The Myth of 'I'm Bad at Math'

Basic ability in the subject isn't the product of good genes, but hard work.

"How to Fall in Love With Math"

As a mathematician, I can attest that my field is really about ideas above anything else. Ideas that inform our existence, that permeate our universe and beyond, that can surprise and enthrall. Perhaps the most intriguing of these is the way infinity is harnessed to deal with the finite, in everything from fractals to calculus. Just reflect on the infinite range of decimal numbers — a wonder product offered by mathematics to satisfy any measurement need, down to an arbitrary number of digits. 

Despite what most people suppose, many profound mathematical ideas don’t require advanced skills to appreciate. One can develop a fairly good understanding of the power and elegance of calculus, say, without actually being able to use it to solve scientific or engineering problems. 

Think of it this way: you can appreciate art without acquiring the ability to paint, or enjoy a symphony without being able to read music. Math also deserves to be enjoyed for its own sake, without being constantly subjected to the question, “When will I use this?”

"Mathematician’s trick: I will catch you with my psychic rooster"

John Napier is most famous for inventing logarithms in the early 1600s, which greatly simplified calculations. 
But this post is about a lesser known rumored story about Napier.

Once, Napier suspected one of his servants was stealing kitchen utensils from him. But how to prove it, and how to figure out which one was pilfering from him?

Napier devised a scheme that would out the guilty party.

A Matemática na Natureza

«Modelos matemáticos usados por uma equipa de cientistas do Centro John Innes, em Norwich (Reino Unido), provaram que a quantidade de amido consumido pelas plantas de noite é calculada com precisão através de operações de matemática.

(...) Enquanto os mecanismos do interior das folhas medem a quantidade de amido armazenado, a informação sobre o tempo tem origem num relógio interno semelhante ao relógio interno do corpo humano.

(...) "É o primeiro exemplo concreto na biologia de tal cálculo aritmético sofisticado", afirmou à BBC o matemático Martin Howard, um dos membros da equipa.

"Estas experiências não provam que as plantas têm inteligência, mas sugerem antes que as plantas têm um mecanismo concebido para regular automaticamente o ritmo a que queimam os hidratos de carbono de noite", avisa Richard Buggs.

O investigador da Universidade de Londres acrescenta que "as plantas não fazem operações matemáticas de forma voluntária e com um objetivo em mente como nós, humanos, fazemos".

Aliás, os cientistas dizem que os pássaros usam métodos matemáticos semelhantes para conservar os seus níveis de gordura durante as migrações.»

Ler aqui.

A Conjectura de Goldbach

«Todo o número par maior que dois pode escrever-se como soma de dois números primos.»

"Unheralded Mathematician Bridges the Prime Gap"

«On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.»

Ler na íntegra aqui.

"Ants 'Use Math' to Find Fastest Route"

«Just as light does, ants traveling through different materials follow the fastest path, not the shortest one.

A recent study found that when fire ants (Wasmannia auropunctata) crossed different surfaces, the insects chose the route that would minimize their total walking time, rather than the distance traveled. The ants' behavior offers a window into how groups of social insects self-organize, the scientists say.»

Ler na íntegra aqui.

"There’s more to mathematics than rigour and proofs"

«One can roughly divide mathematical education into three stages:

1- The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years.

2- The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner (e.g. re-doing calculus by using epsilons and deltas all over the place). The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. This stage usually occupies the later undergraduate and early graduate years.

3- The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond.»

Ler na íntegra aqui.

Ano Internacional da Matemática do Planeta Terra

«A matemática rodeia-nos. Há matemática no crescimento das células, que, partir de certo tamanho, são obrigadas a dividir-se. Há matemática na rua, nas calçadas portuguesas, mas ela também cresce para outras escalas: económicas, planetárias, cósmicas. Ajuda a calcular a trajectória de asteróides, a construir modelos climáticos, a antecipar problemas económicos. É por tudo isso que se pretende uma maior ligação entre esta disciplina e o mundo no Ano Internacional da Matemática do Planeta Terra 2013 (MPT2013), que é lançado esta terça-feira às 19h no Pavilhão do Conhecimento, em Lisboa.»

O Paradoxo da Amizade

«It’s a mathematical fact. Your friends are probably more popular than you are.

Don’t believe it? Consider this: the average Facebook user has 245 friends, but the average friend on Facebook has 359 friends, according to a 2011 Pew survey.

That’s right. The average person on Facebook has fewer friends than their friends do.

But how can that be? It definitely seems weird, and that’s why this phenomenon is known as “friendship paradox,” described in a 1991 paper by Scott L. Feld amusingly titled “Why Your Friends Have More Friends Than You Do.”

It turns out the paradox is not so mysterious, however, when you take a closer look at the math. Below I will give an example of the friendship paradox and then go through the math of why it happens.»

Ler na íntegra aqui.

Alan Turing

«Alan Mathison Turing ha sido uno de los matemáticos más excepcionales del siglo XX. Sin embargo, el éxito de la aplicación de sus resultados a las ciencias de la computación ha sido tan grande que ha opacado el valor de sus matemáticas. A esto se añade que su trabajo en la descodificación de los códigos de las máquinas Enigma en la Segunda Guerra Mundial ha sido valorado relativamente tarde, y todavía depara sorpresas matemáticas a medida que los materiales secretos del ejército británico van siendo desclasificados.»

