Not going to class isn’t typically something good to boast about. But perhaps the late Vladimir Voevodsky is the exception to the rule.

Voevodsky is credited with founding new fields of mathematics, such as motivic homotopy theory, and a computer tool to help check mathematical proofs, as the New York Times explored in an obituary this week. The latter was a feat that other mathematicians didn’t dare approach, but Voevodsky’s effort has overwhelmingly benefited the industry — and everyone, really — by allowing mathematicians to fact-check their work.

He died at age 51 on Sept. 30, at his home in Princeton, New Jersey from unknown causes. He leaves behind his former wife Nadia Shalaby and their two daughters.

“His contributions are so fundamental that it’s impossible to imagine how things were thought of before him,” Chris Kapulkin, a former colleague at the University of Western Ontario, told the Times.

Among Voevodsky’s achievements was changing the meaning of the equal sign. In 2002, he won the Fields Medal for discovering the existence of a “mathematical wormhole” that allowed theoretical tools in one field of mathematics to be used in another field.

He wasn’t a top student of the traditional, rule-abiding sense. According to the Times, Voevodsky was kicked out of high school three times. He was also kicked out of Moscow University after failing academically. He later attended Harvard. Despite neglecting to attend lectures, he graduated in 1992.

He worked through it all, and all present and future mathematicians have him to thank.

Shocked by the untimely death of Vladimir Voevodsky, a great mathematician and a wonderful human. R.I.P. dear friendhttps://t.co/wtvbjxtNZ9

If mathematics isnt your strong suit, this equation that went viral in Japan may just trip you up. According to the YouTube channel MindYourDecisions, a study found that only 60 percent of individuals in their 20s could get the right answer. This is significantly lower than the 90 percent success rate in the 1980s.

To learn which common mistake people are making, check out the video below.

The latest news from the none-of-your-thoughts-are-original department comes from mathematics blogger Alex Bellos, who set out to determine the worlds favorite number.

Bellos apparently doesnt have a favorite number himself, but as Nautiluswrites, he began asking people about their favorite numbers a few years ago, and after setting up theFavourite Numberwebsite, more than 44,000 people voted for the numeral they liked best and explained why.

Heres what Bellos discovered.

The third-most popular number is 8, because, as some of Belloss respondents wrote, In Japan, eight is a lucky number, because the Japanese character for eight means an opening to the future and because of its symmetrical and round shape and because it has always given me a sense of friendliness and warmth (unlike, for example, 9 which looks bossy or 6 which appears to me a bit submissive).

No. 2 on the list is the No. 3, because Its curly, but not pretentious curly like eight and because in Chinese, “3 means alive.

But the worlds most favorite number is No. 7. As for why, Bello will explain that himself. Its basically because of our desire to be outliers when it comes to arithmetical patterns.

As for why numbers ending in 0 or 5 are unpopular, Bellos said its because we use those numbers as approximations more than, say, 7 or 9.

When we say 100, we dont usually mean exactly 100, we mean around 100, Bellos told Nautilus. So 100 seems incredibly vague. Why would you have something as your favorite that is so vague?It seems that we like our numbers to be somewhat unique, which may be why prime numbers are popular. They arent divisible by any smaller numbers (aside from 1).

Belloss research was revealed in 2014, but his conclusion has been bouncing all over the internet for the past few days. Because your favorite number is probably a topic thatis endlessly fascinating.

Interestingly, the number 13 isnt as unpopular as you might think. It ranks as the sixth-most popular favorite number (but decidedly lower among people who have been hacked to pieces by Jason Voorhees).

As he was brushing his teeth on the morning of July 17, 2014, Thomas Royen, a little-known retired German statistician, suddenly lit upon the proof of a famous conjecture at the intersection of geometry, probability theory, and statistics that had eluded top experts for decades.

