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Posts Tagged ‘Imaginary Numbers

“Why, sometimes I’ve believed as many as six impossible things before breakfast”*…

Imaginary numbers were long dismissed as mathematical “bookkeeping.” But now, as Karmela Padavic-Callaghan explains, physicists are proving that they describe the hidden shape of nature…

Many science students may imagine a ball rolling down a hill or a car skidding because of friction as prototypical examples of the systems physicists care about. But much of modern physics consists of searching for objects and phenomena that are virtually invisible: the tiny electrons of quantum physics and the particles hidden within strange metals of materials science along with their highly energetic counterparts that only exist briefly within giant particle colliders.

In their quest to grasp these hidden building blocks of reality scientists have looked to mathematical theories and formalism. Ideally, an unexpected experimental observation leads a physicist to a new mathematical theory, and then mathematical work on said theory leads them to new experiments and new observations. Some part of this process inevitably happens in the physicist’s mind, where symbols and numbers help make invisible theoretical ideas visible in the tangible, measurable physical world.

Sometimes, however, as in the case of imaginary numbers – that is, numbers with negative square values – mathematics manages to stay ahead of experiments for a long time. Though imaginary numbers have been integral to quantum theory since its very beginnings in the 1920s, scientists have only recently been able to find their physical signatures in experiments and empirically prove their necessity…

Learn more at “Imaginary numbers are real,” from @Ironmely in @aeonmag.

* The Red Queen, in Lewis Carroll’s Through the Looking Glass


As we get real, we might spare a thought for two great mathematicians…

Georg Friedrich Bernhard Riemann died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.


Andrey (Andrei) Andreyevich Markov died on this date in 1922.  A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors.  (For example, the probability of winning at the game of Monopoly can be determined using Markov chains.)  His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological, physical, and social sciences, and is widely used in Monte Carlo simulations and Bayesian analyses.


Pictures worth a million words…

In his great opus De Revolutionibus Orbium Coelestium published shortly before his death in 1543, Copernicus takes 405 pages of words, numbers and equations to explain his heliocentric theory. But it is the diagram that he draws at the beginning of the book that captures in a simple image his revolutionary new idea: it is the Sun that is at the centre of the Solar System, not the Earth.

A diagram has the power to create a whole new visual language to navigate a scientific idea. Isaac Newton’s optics diagrams [Opticks, 1704] for example transform light into geometry. By representing light as lines, Newton is able to use mathematics and geometry to predict the behaviour of light. It was a revolutionary idea.

Mathematicians had been struggling with the idea of the square root of minus one. There seemed to be no number on the number line whose square was negative. Experts knew that if such a number existed it would transform their subject. But where was this number? It was a picture drawn independently by three mathematicians at the beginning of the 19th Century that brought these numbers to life. Called the Argand diagram after one of its creators, this picture… was a potent tool in manipulating these new numbers [Imaginary Numbers] since the geometry of the diagram reflected the underlying algebra of the numbers they depicted.

Although better known for her contributions to nursing, Florence Nightingale’s greatest achievements were mathematical. She was the first to use the idea of a pie chart to represent data.  Nightingale’s diagrams were designed to highlight deaths in the Crimea. She had discovered that the majority of deaths in the Crimea were due to poor sanitation rather than casualties in battle. She wanted to persuade government of the need for better hygiene in hospitals. She realised though that just looking at the numbers was unlikely to impress ministers. But once those numbers were translated into a picture – her “Diagram of the Causes of Mortality in the Army in the East” – the message could not be ignored.

Read more (and find links to enlarged versions of the images above) at BBC.com, in “Diagrams that Changed the World,” a teaser for new BBC TV series, Marcus du Sautoy’s six-part The Beauty of Diagrams (on air now, and available via iPlayer to readers in the U.K… and readers with VPNs that can terminate in the U.K.)

As we marvel at the power of pictures, we might recall that it was on this date in 1997 that eight planets in our Solar System lined up from West to East– beginning with Pluto, followed by Mercury, Mars, Venus, Uranus, Neptune, Saturn and Jupiter, with a crescent moon alongside– in a rare alignment visible from Earth.  Mercury, Mars, Venus, Jupiter and Saturn were visible to the naked eye; the small blue dots that are Uranus and Neptune, with binoculars.  Pluto was visible only by telescope (but has subsequently been demoted from “planet” anyway…). The planets also aligned in May 2000, but too close to the sun to be visible from Earth.

Readers who missed it have a long wait for the reprise: it will be at least another 100 years before so many planets will be so close and so visible.


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