Imaginary numbers are real

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From Aeon:

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.

In December of 2021 and January of 2022, two teams of physicists, one an international collaboration including researchers from the Institute for Quantum Optics and Quantum Information in Vienna and the Southern University of Science and Technology in China, and the other led by scientists at the University of Science and Technology of China (USTC), showed that a version of quantum mechanics devoid of imaginary numbers leads to a faulty description of nature. A month earlier, researchers at the University of California, Santa Barbara reconstructed a quantum wave function, another quantity that cannot be fully described by real numbers, from experimental data. In either case, physicists cajoled the very real world they study to reveal properties once so invisible as to be dubbed imaginary.

For most people the idea of a number has an association with counting. The number five may remind someone of fingers on their hand, which children often use as a counting aid, while 12 may make you think of buying eggs. For decades, scientists have held that some animals use numbers as well, exactly because many species, such as chimpanzees or dolphins, perform well in experiments that require them to count.

Counting has its limits: it only allows us to formulate so-called natural numbers. But, since ancient times, mathematicians have known that other types of numbers also exist. Rational numbers, for instance, are equivalent to fractions, familiar to us from cutting cakes at birthday parties or divvying up the cheque after dinner at a fancy restaurant. Irrational numbers are equivalent to decimal numbers with no periodically repeating digits. They are often obtained by taking the square root of some natural numbers. While writing down infinitely many digits of a decimal number or taking a square root of a natural number, such as five, seems less real than cutting a pizza pie into eighths or 12ths, some irrational numbers, such as pi, can still be matched to a concrete visual. Pi is equal to the ratio of a circle’s circumference and the diameter of the same circle. In other words, if you counted how many steps it takes you to walk in a circle and come back to where you started, then divided that by the number of steps you’d have to take to make it from one point on the circle to the opposite point in a straight line passing through the centre, you’d come up with the value of pi. This example may seem contrived, but measuring lengths or volumes of common objects also typically produces irrational numbers; nature rarely serves us up with perfect integers or exact fractions. Consequently, rational and irrational numbers are collectively referred to as ‘real numbers’.

Negative numbers can also seem tricky: for instance, there is no such thing as ‘negative three eggs’. At the same time, if we think of them as capturing the opposite or inverse of some quantity, the physical world once again offers up examples. Negative and positive electric charges correspond to unambiguous, measurable behaviour. In the centigrade scale, we can see the difference between negative and positive temperature since the former corresponds to ice rather than liquid water. Across the board then, with positive and negative real numbers, we are able to claim that numbers are symbols that simply help us keep track of well-defined, visible physical properties of nature. For hundreds of years, it was essentially impossible to make the same claim about imaginary numbers.

In their simplest mathematical formulation, imaginary numbers are square roots of negative numbers. This definition immediately leads to questioning their physical relevance: if it takes us an extra step to work out what negative numbers mean in the real world, how could we possibly visualise something that stays negative when multiplied by itself? 

Link to the rest at Aeon

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19 thoughts on “Imaginary numbers are real”

  1. Great links for my Story folders.

    Thanks…

    I love the novel Contact by Carl Sagan, but it is pure fantasy. Think Dragons and Elves and that’s how I see his description of detecting the “Alien” signal, deciphering it using “Universal Math”, having “plans” in the signal to build the device. Yikes! I need to read it again.

    – Math is an artificial construct that few people actually understand.

    – Math is not a “Universal Language”.

    I have so many characters utterly destroyed on contact with aliens because of their delusion that Math is a “Universal Language”.

    – “It’s a cook book!”

    I love the way the movie Arrival had the aliens understand complex things easier than simple things like 2 + 2 = 4.

    Look at C.J. Cherryh’s stories about The Alliance-Union and their contact with the Hydrogen based aliens. The translations are all probabilities of meaning. Or the Jovians in John Varley’s Eight World stories where no communication is possible.

    BTW, Just finished reading Ophiuchi Hotline again after decades of the book sitting on my shelf as a “source book”. I need to read it again, and hopefully remember the non-linear story telling.

    Solaris is another good example, along with the movie Annihilation, of how actual alien contact would work. Arrogant people trying to impose their “world view” on The Other usually get them eaten, or worse.

    Annihilation (2018) Josie`s choice
    https://www.youtube.com/watch?v=eyGK-25Z248

    • There’s also THE DEEP where aliens come to colonize water world Earth.

      David Gerrold THE WAR AGAINST THE CHTORR

      https://en.wikipedia.org/wiki/The_War_Against_the_Chtorr

      And Gordon R. Dickson’s WAY OF THE PILGRIM.

      As Hawking said, advertising our position to all comers isn’t wise; the only thing we can count on is aliens being alien. After all different human societies are.

      • The Chtorr books are intense. I need to read them again as well.

        Then there is Ian McDonald:

        Chaga (McDonald novel)
        https://en.wikipedia.org/wiki/Chaga_(McDonald_novel)

        And Paul J. MacAuley:

        Something Coming Through
        https://en.wikipedia.org/wiki/Something_Coming_Through

        That one is a useful template, where aliens transport people to deserted worlds. That brings up the classic concept of Forerunners by Andre Norton. Where people find technologies that are left over from Singularities.