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Original story reprinted with permission from Quanta Magazine, an editorially independent division of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences

Known as the Gaussian correlation inequality (GCI), the conjecture originated in the 1950s, was posed in its most elegant form in 1972 and has held mathematicians in its thrall ever since. I know of people who worked on it for 40 years, said Donald Richards, a statistician at Pennsylvania State University. I myself worked on it for 30 years.

Royen hadnt given the Gaussian correlation inequality much thought before the raw idea for how to prove it came to him over the bathroom sink. Formerly an employee of a pharmaceutical company, he had moved on to a small technical university in Bingen, Germany, in 1985 in order to have more time to improve the statistical formulas that he and other industry statisticians used to make sense of drug-trial data. In July 2014, still at work on his formulas as a 67-year-old retiree, Royen found that the GCI could be extended into a statement about statistical distributions he had long specialized in. On the morning of the 17th, he saw how to calculate a key derivative for this extended GCI that unlocked the proof. The evening of this day, my first draft of the proof was written, he said.

Not knowing LaTeX, the word processer of choice in mathematics, he typed up his calculations in Microsoft Word, and the following month he posted his paper to the academic preprint site arxiv.org. He also sent it to Richards, who had briefly circulated his own failed attempt at a proof of the GCI a year and a half earlier. I got this article by email from him, Richards said. And when I looked at it I knew instantly that it was solved.

Upon seeing the proof, I really kicked myself, Richards said. Over the decades, he and other experts had been attacking the GCI with increasingly sophisticated mathematical methods, certain that bold new ideas in convex geometry, probability theory or analysis would be needed to prove it. Some mathematicians, after years of toiling in vain, had come to suspect the inequality was actually false. In the end, though, Royens proof was short and simple, filling just a few pages and using only classic techniques. Richards was shocked that he and everyone else had missed it. But on the other hand I have to also tell you that when I saw it, it was with relief, he said. I remember thinking to myself that I was glad to have seen it before I died. He laughed. Really, I was so glad I saw it.

Richards notified a few colleagues and even helped Royen retype his paper in LaTeX to make it appear more professional. But other experts whom Richards and Royen contacted seemed dismissive of his dramatic claim. False proofs of the GCI had been floated repeatedly over the decades, including two that had appeared on arxiv.org since 2010. Boaz Klartag of the Weizmann Institute of Science and Tel Aviv University recalls receiving the batch of three purported proofs, including Royens, in an email from a colleague in 2015. When he checked one of them and found a mistake, he set the others aside for lack of time. For this reason and others, Royens achievement went unrecognized.

Proofs of obscure provenance are sometimes overlooked at first, but usually not for long: A major paper like Royens would normally get submitted and published somewhere like the Annals of Statistics, experts said, and then everybody would hear about it. But Royen, not having a career to advance, chose to skip the slow and often demanding peer-review process typical of top journals. He opted instead for quick publication in the Far East Journal of Theoretical Statistics, a periodical based in Allahabad, India, that was largely unknown to experts and which, on its website, rather suspiciously listed Royen as an editor. (He had agreed to join the editorial board the year before.)

With this red flag emblazoned on it, the proof continued to be ignored. Finally, in December 2015, the Polish mathematician Rafa Lataa and his student Dariusz Matlak put out a paper advertising Royens proof, reorganizing it in a way some people found easier to follow. Word is now getting around. Tilmann Gneiting, a statistician at the Heidelberg Institute for Theoretical Studies, just 65 miles from Bingen, said he was shocked to learn in July 2016, two years after the fact, that the GCI had been proved. The statistician Alan Izenman, of Temple University in Philadelphia, still hadnt heard about the proof when asked for comment last month.

No one is quite sure how, in the 21st century, news of Royens proof managed to travel so slowly. It was clearly a lack of communication in an age where its very easy to communicate, Klartag said.

But anyway, at least we found it, he addedand its beautiful.

In its most famous form, formulated in 1972, the GCI links probability and geometry: It places a lower bound on a players odds in a game of darts, including hypothetical dart games in higher dimensions.

