Everything and More
Everything and More: A Compact History of Infinity is the first in the series of W.W. Norton's "Great Discoveries" series, which are math- and science-focused books written mostly by authors outside the scientific field. Throughout this "booklet" (3), as the narrator calls it, Wallace explains and examines the complex history of infinity in the context of mathematics. In particular, he focuses on nineteenth-century mathematician Georg Cantor, the inventor of set theory. Wallace traces the concept of infinity (as well as resistance to it) from the Greeks through the twentieth century, paying special attention to the beauty and often counter-intuitive nature of mathematics. Although Wallace states that Everything and More is a "piece of pop technical writing" (1) designed to appeal to literature enthusiasts, he also includes phrases titled "IYI" ("If You're Interested") to appeal to those with more of an interest in or knowledge of mathematics.
Everything and More exists as a fusion of historical account, mathematical textbook, and story-telling. Wallace combines the story of math’s journey toward understanding infinity with textbook-style symbols, glossaries, explanations, and proofs—all wrapped up in a delightfully conversational tone, as if Wallace were the teacher, the text his classroom, and the readers his students.
Everything and More does not include an explicit definition of infinity, but does bestow infinity a central position in the story of mathematics and figures infinity as a problem, a solution, a controversy—whose story parallels the story of human progress.
Wallace identifies infinity as both a primary source of problems for mathematics as well as an integral solution that moves math forward. On the one hand, Wallace claims that “nothing has caused math more problems—historically, methodologically, metaphysically—than infinite quantities” (32). In fact, according to Wallace, the story of math is the history of infinity-related problems (32). On the other hand, a careful parsing of the different types of infinity—infinity as quantity and infinity as procedure—leads to solutions and advancement in mathematics. For instance, discerning the nuances of infinity led mathematicians to a solution for Zeno’s Paradox: “The Dichotomy’s central confusion is now laid bare: the task of moving from point A to point B involves not [an infinite-number] of necessary subtasks, but rather a single task whose ‘1’ can be validly approximated by a convergent infinite series” (195).
Furthermore, in expounding “math’s battle over [infinity]” (205), Everything and More captures infinity as controversial, as a concept whose existence was constantly threatened throughout the history of math in spite of, or perhaps because of, all the problems and solutions it created. While Cantor’s work attests that “actually-infinite sets can be understood and manipulated, truly handled by the human intellect” (205), Wallace nonetheless posits infinity as “a concept it [math] couldn’t really ever handle anyway” (68)—because with each paradox that infinity solves and each testament to its prospect of being “handled,” infinity introduces a new wave of paradoxes and problems that dashes previous hopes.
Wallace summarizes: “the Story of [infinity’s] overall dynamic, whereby certain paradoxes give rise to conceptual advances that can handle those original paradoxes but in turn give rise to new paradoxes, which then generate further conceptual advances, and so on” (278). The way Wallace figures it—the story of infinity mirrors the story of human progress, where problems lead to solutions that lead to new problems and so on, infinitely.
Infinity and Mental Illness
Everything and More forges a strong link between infinity and mental illness and argues that the origin of this link lies in abstraction. Wallace observes that “Historians and pop scholars tend to spend a lot of time on Cantor’s psychiatric problems and on whether and how they were connected to his work on the mathematics of [infinity]” (6), but he goes further to investigate the reasons for this connection—between psychiatric problems and the mathematics of infinity. Wallace concludes that the instigator of madness is the abstract way of thinking. He qualifies G.K. Chesterton’s claim that logic induces madness and instead asserts that “what Chesterton’s really trying to talk about is one of logic’s main characteristics—and mathematics’. Abstractness. Abstraction,” for “logic is just a method, and methods can’t unhinge people” (8). In so doing, Wallace implies that abstraction can “unhinge people.” Moreover, Wallace uses the ability to end abstract thinking as a means to categorize “sane, functional people…from the unhinged” (17). By conjecturing that “the dreads and dangers of abstract thinking are a big reason why we now all like to stay so busy and bombarded with stimuli all the time,” since “abstract thinking tends most often to strike during moments of quiet repose” (13), Wallace conveys that the seeds of mental illness are not inherent within us, but rather planted by abstract thinking. Thus, Wallace warns the reader against the dangers of abstract thinking and its ability to “unhinge,” but on the other hand, he does not portray mental illness or the state of being “unhinged” (17) as negative. In fact, throughout the booklet, Wallace glorifies the Mentally Ill Mathematician as a hero of our times.
The Language or Map of Math
The abstractness of math renders math ostensibly a world of its own, but Everything and More illuminates math as more than just a solipsistic inhabitant of its own world. Everything and More conceives of math as a map—a map that translates problems, ideas, and concerns from “the world we all really live in” (71) to a world of symbols and abstractions. In this process of translation, math becomes more than just a set of isolated symbols and abstractions; math becomes a language—a language that maps one world into another. After acknowledging the status of math as a real language, Wallace proceeds to remind the reader to keep in mind the principal traits of language: “both a map of the world and its own world” (30).
Math acts as a map by connecting and translating the problems of humanity into its own problems. Wallace explains that “most of math’s definitions are built up out of other definitions; it’s the really root stuff that has to be defined from scratch. Hopefully…that scratch will have something to do with the world we all really live in” (71). Through the vocabulary of math presented in Everything and More, one can see that the “root stuff” of mathematics is indeed defined by the world we all really live in. Intersections between the vocabulary of math and our daily vocabulary abound—for example, real, imaginary, rational, irrational, identity, existence, uniqueness, and sensitivity—just to name a few. In fact, Everything and More portrays the function of math as the endeavor to represent real world; the text identifies calculus as “a seminal advance” because calculus gives math the “ability to represent continuity and change and real-world processes” (126). The roots of math are therefore intertwined and scratched in the world we all really live in. The language of math seems to capture the same issues that our world cares about. Sometimes, math simplifies the issues that we struggle with day-to-day and offers complex concepts the well-defined and stable type of definitions that we cannot manage to offer; other times, math mirrors or complicates these concepts.
An instance of math mapping principles from our world to its own world occurs in the battles between the Platonists and Intuitionists. During these battles, notions of morality slither into the realm of math. In using words such as “delusory” and “evil” to describe continuous functions (216), Wallace renders the world of math reflective of the morality, politics, and divisions extant the human world.
While we know the humanities for the endless and answerless questions it asks, we know math for its ability to provide each problem with a solution. In Everything and More, however, Wallace adduces a counterintuitive instance, where math, like the humanities, raises questions it cannot answer: “In the words of one math-historian, Fourier’s techniques ‘raised more questions than he was interested in answering or capable of solving.’ Which is both tactful and true” (163). Thus, at times, like in the humanities and in the rest of our world, there is no answer in math.