        Forbidden Planet Official Trailer #1 – Leslie Nielsen Movie (1956) HD
        https://www.youtube.com/watch?v=AxQ9GG6hUDM

        Monsters from the “id”. (Notice the “i” in that. That’s a clue to what went wrong. HA!)

      • I feel the need to remind people that the inverse square law is a thing. The fears expressed by the Dark Forest hypothesis doesn’t stand up to scrutiny

        All our transmissions rapidly become indistinguishable from the background radiation, decoherence prevents measurement.

        It’s all about distance and thermodynamics.

        • That’s what’s so fun, all of the “fears” that people have written over the decades fail Reality.

          – Picking up “I Love Lucy” episodes.

          – Detecting atomic bombs detonating.

          And the latest:

          – Detecting civilization on other planets by seeing their lights on the night side.

          I found this paper that helps explain:

          The Dark Forest Rule: One Solution to the Fermi Paradox
          https://www.researchgate.net/publication/283986931_The_Dark_Forest_Rule_One_Solution_to_the_Fermi_Paradox

          I read Liu Cixin’s The Three-Body Problem when it first came out, then never bothered with the rest of his stuff, so I never heard of The Dark Forest Rule.

          The SF in the book was a “MacGuffin” to distract from the history of the Cultural Revolution that he was really writing about. That was interesting, but the SF was ridiculous.

          If they could surround the Earth with a “proton”, then they could literally cut us off from the Sun and freeze the planet, killing us all. Then they could remove the “proton” and let the surface warm up again for colonization. That’s classic.

          When I read the Plot synopsis on the Wiki page for The Dark Forest, I’m going “What? No thanks. I’m not interested”, and that is saying something because I will read any wacky story[1]. HA!

          Two books that are right on the edge, yet hold up under multiple reads, that people never heard of:

          Eric S. Nylund

          – Signal to Moise

          Jack Potter puts computer cryptography to work for the highest bidder: sometimes for private corporations, sometimes for the government. Sometimes the work is legal; if not, Jack simply raises his price. But one day, Jack discovers something cloaked in the hiss of background radiation streaming past the Earth from deep space: a message from an alien civilization. One that’s eager to do business with humanity — and its representative.

          Before he knows it, Jack has entered into a partnership that will open a Pandora’s Box of potential profit and loss. The governments, the multinationals, and mysterious players more powerful still, all want a piece of the action — and they’re willing to kill, even wage war, to get it. Now Jack is entangled shifting web of deceit and intrigue in which no one, not even his closest friends, can be trusted. For Earth’s cloak-and-dagger business practices are writ large in the heavens…and hostile takeovers are just as common across light years as they are across boardroom tables.

          – A Signal Shattered

          Jack Potter’s and his friends narrowly avoided disaster in the last novel (although most other people on the Earth did not!). Jack finds himself victim of alien technologies…evolving into something not quite alien…not quite human, either. With galactic corporate raiders still looking to double deal him, Jack has to outwit them all—to save himself…and what’s left of humanity.

          Sadly, you have to find them used.

          [1] Seveneves and Anathem by Neal Stephenson totally failed and I will not read them again. Which is saying a lot since I read his books over and over.

          Anything by David Brin is right on the edge but they work. Earth and Existence are good examples of pushing the limits. I need to read them again.

          • To an extent the whole “dangerous aliens” meme is a reaction to the decades old popular media meme (a favorite of UFOlogists) that aliens, being more technologically advanced, must naturally be more ethical and benevolent. They’ll come to save us from ourselves. Which drives many authors to take the contrarian position.

            Along the lines of the Nylund book:

            Greg Costikyan’s FIRST CONTRACT:

            https://www.amazon.com/First-Contract-Greg-Costikyan/dp/0312873964

            Highly recommended to anybody interested in first contact or encounters between civilizations.

            Also, in a simiar vein, the first section of John Ringo’s TROY RISING: the not entirely tongue in cheek MAPLE SYRUP WAR. (Now if only he’d do the final volume in the series…)

            • Was the “ethical and benevolent” meme itself a reaction to H G Wells’ Martians and the pulps BEMs? You know the idea: because the aliens are more technologically advanced, they will naturally stomp us.

              • More likely it was a reaction to MAD during the cold war.

                Between Wells and 1960 there wasn’t a dominant meme,we got both INVASION OF THE BODY SNATCHERS and THE DAY THE EARTH STOOD STILL. (And MARTIANS GO HOME). Afterwards there were less INDEPENDENCE DAY’s and more STARMAN’s and CLOSE ENCOUNTERS. Humans were the real danger.

                Ever run into this?

                https://www.youtube.com/watch?v=fI-i2cpGRfc

                The meme is at heart another version of the rapture and other divine/angelic interventions saving “us” from evil. Its easier than trying to solve our tribalism ourselves.