Imagine two convex polygons, such as a rectangle and a circle, centered on a point that serves as the target. Darts thrown at the target will land in a bell curve or Gaussian distribution of positions around the center point. The Gaussian correlation inequality says that the probability that a dart will land inside both the rectangle and the circle is always as high as or higher than the individual probability of its landing inside the rectangle multiplied by the individual probability of its landing in the circle. In plainer terms, because the two shapes overlap, striking one increases your chances of also striking the other. The same inequality was thought to hold for any two convex symmetrical shapes with any number of dimensions centered on a point.

Special cases of the GCI have been provedin 1977, for instance, Loren Pitt of the University of Virginia established it as true for two-dimensional convex shapesbut the general case eluded all mathematicians who tried to prove it. Pitt had been trying since 1973, when he first heard about the inequality over lunch with colleagues at a meeting in Albuquerque, New Mexico. Being an arrogant young mathematician I was shocked that grown men who were putting themselves off as respectable math and science people didnt know the answer to this, he said. He locked himself in his motel room and was sure he would prove or disprove the conjecture before coming out. Fifty years or so later I still didnt know the answer, he said.

Despite hundreds of pages of calculations leading nowhere, Pitt and other mathematicians felt certainand took his 2-D proof as evidencethat the convex geometry framing of the GCI would lead to the general proof. I had developed a conceptual way of thinking about this that perhaps I was overly wedded to, Pitt said. And what Royen did was kind of diametrically opposed to what I had in mind.

Royens proof harkened back to his roots in the pharmaceutical industry, and to the obscure origin of the Gaussian correlation inequality itself. Before it was a statement about convex symmetrical shapes, the GCI was conjectured in 1959 by the American statistician Olive Dunn as a formula for calculating simultaneous confidence intervals, or ranges that multiple variables are all estimated to fall in.

Suppose you want to estimate the weight and height ranges that 95 percent of a given population fall in, based on a sample of measurements. If you plot peoples weights and heights on an xy plot, the weights will form a Gaussian bell-curve distribution along the x-axis, and heights will form a bell curve along the y-axis. Together, the weights and heights follow a two-dimensional bell curve. You can then ask, what are the weight and height rangescall them w < x < w and h < y < hsuch that 95 percent of the population will fall inside the rectangle formed by these ranges?

If weight and height were independent, you could just calculate the individual odds of a given weight falling inside w < x < w and a given height falling inside h < y < h, then multiply them to get the odds that both conditions are satisfied. But weight and height are correlated. As with darts and overlapping shapes, if someones weight lands in the normal range, that person is more likely to have a normal height. Dunn, generalizing an inequality posed three years earlier, conjectured the following: The probability that both Gaussian random variables will simultaneously fall inside the rectangular region is always greater than or equal to the product of the individual probabilities of each variable falling in its own specified range. (This can be generalized to any number of variables.) If the variables are independent, then the joint probability equals the product of the individual probabilities. But any correlation between the variables causes the joint probability to increase.

Royen found that he could generalize the GCI to apply not just to Gaussian distributions of random variables but to more general statistical spreads related to the squares of Gaussian distributions, called gamma distributions, which are used in certain statistical tests. In mathematics, it occurs frequently that a seemingly difficult special problem can be solved by answering a more general question, he said.

Royen represented the amount of correlation between variables in his generalized GCI by a factor we might call C, and he defined a new function whose value depends on C. When C = 0 (corresponding to independent variables like weight and eye color), the function equals the product of the separate probabilities. When you crank up the correlation to the maximum, C = 1, the function equals the joint probability. To prove that the latter is bigger than the former and the GCI is true, Royen needed to show that his function always increases as C increases. And it does so if its derivative, or rate of change, with respect to C is always positive.