Furthermore, math, like any other language, cannot escape the difficulties in communication—in expressing the self and in translating the self into a language that others can understand. In spite of its ambition to be crystal clear and “clean” (34), math falls prey to the inescapable inability to communicate. In particular, math suffers from an inability to articulate its difficulties. Wallace quotes Russell, “To state clearly the difficulties involved, was to accomplish perhaps the hardest part of the philosopher’s task” (52) and then revises Russell’s claim into a stronger one: “Stating these difficulties clearly is, in fact, the whole and only difficulty involved here” (53). Just as the inability to communicate becomes central to nearly every one of Wallace’s characters, math is no exception. Wallace reminds us that even though “the power and maybe even whole raison d’etre of the language of math is that it’s designed to be so clean and nonconnotative that it avoids ambiguities,” the attempt to “express numerical quantities and relations in natural language—to translate mathematical propositions into English and vice versa—often causes trouble” (34). In a way, Everything and More exists as Wallace’s attempt to tackle this difficulty of translation—between numerical relations and our natural language, between the world of math and “the world we all really live in.”
Footnotes and Abbreviations
As is typical to much of Wallace's writing, he uses footnotes regularly throughout Everything and More. One of the most frequent footnotes that he employs is the "If You're Interested" note, abbreviated "IYI." According to Wallace, "the boldface 'IYI' designates bits of material that can be perused, glanced at, or skipped altogether if the reader wants. Meaning skipped without serious loss . . . most IYI-grade chunks, though, are designed for readers with strong technical backgrounds,or unusual interest in actual math, or preternatural patience, or all three; they (the chunks provide a more detailed look at the stuff that the main discussion glosses or breezes through" (2-3). In fact, this type of footnote is so prevalent throughout the booklet that Wallace states, "over half the document's footnotes are probably IYI, as well as several different paragraphs and even a couple subsections of the main text" (2). One purpose of IYI is to make the actual book more readable for those with weaker mathematical backgrounds--they can merely skip the notes that do not, in fact, interest them. Another purpose is due to the "strange stylistic problem in tech writing, which is that the same words often have to get used over and over in a way that would be terribly clunky in regular prose" (3), thus resulting in the need for an abbreviation in order to "achieve any kind of variation at all" (3).
The Beauty of Math
One of the more prevalent themes in Everything and More is the notion that beauty is an integral aspect to math. On the first page of the forward, for example, Wallace writes that the booklet's subject is "a set of mathematical achievements that are extremely abstract and technical, but also extremely profound and interesting, and beautiful" (1). In the same paragraph, he states that the aim of Everything and More is to "make the math beautiful--or at least to get the reader to see how someone might find it so" (2). Although Wallace clearly recognizes that not all readers will inherently see the beauty of math, he does hope to make readers see the possibility of beauty in the more complex achievements he details throughout the booklet. Wallace also frequently refers to math as being "sexy" throughout the course of Everything and More; the first example comes when he states that the "sexy math terms don't matter for now"(5). Thus, Wallace attempts to transform numbers and equations into things with aesthetic value, such as beauty or sexiness, to illustrate one way by which to view mathematics and make sense of complex problems.
A Word on Sexy Math
Wallace’s use of the word sexy in descriptions of math concepts stands out to some degree and becomes a memorable minutia of the text. For some readers, the use of sexy stimulates a newfound interest in math or causes them to pay more attention to the math; for others, it does not. It is possible that the use of sexy to describe math is just another element of Wallace’s distinct voice and humor.
It is also possible that the use of "sexy" can be interpreted in another way. In Everything and More, sexual imagery mingles with mathematical concepts. Throughout the text, Wallace litters variations of the word sex in his descriptions of math, and at one point, the booklet even deploys sex as a metaphor for math.
To illustrate, after compiling a list of terms, Wallace declares that “the sexy math terms don’t matter for now” (5). When explaining Zeno’s paradox, Wallace insists, “Put a little more sexily, the paradox is that a pedestrian cannot move from point A to point B without traversing all successive subintervals of AB” (49). When describing the difference between the Number Line and the Real Line, Wallace mentions, “For sexy technical reasons that we’ll get to, the Number Line is more properly called the Real Line if it also maps the irrational numbers” (73). Why are these terms or explanations or reasons sexy? The text provides no clear answer. The text, however, does imply that the language of math can be translated into the language of sex. Wallace explains the derivative: “In sexual terms, it’s an expression of the rate of change of a function with respect to the function’s independent variable” (116). Although the text does not clarify the way in which these explanations are sexual, the text does establish an explicit connection between math and sex. In one case, the text even figures sex as a mathematical activity: “The thing to appreciate about the humble Number Line’s marriage of math and geometry is that it’s also the perfect union of form and content” (72). With the “marriage of math and geometry”—a “perfect union of form and content”—Everything and More imbues mathematical activities with images of sex. In one revealing footnote, Wallace informs the reader: “If you’re innocent of college math, this def. will make more sense when we look at classical calc” (116). In this phrase, sex becomes a metaphor for math. The phrase insinuates that doing college math is like having sex; not having done college math, like being a virgin. But how can doing math possibly be like having sex?
Perhaps they have a similar function? If sex serves to propagate the human species, then math may serve to propagate another species. In Everything and More, Wallace claims “there is more than one species of infinity” (72). Thus, infinity in Everything and More occupies the status of a species. In fact, infinity is the species that mathematics, throughout the course of its history (and through Georg Cantor), has disseminated. Then based on this analysis, math becomes a metaphor for sex, and infinity a metaphor for the human race.
In what way do abstractions exist?
Throughout Everything and More, Wallace posits this question. Wallace wonders about different types of existence. For instance, does a unicorn exist in the same way that people or integers exist? Do abstract entities exist outside of human minds? Are they discovered, or created, or somehow both (20)?
The text suggests that abstractions owe their existence to mathematical proofs. Essentially, abstractions are constructed out of proofs. In discussing abstractions in the context of set theory, Wallace insists that “we are proving, deductively and thus definitively, truths about the makeup and relations of such things” (256). The necessity of proofs indicates that without proofs, the “truths about the makeup and relations of such things” falter with uncertainty. Further, the writing suggests that the proof literally gives existence to abstractions: “So one thing to appreciate up front is that, however abstract infinite systems are, after Cantor they are most definitely not abstract in the nonreal/unreal way that unicorns are” (205). Wallace does not assert that infinite systems have always existed; to the contrary, by writing “after Cantor, they are most definitely not abstract in the nonreal/unreal way that unicorns are,” Wallace suggests that Cantor’s mathematical discoveries (i.e. proofs) somehow confer a more real existence to infinite systems. Through proofs, mathematicians construct and validate the existence of abstract entities.
The actual existence led by these entities, however, seems to be replete with distress, haunted by illness. Just as Wallace’s other works capture the pain and suffering enmeshed in human existence, Everything and More detects the vulnerability of abstract existence. Wallace laments, “And, as true numbers, transfinites turn out to be susceptible to the same kinds of arithmetical relations and operations as regular numbers” (243). The word “susceptible” renders the existence of abstractions as a pathological and tragic one.