                • Oh, dear. System overload. Flooded with too many references. Overwhelmed.

                  Thanks for the link.

                  The following links only need to be reassembled for access.

                  This of course may still cause the post to be held for review despite my efforts. HA!

                  Who’s Behind the Door? by Zebra HD
                  https:/
                  /www.youtube.com/watch?v=Ofz0iJQd-xU

                  Children Of The Sun
                  https:/
                  /www.youtube.com/watch?v=_YL7ALghfi8

                  Styx – Come Sail Away (Official Video)
                  https:/
                  /www.youtube.com/watch?v=e5MAg_yWsq8

                  Hypnotized
                  https:/
                  /www.youtube.com/watch?v=fDzXbdxeeHI

                  After the Gold Rush
                  https:/
                  /www.youtube.com/watch?v=gGEhbRvfNS4

                  Prelude – After the Goldrush
                  https:/
                  /www.youtube.com/watch?v=3oO8h4hGkjo

                  Trio – After The Gold Rush (Official Music Video)
                  https:/
                  /www.youtube.com/watch?v=ykMzS6ugnPI

                  88:88
                  https:/
                  /www.youtube.com/watch?v=YgnmtKWvErQ

  2. As Alicia intimates, calling them “imaginary” was probably a mistake. It seems to have helped sow confusion in the writer of the article and caused her to overstate the importance of the results she reported, which were not remotely surprising and, to my mind, not very interesting.

    Alicia calls the second axis a phase shift, which is one way to look at it, especially if you are into electronics or engineering. To me though it is just a formal system of mathematics involing ordered pairs of real numbers, with the rule for the multiplication of those pairs defined in a way which is extremely fruitful for mathematicians (as well as being useful in more applied fields). The number i that people call imaginary is, of course, just the ordered pair (0,1).

    Anything that can give you a result as beautiful as Euler’s Identity (e raised to the power of i times pi equals minus 1) needs no other justification.

    • Interesting article but I think his arguement falls apart when he says that “all mathematical models are approximations of reality”. This is of course true but irrelevant as it confuses mathematics as an artifact discovered or invented by the human mind – which has no connection to empirically observed “reality” – with the application of mathematics for practical purposes.

      As for the responses to Einstein’s question, I find none of them particularly convincing. No. 1 makes an assumption about the universe for which there is no evidence, and No. 2 makes an assumption about the way mathematics is invented/discovered that is probably not true, and pretty much misses Einstein’s point in that far from being invented to describe the universe Riemannian geometry was already there waiting when it was needed for General Relativity.

      But then I’m a somewhat reluctant Platonist so I’m not likely to agree.

  4. Imaginary numbers are not “Real” they are “Imaginary”. People hear that and think, “Then they do not exist.” Wrong, “Imaginary” numbers do exist. They are the heart of modern technology, and those Physicists would know that if they ever worked in the real world.

    Electrical Engineers use Imaginary number all of the time for circuits, large and small. The computer, the power grid, etc…, all use Imaginary numbers.

    You have a complex number:

    a + bi = c

    “i” is the square root of negative one. Grade school kids learned this, along with Mobius strips and Klein bottles, at least I did in the Long Ago and the Far Away.

    You create a graph, with one axis labeled “Real”, the other at right angles to it as “Imaginary”.

    “a” and “b” are Real numbers. “b” is rotated through “Imaginary” space to transform it and then is added to “a”. This creates the Real result “c”.

    Complex numbers
    https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-ac-analysis/v/ee-complex-numbers

    If that video does not burn your brain, then you have no “Imagination”[1].

    Get it? “Imaginary”, “Imagination”. HA!

    What has this to do with Story? I’m glad you asked.

    You write Story by taking some “Real” event, and rotating it through “Imaginary space”. i.e., your “Imagination”, and the “Real” result cascades out onto the page, utterly transformed by your “Imagination”.

    That’s why writing seems “complex” when it’s not. You just have to understand that the “i” in the word “Write” needs that “Imaginary” component to make it work.

    [1] The same applies to the article when they start discussing the quantum experiments.

    – As the article essentially points out, all Reality is Imaginary.

    This will also burn your brain.

    Double slit experiment explanation macro quantum from Through The Wormhole S02E06
    https://www.youtube.com/watch?v=sGCtMKthRh4

    Whew! I need to go lay down now. That was intense.

  5. Might have been easier for non-math types had ‘imaginary’ – a word with odd connotations as to reality – not been chosen to represent a concept which is best represented graphically (a 2-D representation is very clear for many physics and engineering concepts).

    The second axis is a phase shift. Many engineering concepts require a way to express that.

    The mathematics are quite elegant – and stick with those who learn them in similar ways to a lot of other concepts, as a shortcut. I’m sure there was a wrench somewhere back in my youth when exposed to them for the first time, but I’ve used them since prep school, so they are as natural and convenient as the letters of the alphabet are for readers.

    But they may be like swimming: you can learn to swim as an adult, but it is said that you will never frolic the same way as if you had learned as a kid.

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