His familiarity with gamma distributions sparked his bathroom-sink epiphany. He knew he could apply a classic trick to transform his function into a simpler function. Suddenly, he recognized that the derivative of this transformed function was equivalent to the transform of the derivative of the original function. He could easily show that the latter derivative was always positive, proving the GCI. He had formulas that enabled him to pull off his magic, Pitt said. And I didnt have the formulas.

Any graduate student in statistics could follow the arguments, experts say. Royen said he hopes the surprisingly simple proof might encourage young students to use their own creativity to find new mathematical theorems, since a very high theoretical level is not always required.

Some researchers, however, still want a geometric proof of the GCI, which would help explain strange new facts in convex geometry that are only de facto implied by Royens analytic proof. In particular, Pitt said, the GCI defines an interesting relationship between vectors on the surfaces of overlapping convex shapes, which could blossom into a new subdomain of convex geometry. At least now we know its true, he said of the vector relationship. But if someone could see their way through this geometry wed understand a class of problems in a way that we just dont today.

Beyond the GCIs geometric implications, Richards said a variation on the inequality could help statisticians better predict the ranges in which variables like stock prices fluctuate over time. In probability theory, the GCI proof now permits exact calculations of rates that arise in small-ball probabilities, which are related to the random paths of particles moving in a fluid. Richards says he has conjectured a few inequalities that extend the GCI, and which he might now try to prove using Royens approach.

Royens main interest is in improving the practical computation of the formulas used in many statistical testsfor instance, for determining whether a drug causes fatigue based on measurements of several variables, such as patients reaction time and body sway. He said that his extended GCI does indeed sharpen these tools of his old trade, and that some of his other recent work related to the GCI has offered further improvements. As for the proofs muted reception, Royen wasnt particularly disappointed or surprised. I am used to being frequently ignored by scientists from [top-tier] German universities, he wrote in an email. I am not so talented for networking and many contacts. I do not need these things for the quality of my life.

The feeling of deep joy and gratitude that comes from finding an important proof has been reward enough. It is like a kind of grace, he said. We can work for a long time on a problem and suddenly an angel[which] stands here poetically for the mysteries of our neuronsbrings a good idea.

Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

The latest news from the none-of-your-thoughts-are-original department comes from mathematics blogger Alex Bellos, who set out to determine the worlds favorite number.

Bellos apparently doesnt have a favorite number himself, but as Nautiluswrites, he began asking people about their favorite numbers a few years ago, and after setting up theFavourite Numberwebsite, more than 44,000 people voted for the numeral they liked best and explained why.

Heres what Bellos discovered.

The third-most popular number is 8, because, as some of Belloss respondents wrote, In Japan, eight is a lucky number, because the Japanese character for eight means an opening to the future and because of its symmetrical and round shape and because it has always given me a sense of friendliness and warmth (unlike, for example, 9 which looks bossy or 6 which appears to me a bit submissive).

No. 2 on the list is the No. 3, because Its curly, but not pretentious curly like eight and because in Chinese, “3 means alive.

But the worlds most favorite number is No. 7. As for why, Bello will explain that himself. Its basically because of our desire to be outliers when it comes to arithmetical patterns.

As for why numbers ending in 0 or 5 are unpopular, Bellos said its because we use those numbers as approximations more than, say, 7 or 9.

When we say 100, we dont usually mean exactly 100, we mean around 100, Bellos told Nautilus. So 100 seems incredibly vague. Why would you have something as your favorite that is so vague?It seems that we like our numbers to be somewhat unique, which may be why prime numbers are popular. They arent divisible by any smaller numbers (aside from 1).

Belloss research was revealed in 2014, but his conclusion has been bouncing all over the internet for the past few days. Because your favorite number is probably a topic thatis endlessly fascinating.

Interestingly, the number 13 isnt as unpopular as you might think. It ranks as the sixth-most popular favorite number (but decidedly lower among people who have been hacked to pieces by Jason Voorhees).