Interestingly, as “both a map of the world and its own world” (30), the abstractions of mathematics exist both on their own in their own world and as a copy of themselves in our world. In the final line of the booklet, Wallace declares, “Mathematics continues to get out of bed” (305), and in so doing, conveys that in spite of the coming and going of mathematicians, mathematics, on its own, continues to live, to exist. On the other hand, for everything in the land of abstractions, there exists a copy of it in our land. For instance, while the math world eventually constructs a proof of the unprovability of the Continuum Hypothesis, the human world receives a version of that proof in the form of Georg Cantor’s inability to prove the C.H. Everything and More narrates a connection, a correspondence between the existence of abstractions and existence in our world.
Everything and More and Classroom Discussion
With our class, Everything and More has proven to be not conducive to class discussion. Why not? Here is one theory. The definitions of abstraction proffered by the text in many cases also describe the text itself. Wallace spends over four pages explaining the meaning of abstraction. He writes that the O.E.D. has nine major definitions of the adjective abstractus, but he quotes only the most relevant ones, which include “‘Withdrawn or separated from matter, from material embodiment, from practice, or from particular examples’” (8). Like this definition of abstract, Everything and More, by virtue of being a book, is separated from the actual matter, material embodiment, and practice of math. If “abstraction proceeds in levels, rather like exponents or dimensions” (11), and if math is already abstract, then Everything and More, in being a book about math, becomes one level more abstract. Everything and More also presents the definition of abstract as a verb: “Abstracting something can mean reducing it to its absolute skeletal essence” (10). The definition matches precisely the way in which the text treats its mathematical concepts and proofs. In covering the entire history of infinity, the text shrinks much of the history, the processes, and the details. In catering to the general reader, the text distills the complexity of the actual concepts and proofs. Essentially, the text reduces the mathematical concepts and proofs to “its absolute skeletal essence.” Finally, the definition of abstract as “‘Abstruse’” (8) is clearly fitting; the text seems to cause “lots of confusion” (from class handout—needs citation) for not just the common readers, but also the PhDs of math. How does its abstractness contribute to this text’s non-conduciveness to discussion? Wallace suggests with respect to abstractions, that “the confusion comes when we try to consider what exactly they [abstract concepts like motion and existence] mean” (11). Coincidentally, English class discussions tend to ask the same question: what exactly does this mean? In the case of Everything and More, the “this” often refers to an abstraction sitting on top of multiple levels of abstractions, and thinking about these abstractions means, in Wallace’s words, “thinking hard about things that for the most part people can’t think hard about—because it drives them crazy” (10), and that is why Everything and More might not be the text most conducive to classroom discussion.
Brief Introduction to Criticism on Everything and More
The writer claims the aim of the booklet to be not only “to discuss these [abstract and technical mathematical] achievements in such a way that they’re vivid and comprehensible to readers who do not have pro-grade technical backgrounds and expertise,” but also “to make the math beautiful—or at least to get the reader to see how someone might find it so” (2).
Given this aim, how successful is this booklet? In a class discussion of what constitutes the beauty in math, one student suggested that the beauty of math derives largely from the actual process and experience of doing math—from the moments in which we figure something out, make a breakthrough, or even advance one tiny step along a problem with a seemingly endless array of steps. Consequently, as a booklet that mainly details another’s mathematical discoveries, Everything and More inevitably denies its readers that experience—the experience of figuring out math—and therefore, a part of math’s beauty. On the other hand, the writer does embed a few problems into the text in which he does not immediately give away the answer but instead gives readers a chance to solve the problem on their own—for instance, missing dollar word problem (34).
Critiques of Everything and More
One frequent complaint about Everything and More is that Wallace's writing style diminishes the mathematical aspects of the booklet, and vice-versa. Because of the difficulty of his subject matter, Wallace puts his fiction-writing skills to work in order to make the material more approachable to those readers with weaker backgrounds in math. Wallace attempts to create the best of both worlds in Everything and More: both a book about serious, complex math, and an easily readable work. Unfortunately, this aim results in phrases like "Let's pause to consider the vertiginous levels of abstraction involved here. If the human CPU cannot apprehend or even really conceive of infinity, it is now apparently being asked to countenance an infinity of infinities, an infinite number of individual members of which are themselves not finitely expressible, all in an interval [0-1] so finite- and innocent-looking we use it in little kids' classrooms. All of which is just resoundingly weird" (80), which is far too informal for a math textbook and too syntactically awkward for a "readable" work.
Another criticism about Everything and More is that the abbreviations are unnecessary and distracting. Indeed, the list of abbreviations in the forward contains 37 terms (3-4), not including the ubiquitous IYI. Although Wallace states that the "booklet's abbreviations are contextualized in such a way that it ought to be totally clear what they stand for" (3), some readers experience difficulty with this claim, and therefore must frequently flip to the front of the book to reread what each abbreviation stands for, or skip that section of the booklet, neither of which is a productive option.
The IYIs have been criticized as being somewhat insidious. Because the reader has bought, and is reading 'Everything and More', the reader is presumably already interested in the book. Therefore it may seem that the If-You're-Interested notes aren't so much considerate asides, but devices to grab the reader's attention.
Errors in Everything and More
A substantial portion of the backlash on Everything and More consists in complaints on the abundance of mathematical errors in the text. The mathematical community has pointed out dozens of errors, ranging from computational errors to mathematical statements that “make no sense” and produce “lots of confusion” (from in class handout—Mathematical errata for Everything and More). There are different ways to interpret these errors. One critic, for example, views the cornucopia of errors as detrimental to the author’s credibility and insists that “any time errors are found in a text that bring the author’s credibility into question, the critic/reader can take two paths, justifying the error’s using the intentional fallacy (or maybe the affective fallacy, or maybe both) or admit the author’s fallibility and humanity.”
Everything and More itself provides another way of looking at mathematical errors. As Wallace enumerates Cantor’s repeated attempts to prove the Continuum Hypothesis, he exposes the unintentional nature and heartfelt grief in Cantor’s errors: “Cantor’s attempts to prove the C.H. went on through the 1880s and ‘90s, and there are some heartbreaking letters to Dedekind in which he’d excitedly announce a proof and then a couple days later discover an error and have to retract it” (293). This passage discerns the inevitability and tragedy in errors: sometimes we make mistakes, and it’s sad when we do—because these mistakes are born not necessarily out of sloppiness, but out of an attempt to reveal some truth. As readers, we can extend and apply Wallace’s outlook on Cantor’s errors to our own outlook on Wallace’s errors. Unlike the above critic, we do not have to let an author’s errors devastate his or her credibility.
