was her visionary insight. She recognized that the Analytical Engine had the potential to do more than just calculations. She envisioned it as a general-purpose machine that could be programmed to perform various tasks beyond number crunching. Ada's notes included algorithms for generating complex graphics and music, making her the first to articulate the concept of software.
Ada's work on the Analytical Engine demonstrated her deep understanding of its potential and laid the foundation for modern computer programming. Her ideas were far ahead of her time, anticipating the development of computer programs and the possibilities they could unlock.
Lovelace's Vision for the Analytical Engine
Ada Lovelace's vision for the Analytical Engine went beyond mere calculations. She saw its potential for creating not only numerical results but also generating complex graphics and music. Her foresight in recognizing the capabilities of the Analytical Engine as a general-purpose computing machine laid the foundation for modern computer programming.
Ada's notes on the Analytical Engine were published in 1843, in an English science journal. However, due to the prevailing gender biases of the time, her work received little attention and was not widely understood or acknowledged.
It was not until the late 20th century that Ada Lovelace's contributions to computer programming were fully recognized. Computer scientists and historians revisited her work and acknowledged her as the world's first computer programmer. They marveled at her ability to grasp the potential of computing machines and her visionary ideas that remain relevant even today.
The Legacy of Ada Lovelace
Ada Lovelace's contributions to the field of computer programming were ahead of her time. Unfortunately, her work went largely unnoticed during her lifetime. It was only in the mid-20th century that her notes on Babbage's Analytical Engine were rediscovered and recognized for their significance.
Today, Ada Lovelace is celebrated as a pioneer and visionary in the world of computer science. Her insights and ideas continue to inspire generations of programmers and technologists. The second Tuesday of October is observed as Ada Lovelace Day in her honor, to recognize the achievements of women in science, technology, engineering, and mathematics (STEM).
Ada Lovelace's remarkable contributions to computer programming laid the groundwork for the digital revolution that followed. Her visionary ideas and analytical prowess continue to shape the world of technology today. Ada Lovelace will forever be remembered as the pioneer who paved the way for future generations of programmers.
Ada Lovelace's remarkable journey from being the daughter of a famous poet to becoming the pioneer of computer programming is a testament to her brilliance, determination, and forward-thinking mindset. She overcame societal barriers and made invaluable contributions that continue to shape the world we live in. Ada Lovelace's legacy serves as an inspiration for aspiring programmers, reminding us of the power of ideas and the impact one person can make on the world.