A new study found that young U.S. girls are less likely than boys to believe their own gender is the most brilliant.

While all 5-year-olds tended to believe that members of their own gender were geniuses, by age 6 that preference had diminished for girls a difference the researchers attributed to the influence of gender stereotypes.

“We found it surprising, and also very heartbreaking, that even kids at such a young age have learned these stereotypes,” said Lin Bian, the study’s co-author and a doctoral candidate at the University of Illinois at Urbana-Champaign.

“It’s possible that in the long run, the stereotypes will push young women away from the jobs that are perceived as requiring brilliance, like being a scientist or an engineer,” she told Mashable.

A growing field

The study, published Thursday in the journal Science, builds on a growing body of research that suggests gender stereotypes can shape children’s interest and career ambitions at a young age.

A global study by the Organization for Economic Cooperation and Development found that girls “lack self-confidence” in their ability to solve math and science problems and thus score worse than they would otherwise, which discourages them from pursuing science, engineering, technology and mathematics (STEM) fields.

A 2016 study suggested a “masculine culture” in computer science and engineering makes girls feel like they don’t belong.

Thursday’s research looks not at specific skills but at the broader concept of high-level intellectual abilities. In short, can girls be geniuses, too?

Sapna Cheryan, a psychology professor at the University of Washington who was not involved in the study, said the results were “super important” because they’re among the first to show us how young children not adults or high-schoolers respond to gender stereotypes.

But she said the findings are just as revealing for young boys as for girls.

“It’s not that girls are underestimating their own gender it’s that boys are overestimating themselves,” she told Mashable. Cheryan was the lead author of last year’s masculine culture study.

“What we want as a society is for people to say boys and girls are equal,” she added.

Stereotyping starts early

Andrei Cimpian, a co-author of Thursday’s study, said his earlier research with adults showed that the fields people associate with requiring a high level of smarts also tend to be overwhelmingly represented by men.

“Across the board, the more that people in a field believe you need to be brilliant, the fewer women you see in the field,” Cimpian, an associate professor of psychology at New York University, told Mashable.

This same idea burrows itself into our brains as children, the study suggests.

Researchers worked with 400 children ages 5, 6 and 7 in a series of four experiments for the new study. (Not every child participated in every experiment for the study.)

In the first experiment, the psychologists wanted to see whether children associate being “really, really smart” with men more than with women.

To answer that question, a researcher told each child an elaborate story about a person who was brilliant and quick to solve problems, without hinting at all at the person’s gender. Next, the children looked at a series of pictures of men and women and were asked to guess who from the line-up was the character in the story.

During a series of similar questions, researchers kept track of how often children chose members of their own gender as being brilliant.

Among 5-year-olds, boys picked boys a majority of the time, while girls picked girls.

“This is the heyday of the ‘cooties’ stage,” Cimpian said. “It’s consistent with what we know about in-group biases in this young age group.”

But among 6- and 7-year-olds, a divide emerged. Girls were significantly less likely to rate women as super smart than boys were to pick members of their own gender.

The age groups were similarly split in a second prompt. Researchers asked kids to pick from activities described as either suited for brilliant kids, or kids who try really hard.

Five-year-old boys and girls both showed interest in the smart-kid activities. But by age 6, girls expressed more interest in the games for hard workers, while boys kept on with the “brilliant” games.

Why is this happening?

Researchers said it’s not entirely clear how these stereotypes form. Certainly marketing towards children lab sets are for boys, dollhouses are for girls plays a role.

And history books are filled with the achievements of white men who, generally speaking, did not face the same systemic discrimination that kept women and people of color out of classrooms and laboratories.

Cimpian and Bian said they are planning a larger, longer-term study to explore how these stereotypes form and stick, and how we can correct them.

In the meantime, they suggested a few ways that parents and teachers of young children could work to dispel the biased idea that men are inherently more prone to brilliance than women.