David Foster Wallace is a recognized mathematical critic as well. For Science Magazine (22 December 2000: Vol. 290. no. 5500, pp. 2263 - 2267) he reviewed two books, one titled The Wild Numbers, by Philibert Schogt, and one titled Uncle Petros & Goldbach's Conjecture, by Apostolos Doxiadis. This is not directly related to Everything and More, but serves to showcase Wallace's mathematical talents and inclinations in a more comprehensive manner. He also muses on how a new wave of mathematical work, perhaps a whole new genre, called "Math Melodrama" by Wallace, is invading the commercial sphere. The whole thing can be read at http://www.sciencemag.org/cgi/content/full/290/5500/2263


I could never understand the supposed gap in math scores between girls and boys either. A recent issue of Science magazine (May 30) should put that notion to rest. A study was done that showed that in countries where gender-equal societies are the norm, girls not only close the gap in math scores, but they also do even better than the boys in reading which was supposed to be their strong area anyway.
Th[e] evidence suggests that intra-gender performance differences in reading versus mathematics and in arithmetic versus geometry are not eliminated in a more gender-equal culture. By contrast, girls' underperformance in math relative to boys is eliminated in more gender-equal cultures. In more gender-equal societies, girls perform as well as boys in mathematics and much better than them in reading.
So the gender gap in geometry (where boys have the advantage) and arithmetic (where girls do better) does not change, but the overall performance of girls is improved if they live in a society that values their brains. Gee, geometry was my favorite subject.Recently I have become immersed in two new Teaching Company lecture series. One is An Introduction to Number Theory, taught by Edward B. Burger in which he actually uses equations but keeps assuring us that we need not get into a panic over them. The other is a fantastic series on Dark Matter, Dark Energy: The Dark Side of the Universe taught by Sean Carroll of Cal Tech. This one is a winner. Professor Carroll speaks clearly, but rapidly, and non-stop. He is not reading from a prompter and only occasionally glances at his notes. He really knows his stuff. And he is not afraid to give us the nitty-gritty. Although he does not go into the details of Einstein's equation of general relativity or the Friedmann Equation which is the application of general relativity to a rapidly expanding universe, he does give the equations and explains what each term means.
I am learning all kinds of new stuff and understanding it better than when I read books by people like Brian Greene and even Stephen Hawking. But then I believe in the spiral theory of learning. You go round and round, repeating the same material over and over that you learned before, but each time at a much higher level. I'd like to tackle a book with all the gory equations included next to see if I can follow it.
P.S. These courses were on sale when I bought them. If they are no longer on sale, just wait. At some point during the year they will be back on sale.


This advanced course will cover modern methods in nonlinear systems theory and applications. Basic feature of nonlinear systems theory is that state space approach is prevailing and frequency response methods are not generally applicable. The course will cover the topic: state model of nonlinear dynamical systems and its analysis, Lyapunov's stability, asymtotic stability and Lyapunov's methods, control synthesis via approximate linearization, high gain observers, gain scheduling. The main stress is on the so-called structural methods for the nonlinear control design, i.e. study of the system structural properties allowing easier applications of known control methods. That would consist of basics of differential geometry, Lie derivative, various types of exact feedback linearization, input-output linearization, zero dynamics, minimum phase systems, MIMO systems, decoupling. Last but not least, the course will provide verz basci information about chaotic systems including some motivating examples.
1. State space model of nonlinear dynamical system, typical nonlinear phenomena, examples.State space model of nonlinear dynamical system, typical nonlinear phenomena, examples.
2. Further practical examples of nonlinear dynamical systems. Chaotic systems.
3. Mathematical basics. Existence and uniqueness of solutions, dependence on initial conditions and parameters..
4. Definitions and methods of stability analysis. Ljapunov's function method and approx. linearization method.
5. Analysis of stability of perturbed asymptotically and exponentially stable systems.
6. Control synthesis via approximate linearization and obust linear methods. High gain observers.
7. Control synthesis via approximate linearization and gain scheduling.
8. Structural methods in nonlinear control synthesis - basic notions, exact transformations of nonlinear systems.
9. Structural methods and various types of exact linearization. Zero dynamics and minimum phase.
10. Structural methods and some basics of differential geometry and advanced analysis.
11. SISO systems. Relative degree. Input-output linearization. Zero dynamics, minimum phase systems.
12. MIMO systems I. Vector relative degree. Input-output linearization.
13. MIMO systems. Zero dynamics, minimum phase systems.
14. MIMO systems. Decoupling.
1. Example of nonlinear models and their simulations.
2. Laboratory examples of nonlinear systems.
3. Setting of individual long term homeworks for analysis and design of nonlinear systems control.
4. Sustems modelling
5. Modeling and simulation model design
6. Nonlinear systems simulations
7. Analysis of stability
8.Desing tasks and algorithms of control
9. Realization of control algorithms
10. Verification of control algorithms on simulation models
11. Exact feedback linearization design
12. Exact feedback linearization design - MIMO systems
13. Realization of controllers based on Exact feedback linearization design
14. Defense of results of long term homework


- ordinates axis of the plot
- ось ординат графика
English-Russian dictionary of telecommunications. 2015.
Смотреть что такое "ordinates axis of the plot" в других словарях:
Phase space — In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For… … Wikipedia
Earth in fiction — An overwhelming majority of fiction is set on or features the Earth. However, authors of speculative fiction novels and writers and directors of science fiction film deal with Earth quite differently than authors of conventional fiction. Unbound… … Wikipedia
Ordinate — Or di*nate, n. (Geom.) The distance of any point in a curve or a straight line, measured on a line called the axis of ordinates or on a line parallel to it, from another line called the axis of abscissas, on which the corresponding abscissa of… … The Collaborative International Dictionary of English
Complex number — A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the square root of –1. A complex… … Wikipedia


The elementary math program is aligned to the Minnesota Academic Standards and those provided by the National Council of Teachers of Mathematics. Computational skills, number concepts, and problem solving skills are essential components with emphasis placed on the mastery of basic facts, in relation to four mathematical strands; Numbers and Operations, Algebraic Concepts, Geometry and Measurement, and Data Analysis. Students engage in a variety of hands-on, individual, and collaborative activities as they formulate, analyze, and solve problems proficiently in order to develop a solid foundation for further mathematics education and use in everyday life. It is understood that students use diverse strategies to solve problems and teachers recognize and take advantage of these alternative approaches to help students develop a better understanding of mathematics. Support and enrichment opportunities complement the learning styles and development of all learners.


- 1 min read
Cuemath ropes in Vidya Balan as brand ambassador
The ad campaign features the actress engaging in the Cuemath ways of learning maths...
Created by Simple Tribe, the TVC featuring Balan along with Nitesh Pandey and child actor Purab Mody, tackles Math in a fun manner by laying out a situation where a child is able to order more pizza for less money due to the application math he's been learning with Cuemath, hence making better decisions and maximizing fun. The campaign slogan aptly backs it up – “Math Karo, Aage Badho.”