Ler na íntegra aqui.

Porque ficamos com os dedos enrugados na água?

«Investigadores de la Universidad de Newcastle han indagado sobre el asunto y han llegado a la conclusión de que este efecto tiene una explicación que tiene que ver con nuestra propia evolución. Los dedos arrugados mejoran nuestro agarre de objetos mojados o que se encuentran bajo el agua, de la misma forma que un neumático con surcos se aferra mejor a la carretera. Es probable que esta capacidad les viniera estupendamente a nuestros antepasados dedicados a recolectar frutos en entornos húmedos.»

Ler na íntegra aqui.

"Disputed Themes in Mathematics, 2012"

«Mathematics is ‘easy’ and agreeable for some people, but it is ‘hard’ or even incomprehensible for others; it deals with discrete objects (numbers, points, lines, sets) but also with the notion of continuum; it deals with finitude but also with the infinite; with certainty but also with uncertainty, probability, and chance; with the most general ideas but also with particular cases… and on, and on …»

Ler na íntegra aqui.

"Is zero an even number?"

«According to Dr James Grime of the Millennium Maths Project at Cambridge University, reaction time experiments in the 1990s revealed people are 10% slower at deciding whether zero is odd or even than other numbers.

Children find it particularly difficult to recognise if zero is odd or even. "A survey of primary school children in the 1990s showed that about 50% thought zero is even, about 20% thought it was odd and the remaining 30% thought it was neither, both, or that they don't know," explains Dr Grime.

"It appears that we may file numbers mentally into lists such as the even numbers two, four, six, eight or numbers to the power of two which would include two, four, six, eight or two, four, eight, 16. Zero is not on these lists so it takes us longer to work out."

So why, mathematically, is zero an even number? Because any number that can be divided by two to create another whole number is even.

Zero passes this test because if you halve zero you get zero. Zero also has odd numbers either side of it - minus one and one - and so this is another test it passes to be classified as an even number.»


Ler na íntegra aqui.

Pinguins e Matemática

«Don't let the adorable mini-orchestra-conductor look fool you: penguins aren't that nice. When emperor penguins huddle together during Antarctic storms, they act like they're all in it together. But a new mathematical model shows just how the clusters of birds keep warm, accounting for everything from their geometry to the speed of the wind. Concern for one's fellow bird, it turns out, isn't a factor.

(...) 

To pack their penguin huddle as tightly as possible, the mathematicians imagined the birds on a grid of hexagons. This is the best way for circles to squeeze into a plane (think of a honeycomb), and scientists in the field have observed that real penguins arrange themselves roughly this way. The researchers also assumed that "penguins in this huddle have uniform size and shape."

Next, they added wind to the model, which flowed around the huddle differently depending on its overall shape. Then they calculated the rate at which each computerized penguin was losing body heat. They sent the coldest penguin shuffling around the outside of the huddle until it found the warmest spot it could stand in, then started over with the new coldest penguin.

The simulated penguins constantly shifted positions within the huddle, just as real penguins do. Over time, the model huddle tended to take on the shape of a flat-sided oval and travel slowly downwind (as penguins on the windward side continuously moved away from it).

When they calculated the flock's heat loss, the authors discovered that their model huddle was very fair: every penguin lost approximately the same amount of body heat. But these model penguins were only programmed to maximize their own warmth, not to consider the warmth of other penguins or the group as a whole. This means that even if penguins are only looking out for themselves, the whole huddle stays warm (...)»

Ler na íntegra aqui.

"The Unconscious Brain Can Do Math"

«In the second part of the study, the scientists examined how the unconscious brain processes math problems. Using the CFS technique again, the researchers subliminally exposed the participants to three-digit equations, such as "9 − 3 − 4," for two seconds or less. Then, the participants were shown a number (without CFS masking it) and told to say it out loud. The students were quicker to read aloud a number that was the right answer to the equation they had just subconsciously seen. For example, after being exposed to "9 − 3 − 4," they were quicker to pronounce "2" than "3." This suggests they subconsciously worked out the problem and had the answer on their lips. Other recent studies have shown that humans might be able to unconsciously perform tasks that have typically been associated with consciousness, such as learning and forming intuitions. The new study adds complex, rule-based operations to that list.»

Ler na íntegra aqui.

"¿Por qué los huracanes tienden a formar una espiral logarítmica?"

«Las espirales de los huracanes y de las tormentas tropicales se parecen bastante a una espiral logarítmica, aunque no lo son, y el ángulo a veces coincide con el de la espiral áurea, pero no siempre.»

Porque não há uma solução geral para equação de 5.º grau?

«La resolución de ecuaciones polinomicas es un tema al que quien más quien menos se ha acercado en su etapa de estudiante. Todos sabemos cómo resolver una ecuación de primer grado (despejando la incógnita) y la gran mayoría recordamos la famosa fórmula para resolver ecuaciones de segundo grado. También muchos, aunque posiblemente menos, sabrán que hay fórmulas del mismo tipo para las ecuaciones de grados tres y cuatro. Y algunos menos que no se puede resolver de manera general la ecuación de quinto grado. Pero, ¿sabemos por qué?»

Ler na íntegra aqui.