Bian noted that previous research has shown that girls respond better to what psychologists call a “growth mindset” the idea that studying, learning and making an effort are the key ingredients for success, not a stroke of genetic luck.

“We should recommend the importance of hard work, as opposed to brilliance,” she said.

Sharing and touting the achievements of women can also help counter the stereotypes that genius is reserved for men. Cimpian cited the book and movie Hidden Figures, about the women scientists who helped NASA astronauts get to space for the first time, as a prime example.

Cheryan, the UW psychologist, said including young boys in such efforts is critical.

“There’s a societal message that if there’s a gender gap, it’s the girls we need to fix,” she said. “We have to be careful with that message, because it just reinforces the similar hierarchy that the boys are always doing the right thing. In reality, there’s probably things that could happen on both sides.”

BONUS: 5 Gender Stereotypes That Used To Be The Opposite

If (cos(6k/2000)-i cos(12k/2000))e^(3i/4) means nothing to you, then you’re probably like the rest of us. Normal.

The last time “cos” resonated with you, was during high school math– when it meant “cosine,” a trigonometric function…of some sort.

But to 25-year-old Iranian student Hamid Naderi Yeganeh, using cosines are a part of daily life — what you would expect of a mathematics major and award-winning mathlete.

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Math genius and artist: Is this the next da Vinci?

Yeganeh’s work with circles and line segments is expanding to include animations. Beyond that, he’s beginning to think in 3-D, creating sculptures made of fractals.

“The power of mathematics is unlimited. There’s an infinite number of great artworks that we can create,” he says.

If mathematics isnt your strong suit, this equation that went viral in Japan may just trip you up. According to the YouTube channel MindYourDecisions, a study found that only 60 percent of individuals in their 20s could get the right answer. This is significantly lower than the 90 percent success rate in the 1980s.

To learn which common mistake people are making, check out the video below.

It is possible to nurture young minds by making them to go through mathematical activities at a very young age. By exposing the little minds to various activities like counting, number sequencing and patterns, it is possible to train up their minds. There are various kinds of activities and they should be chosen as per the age groups. If you go through online, you can access free as well as premium sites which offer preschool math activities.

How to make the most from the activities?

It is possible to make the most from preschool math activities by exposing the little minds in a systematic way. They should not be overburdened by dumping too many exercises and making them to go through lessons which are beyond their capacity. All the activities should create interest in them. They should have fun with and every activity. If you include everyday aspects in these activities, it is very easy to grasp the concept. The child will be immersed in these activities automatically when you give proper direction and encouragement.

Examples

Some of the everyday examples are ‘counting of stairs’ at home or school. The ingredients that are used in the cooking process can be counted. You can ask the child to prepare groups of various kinds of items of play. Children can remember various kinds of shapes like circles, rectangles, squares and triangles and they will also be able to form a shape by joining various items together.

Methodologies

It is true that the ways math are taught today are completely different from yesteryears. There are drastic changes and children have ample opportunities to explore the real world in the way it is present. Instead of teaching only one way to solve mathematical puzzles, children are encouraged to explore new ways in an open way.

The role of teacher or trainer in delivering the right kind of education is very high. If the teacher is aware of the multiple ways through which a mathematical problem can be resolved, he will encourage students to come with various solutions. The teacher should be resourceful and willing to learn and implement new strategies so that the learning process will be more intuitive. If you ask a child about the way he got the answer to a particular mathematical issue, you will understand his or her way of thinking. Instead of teaching solutions, it is required to show the way to reach those solutions. When a proper platform is created to bring out the best present in children, it is possible to teach complex math in simple ways.

When the child’s mind is molded in a proper way, the child will be able to learn the basics in an appropriate way. The child should have lots of mathematical materials such as beads and blocks. They should be encouraged to use their fingers and the body to begin the counting journey. The physical surroundings should be treated in such a way so that an interesting atmosphere is created to learn mathematics subconsciously.