Manan Khurma, chief executive officer, Cuemath, said, “We help children learn Math organically by visualising concepts from different points of view, building their own mental models and algorithms, developing logical reasoning by finding multiple answers, and applying math in everyday thinking and problem solving. Our aim is to train our future generation at not just solving Math problems easily, but also tackle the challenges that life throws up with confidence and logic.”
"I attended a Cuemath demo class while researching for my last film. Imagine my surprise when 45 minutes into the demo, I learnt to visualise Math concepts with simulations. I never thought I could see Math! It’s this Cuemath way of making Math fun and taking the anxiety away from it that appealed to me immensely and is the future of learning in the 21st Century," Balan said.


In fiction, the use of Math is heavily romanticized. Like in the TV show “Numb3rs”, where Math is used to aid the FBI against crime. Then again, I have read that such practices are actually being done in real life! Also, in Isaac Asimov’s “Foundation” books, there is a science called Psychohistory (which I had already discussed in a list of thought provokers), in which actual sociological and historical factors are converted to statistical-probability equations that would make it possible for the prediction of the future of humanity. Sounds absurd? But consider that others are actually venturing on mathematical study that are somewhat parallel to some Psychohistorical concepts!
Clearly, Math is badass. And, believe me, not only in fiction, but in real life as well.
When I was a kid, I hated Math. And I grew up hating it all throughout my academic life. (Though, ironically, I was “Best in Math” in Nursery. LOL) However, I have loved mysteries and puzzles and logic (long before I learned that that’s the word for that process or exercise), but have not realized that Math was the most perfect avenue where I can work them out. When I was about a young adult, I started to see how special Math real is. And I felt regret, since I haven’t started loving it when I was a kid, when it’s the best time to learn its skills.
I know no real Math. Yeah, I know basic algebra and such. I mean the REAL Math. The mind-blowing Math. The Math that appears to be the key to the mysteries of the universe. Math, as much as Science, is not a panacea, nor it can answer everything. But, still, it is a very invaluable tool in expressing or interpreting seemingly incomprehensible facts of reality into some form of comprehension. The Mathematical World is magical. I find it amazing how numbers and equations can interpret reality, even across its visible boundaries; and how numbers and equations can prove a fact before an actual hands-on scientific experimentation, when experimenting about it is still impossible or it hasn’t been figured out how to carry out the experiment. There are concepts of Quantum Physics (and also in other sciences, but mostly this) that only Math can touch. Sci-fi concepts like time-travel and alternate universes are not laughable ideas at all because of Math. Besides, Math can and have proven things that were thought unthinkable before, like for example, establishing that, at the essence, matter and energy are the same thing.
But there are also cases when something that is impossible in the world of reality is only possible in the world of math. Examples are mindblowing paradoxes (e.g. Zeno’s paradox or the “Achilles and the Tortoise” paradox). They are sound in paper and equations, but in reality, they are impossible. Below is an example of a geometrical paradox.
Also consider how Math, without any logical or mathematical fallacy, proved that 1 and 0.9999… are equal.
x = 0.999999...
multiply both sides by 10
10x = 9.99999...
subtract both sides by x (which is 0.9999999...)
10x - x = 9.999999... - x
10x - x = 9.99999... - 0.99999...
9x = 9
divide both sides by 9
9x/9 = 9/9
which results to,
1 = 0.999999...
And as we continue to learn more about the world of Math, we see how really mindblowing it is.
Of course, there will be times when it’s impossible for numbers and equation to tell the whole story in a particular case or system (e.g. basketball statistics), since there are some other aspects or factors of the system that can’t be put into equation. However, there might come a day when someone will build a perfect equation that would be able to successfully translate ALL the factors and aspects of a case or study, allowing it to become the perfect and final word on the subject. As I’ve said, Math has already proven some things to be possible which had been thought impossible before. Who knows, maybe in a century or two, people would be doing real-life Psychohistorical analyses.


Pittsburg IMPACT Overview
IMPACT (Improving Mathematics through Pittsburg's Active Collaboration of Teachers) is a teacher professional development grant for mathematics, grades 3-8. It will be funded through June 2014 by the CA Department of Education, CaMSP grant, which was awarded to P.U.S.D. in January 2011. The goal of the grant is to increase mathematic content knowledge by giving teachers a greater understanding of Algebra by highlighting its connection to Number Sense and articulating Algebra concepts from K-Algebra 1 standards. Also, teachers will learn more effective delivery of curriculum through stronger pedagogical strategies, lesson planning, observation, and coaching of the three phase model of instruction with the connection to Algebra.
Our 48 participating teachers attend a 60 hour intensive summer course during the summer, which is taught by our partnering professors from CSU East Bay and Los Medanos College. The strategies they learn during the institute will follow them into the classroom. Teachers will also be coached during the year to assist them with the implementation of the strategies. (See our Partnership Profile for more information.)
We are very excited about IMPACT and how it will not only assist the teachers in building their understanding of math and provide them with more pedagogical strategies, but how it will directly impact the students of P.U.S.D. to become better math students.
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Impact Contact Us
Hillview Junior High School
(925) 473 - 2380, x 5610


Basics of Data Science and Machine learning in R
Course code: DAR001
Duration: 44 Hours
About this course
This course is designed and delivered to a large extent with practice sessions. Data science goes beyond traditional analytics and statistical models and facilitates building predictive models. Participants will learn the concepts and motivation to apply data science and interdisciplinary programs. This also provides you the foundation skills to produce various types of predictive models for machine learning, the algorithms involved to build such models and how predictive modelling and machine learning are interrelated. The key aspect of this course is that, participants are exposed to sufficient amount of real time use cases, for each of the machine learning algorithms learnt, in order to analyse, conceptualize and build various types of predictive models, on their own.
Students and Working Professionals who want to learn Data Science, Machine Learning & R. This is a foundation course in data science imparting necessary skills to begin working on machine learning projects.
Understanding of basic mathematics and analytical thinking are required.
This course provides the foundation knowledge needed in analytics and data science:
- - Introduction to analytics and data science and their applications today
- - Using R software and its facilities
- - Producing reports using R
- - Basic mathematics
- - Various Machine learning algorithms
- - Predictive models
- - Project works for applying each of the machine learning algorithms learnt.
1. Welcome to the Course - 4 hours
Introduction to Data Sciences, Machine Learning, Applications, R, Data Collection, Data Types & R Studio.
2. Basic Statistics and other fundamentals - 4 hours
Basic Statistics, Measure of Spread, Measure of Dispersion, Outlier Treatment, Data visualization.
Probability Theory, Probability Distribution – Binomial distribution, Normal distribution.
3. Starting using R - 4 hours
R Installation & Introduction, Process of Cleaning datasets, Treating Missing value, Transforming variables.
4. Linear Regression- 4 hours
Regression – Linear regression, coefficient of determination (R2), Multi linear regression.
5. Making a predictive model using Linear Regression - 4 hours
House Prices Prediction using Linear regression in R (CASE STUDY)
6. Logistic regression - 4 hours
Logistic regression. Logistic regression in R and theory.
7. Logistic regression case study - 4 hours
Titanic Survival prediction in R using Logistic Regression
8. Cluster Analysis
K-means in R and theory.
9. Grouping of Clusters - 4 hours
Grouping of Clusters for Digital marketing (Using K means)
10. Decision Sciences – 4 Hours
Decision Sciences using Decision tree with practical Problem Statement
11. Final Evaluation of Projects based on real world use cases – 4 Hours
Projects are based on real data using applied machine learning


Computational representations of herbrand models using grammars
Springer Berlin Heidelberg
International Workshop on Computer Science Logic, 334-348
Finding computationally valuable representations of models of predicate logic formulas is an important subtask in many fields related to automated theorem proving, e.g. automated model building or semantic resolution. In this article we investigate the use of context-free languages for representing single Herbrand models, which appear to be a natural extension of ``linear atomic representations'' already known from the literature. We focus on their expressive power (which we find out to be exactly the finite models) and on algorithmic issues like clause evaluation and equivalence test (which we solve by using a resolution theorem prover), thus proving our approach to be an interesting base for investigating connections between formal language theory and automated theorem proving and model building.


Table of Contents :
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I am terrible at math. I can do your standard arithmetic, and I can calculate a percentage like nobody’s business (that sweater is 30% off? Sweet!) but once you start talking algebra and calculus and crap, I’m done.
I wasn’t always this way- I started out doing just fine in math, getting As and such through the halcyon days of first and second grade. Then came third grade and Sr. Mary Seton. She was not necessarily a bad teacher, but she was unforgiving and a taskmaster. You would have thought we were a boot camp for the Marine Corps instead of a bunch of 8 year old Catholic kids fresh off our First Communion and therefore still terrified of hell. I was out sick for a couple of days at one point, and I missed the initial explanation of what multiplication was as well as the first five multiplication tables. When I returned she refused to explain what I had missed. I tried my hardest to deconstruct what was going on, but I was eight. And I certainly didn’t want to admit that I didn’t know what was going on- I was a ‘smart kid’ and I liked being a ‘smart kid’ and didn’t want to mess up that impression.
Anyways, I moved on and eventually figured out multiplication and its evil twin division, but I was somehow scarred. In college I took a financial accounting course (don’t ask) and I just could not get my mind to wrap around it. Ask me about the corruption of women’s basic rights as the precursor on The Handmaid’s Tale and I could write you a dissertation- ask me about accrual vs. cash accounting and its impact on the balance sheet and I would look at you like you had nine heads. It was so bad that the professor pulled me aside and asked me what happened around third grade. I was stunned and I sputtered out my tale of multiplication woe and he gave a knowing smile- “I’ve taught plenty of students like you who struggle with this material and they’ve all had some kind of run-in with math right around that age. Don’t worry, we’ll figure it out.” So I’m not alone, and he was tremendously helpful to me in eking out a very painful “C” in that class.
I have always been more attracted to language and words. That’s part of why The Phantom Tollbooth is one of my favorite books- the plot revolves about the tension between math and language, and it is ultimately resolved by the beauty of language (King Azaz the Unabridged and the Mathemagician being forced to finally agree on one thing- to disagree). I love puns and wordplay and how a book is invariably better than the movie. My brain operates best in the colorful, grey areas of words and meaning, not the boring black and white of math, where 2 plus 2 is always 4. Where’s the fun in that?
That’s the thing about math- I can do it, I’ve just never understood it. It is the difference for me between reading words on a page and truly understanding the story. Why on earth did I need to know what a cosign was? Who uses this in real life? (answer: no one. Ever.). By the time I reached high school I would look my algebra teacher in the eye and say “I am going to law school. I don’t need to know this.” She was kindhearted enough to give me extra credit on a final exam basically for spelling my name right. Also, law school was hell, but that’s a topic for another time.
With the wisdom of hindsight, I can now see what was actually being taught. It was not so much “doing the math”, it was problem-solving and critical thinking skills. What is PEMDAS other than a structure under which to analyze and tackle a problem? It is the math equivalent of diagramming a sentence (and I can diagram the shit out of a sentence. Sentences cross the street when they see me coming). Algebra is simply an attempt to solve a crime with only some of the evidence available. Geometry explains how (most) buildings don’t fall down. I don’t recall math ever being presented in that kind of context, it was just something we were supposed to learn based on its intrinsic value. Why do I have to know this? Because I said so.
I know how we teach math has evolved significantly over the years, but with my oldest daughter starting kindergarten this year I’m nervous about helping her with homework as she progresses and it becomes more difficult. She is very interested in science and engineering- loves astronomy, and learning how things are made- and I think I’m overly sensitive to the “girls aren’t good at math” stereotype because I am a walking example of it. We’re cool with counting and basic math, but when we get to manipulating fractions that’s going to be a rough one for mom. I think contextualizing the why? of math is truly the key to understanding it, and that’s the element that was missing for me in the beginning stages of my math education.
So we’ll see what happens over the next couple of years. Thank goodness for online tutorials for parents like me, and my husband’s mathematical capability.
After all, what you learn today, for no reason at all, will help you discover the wonderful secrets of tomorrow.


|Other titles||Money [Canadian].|
|Contributions||Ontario Institute for Studies in Education.|
|The Physical Object|
|Pagination||20 p. :|
|Number of Pages||20|
Math Activities for Kids Money Math Data and Graphing Use this easy activity to help your child make some amazing paintings. They can use their math skills to design patterns for cards, paper, presents and more! Kindergarten. Math. Money Book Activity, Money Worksheet, Money Posters & Songs This is a great page set that would be a fantastic addition to your money unit! It has an interactive book, posters, and worksheets perfect for your first or second graders!Check out the preview for a sneak peek!This helpful set includes: Color half sheet booklet WITHOUT half. The 10 Best Kids Activity Books. We took it all to heart, and rounded up what we think are the best kids activity books. 1. This activity book is filled with facts, coloring areas, and fill-in-the-blank slots that keep your child learning about how the world works. It’s a Author: Fatherly. This product contains 4 place money flip books. It is a a great hands on activity to help students develop conceptual understanding of money. Students model two ways to show the value of money. Students really enjoy this activity. Money Flip Book is licensed under a Creative Commons Attribution-Non.
This workbook is all about helping first graders ace their money math. Counting coins, word problems, and coin matching will help familiarize kids with the basics of money. (23 pages, file size MB) Purchase Individually. Buy Now - $ Unlimited Workbooks. From $/month. Get unlimited workbooks, worksheets, games, and more with an. With Math Games, the learning process is an enjoyable game and math review is something to look forward to. Our free resources for teachers and parents (printable worksheets, apps, progress reports, assignment creation software) make lesson planning and grading a breeze. Math books are excellent supplemental aids for classrooms or primary teaching tools for at-home learning. Many math workbooks provide guides for teachers and tutors that make it easy to verify answers, and Staples offers a variety of calculators to assist in Activity book, problem solving; out of 5 stars with 1 reviews. 1. $ Money Talk and Understanding Money’s Value: Before they understood the value of pennies, nickels, dimes and dollars, I would talk to my boys about what amount of money they had to spend ($, $, $, $). We’d lay out the amount of money on the kitchen table for them to see and talk about -- and then they’d get to put it in.
Fractals, Googols, and Other Mathematical Tales. This book takes math concepts – such as real numbers, exponents, dimensions and the golden rectangle – and explains them through a variety of stories geared especially for kids. Featured math concept: Each tale features a different concept. This book visually explains fractions to kids in. Thanks for making something so organized and easy to use. - Early Years Activity Plans User, Melissa C. This takes down the need to scour the internet for ideas. It's like looking for a recipe on the internet, so many options that often times it is less overwhelming to look in . The Grade 5: Canadian Money No-Prep activity packet is designed to provide students in Grade 5 with practice activities to help them read, count and represent money amounts to $ This packet includes activities with and without the penny. Multiplication is adding over and over again. 5 times 2 is 5 + 5, 5 times 3 is 5 + 5 + 5, 5 times 4 is 5 + 5 + 5 + 5. That’s like counting by fives 4 times. If I put five books on each shelf and there were four shelves, I would have 5 + 5 + 5 + 5 books. 5 x 4 = 20; Anything zero times is zero. A million times zero is zero.


Year 9 Parents' Workshop
Materials from Year 9 Parents' Workshop
Dear Year 9 Parent / Carer,
Thank you for attending the year 9 parents' workshop at the end of September; we hope you found it useful.
Further to the workshop, please find details of the curriculum map for each of the core areas. The map details what your child will be studying in English, mathematics and science during year 9. Please note that the curriculum is subject to change, dependent on the needs of the students. Should you have any questions about these details, please contact your child's English, mathematics or science teacher; or the Head of the relevant department.
Ms Marshall - Assistant Headteacher, Pastoral


Desmos is recognized for its straightforward yet powerful calculators, particularly excelling with its graphing capabilities. As a free-to-use platform, it offers a variety of calculators including graph, matrix, and even 3D calculators. While Desmos shines with its “how to use the tool” guides and a unique classroom option for interactive lessons, it lacks in providing comprehensive educational resources like practice worksheets. Its absence of a photo solving feature might limit its appeal to some users, but its commitment to accessibility and ease of use makes it a valuable tool for basic mathematical explorations.
One-stop solution for all your homework needs. Get the job done.
✅ AI Essay Writer ✅ AI Detector ✅ Plagiarism checker
Quality of Help – 10/50 ⭐
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Desmos appears to focus on simplicity, offering calculators that are best suited for straightforward, mostly one-step tasks. It excels in graph-related queries, delivering accurate results and methodologies. However, the platform falls short when faced with equations and calculus problems, with notifications of unsupported equations or failing to provide answers for more complex tasks. The absence of explanations and related questions significantly limits its usefulness as a comprehensive educational tool.
Value for Money – 15/30 ⭐⭐⭐
Despite being a free service, Desmos’s value is somewhat diminished by its limitations in educational content and problem-solving capabilities outside of graphing and geometry. The platform’s variety of basic calculators and the classroom option do add value, but the lack of advanced math support and additional learning resources might not satisfy users looking for a more holistic math learning experience.
Overall Experience – 18/20 ⭐⭐⭐⭐
Desmos provides a positive user experience with its user-friendly and intuitive website design, making formula entry straightforward. The ability to save graphs and revisit geometry tasks adds convenience, although the platform’s segmented app offering requires users to download separate programs for different calculators. Despite these minor inconveniences, the lack of advertisements and the simplicity of the registration process contribute to an overall pleasant experience.
With an A*Help Score of 43/100, Desmos offers basic but effective tools for certain mathematical tasks, particularly in graphing and geometry. Its user-friendly interface and innovative classroom feature are notable strengths. However, the platform’s limited scope in math levels and absence of comprehensive educational resources hold it back from being a go-to solution for more advanced or diverse math learning needs. Desmos is best suited for users seeking simple, direct solutions in graphing and geometry, with minimal expectations for additional learning support.
This review is built on our hands-on experience with Desmos, complemented by a detailed examination of user feedback across various platforms. Our goal is to provide a balanced perspective on Desmos’s functionality, ease of use, and overall efficacy in serving educational needs.
Is Desmos a Scam?
No, Desmos is not a scam. It is a legitimate, free educational tool that focuses on providing a variety of calculators for mathematical problem-solving, including a notable graph calculator and interactive classroom features.
Is Desmos Legit?
Yes, Desmos is a legitimate platform. It is widely recognized for its graphing calculator and educational resources, offering a legitimate and valuable tool for students and educators alike.
Is Desmos Safe?
Desmos is safe to use. As a free platform that does not require sensitive personal information for basic use, it maintains a high standard of user privacy and security.
Is Desmos Trustworthy?
Desmos is trustworthy for the services it claims to provide. While it may have limitations in the breadth of its mathematical solutions and educational resources, it delivers reliable and accurate graphing capabilities.
Is Desmos a Good Service?
Given the A*Help Score of 43/100, Desmos is a decent service for specific needs—primarily graphing and geometry. While it falls short in offering a comprehensive educational experience, its strengths in creating interactive lessons and providing various calculators make it a useful tool for basic math tasks and classroom engagement.
|> Levels of math
|Pre-Algebra, Algebra 1-2
|Pre-Calculus, Calculus 1-2
|> Other disciplines
|> Additional resources availability
|Guides/ step-by-step explanations
|> Different types of content
|Can build graphs
|Can create table charts
|Can build geometrical figures
|> Technical features
|Solving via photo
|How well it reads handwriting
|> Subscription & limits
|Limits of free use
|Needs registration for usage?
|Work with formulas
|Advertisements on the page
|Why academichelp.net is a credible source of information:
Stay curious with us. Academichelp.net has been a reliable educational resource since 2011, providing students with the latest news, assignment samples, and other valuable materials. Even with the extensive information we process, our quality remains consistent. Each team member has experience in education, allowing us to evaluate new sector offerings critically. Our reviews are up-to-date and relevant, with impartiality ensured by the A*Help score methodology from mystery shopping. We aren’t affiliated with any listed service providers. Our focus remains on providing our audience with reliable and unbiased data.
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Cards with Kaktovik numbers are displayed on a table on October 5, 2022. The cards are used in a game called Tallimakipiaġutaiḷaq, which means “99”. In the game, you add the cards as they are
Nearly 30 years ago, a group of Kaktovik students invented a number system that mirrored the way they counted in Iñupiaq and made math more intuitive for them. Soon anyone in the world will be able to type Kaktovik’s numbers into a computer.
“It’s not just a digital system. It’s not just math. It was created and developed by a teacher from our district and our Iñupiaq students,” said Tennessee Qaġġuna Judkins, Director of Iñupiaq Education at the North Slope Borough School District. “When you use it in a classroom, it’s more relevant to those students, to that population. When it comes to Indigenous ways and understanding, it clicks. It makes a lot more sense.
Today, interest in culturally appropriate education – and the study of Kaktovik numbers in Inupiaq schools – is growing, as is the need for an easy way to use numbers numerically. From September, an international encoding standard, Unicode, included Kaktovik ciphers in its latest version, which means that ciphers will now be universally accepted by computers.
“It means that anyone, anywhere in the world, on a computer can have access to producing things using numbers,” said former school math teacher Harold Kaveolook , William Clark Bartley, whose students invented the system in the 1990s.
Creation of figures
When Bartley was a math teacher at Kaktovik in 1994, his middle school students figured out the numbers to represent the Iñupiaq oral counting system.
“Before doing these numbers, we were using Arabic numerals and visually it didn’t match our language,” said Alicia Solomon, who was in eighth grade at the time. “And we started asking our teacher, ‘What about our own numbers? Our own system?
Most countries use the base-10 Hindu-Arabic numbering system where numbers range from 0 to 9. But in Iñupiaq – as well as other Inuit and Yupik languages - numbers range from 0 to 19, making makes a base 20 system.
“The Iñupiaq word for the number 20 is iñuiññaq, which stands for a whole person,” Judkins said. “You have all 20 appendages – your 10 fingers and 10 toes. A lot of the classroom activities that we use now with this numbering system relate to these body parts and appendages.
Kaktovik students came up with numbers from zero to 19, consisting of straight strokes joined with sharp angles that you can write without lifting the pen.
“We didn’t want them to look like other numbers,” Solomon said. “Our whole math class did it together.”
The Inupiaq counting system didn’t have zeros, so the school district suggested a few names for the digit. A disabled student proposed the visual representation of zero.
“She just raised her hands above her head crossing them,” Bartley said.
Numbers are constructed according to the structure of numbers in Iñupiaq: for example, 16 is akimiaq atausiq in Iñupiaq, which translates to 15 and 1. The Kaktovik number 16 is also composed of 15 and 1.
“The kids really came up with a brilliant system,” Bartley said. “The numbers, you just have to look at them, and you can see what the number means.”
Because Kaktovik numbers visually reflect number composition, using them in math problems was easier than Arabic numbers, Solomon said. Just by looking at Kaktovik’s numbers, students could see how to add, subtract and even divide. For example, for long division, students used colored pencils to match the lines of the divisor in the dividend.
“The numbers almost came through,” Bartley said.
After the students started working with Kaktovik’s numbers, Bartley said their interest in math grew. They would rush to skim through the regular math book to save time working with Kaktovik’s numbers.
“It’s their culture that has made all the kids so much more involved,” he said.
The effort to make Kaktovik figures available on computers began in 2021, when several linguists and language enthusiasts wrote a proposal describing the suitability of the numbering system.
Now, Unicode has a place reserved specifically for Kaktovik ciphers starting with version 15.0 released on September 13. Because the update is so recent, computers, smartphones and other devices today do not come with a font capable of displaying Kaktovik digits, according to Deborah Anderson. , a research linguist at the University of California, Berkeley, who also helps preview new Unicode proposals before they are submitted.
“To access numbers, users need a font with glyphs and an input mechanism, such as a keyboard,” Anderson said.
In the past, Google has developed free fonts for new Unicode characters that are in modern usage. If that’s done for Kaktovik’s digits, a font could become available within the next few months, said Craig Cornelius, software engineer at Google who contributes to Unicode. Eventually, a font will also be available on mobile devices such as Android, but the process will take at least a year, Cornelius said.
“The next step in making this really usable is to create a keyboard that can be used directly on a laptop or desktop,” Cornelius said. “If people can type the characters, someone will be able to see them if the font is installed on their device.”
So far, Cornelius has built a numeric keypad for those who want to start using numbers today. The developers are working on training Iñupiaq Education staff on how to access and use the numbers before the fonts are released to the public.
“The people who worked hard to get the Kaktovik characters into the Unicode Standard started the process,” Cornelius said, “but like any product, it takes time to roll out to all stores.”
North Slope teachers used Kaktovik numerals in math lessons for a period after their invention. Their use has since been reduced, with the influx of out-of-state teachers and the growth of standardized testing in the late 1990s, but more and more teachers in Iñupiaq schools are now considering to restore the system.
Next fall, the North Slope Borough School District hopes to implement several pilot immersion classes that will include teaching students math concepts and math activities, using Kaktovik numbers. Additionally, this immersion program will be built on a culturally appropriate curriculum that will provide students with lessons in Iñupiaq history and local governance.
“There’s this huge wave of Indigenous education and culturally appropriate teaching,” Judkins said.
Kaktovik numbers are part of the Iñupiaq language program in the district, and students learn to count and describe dates and ages using them, said Iñupiaq Department of Education specialist Chrisann Justice.
But creating teaching materials with Kaktovik figures currently means drawing them by hand or, in Justice’s case, copying and pasting screenshots of each figure, which often leads to a pixelated image.
“It would be so convenient to just be able to type them in!” Justice said.
Digitizing Kaktovik numerals may also help preserve the use of numbers in Iñupiaq.
“Our words are long, and it’s just easier to see the numbers than to say the numbers,” Solomon said. “I think it would be great for kids to try to learn numbers, kind of get an idea of themselves and learn our number system. Just to keep in touch with our culture.
• • •


STEM stands for Science, Technology, Engineering and Mathematics.
In 2008 MATRIX first highlighted the importance of innovation, research and development in Northern Ireland’s STEM industries. A year later in 2009, DETI and DEL published the STEM Review and a draft STEM Strategy. Both of these emphasised a need for more links between industry and education.
When considered together, the reports indicate that we need to ensure that Northern Ireland’s young people have the skills, knowledge and understanding in STEM-related subjects in order to take advantage of job opportunities in these fields and contribute to the country’s economy.
The Northern Ireland Executive’s “Programme for Government” and the Skills Strategy for Northern Ireland, “Success through Skills – Transforming Futures”, also recognise that the future success of the Northern Ireland economy will require increased numbers of skilled workers with Science, Technology, Engineering and Mathematics (STEM) qualifications.
MATRIX is helping to champion the role of science and technology as one of the key drivers of economic growth in Northern Ireland.