Throughout the history of math and science, supposedly impossible negative things have repeatedly turned out to be important both mathematically and physically. Here is my unofficial (no, let’s make that official) Top 10 list of “negative” inventions.

**10. Negative refraction: Victor Veselago, 1967**

Refraction refers to how much light slows down (and therefore appears to be bent) when it passes through some medium. Refraction is quantified by an index relative to the refractive index of the vacuum, which is equal to 1. All natural materials have a positive index of refraction, which means light is always bent in the same direction. Veselago, a Russian physicist, figured out that a refraction index of less than zero was possible in theory, meaning light would bend in the opposite direction from the usual. Three decades later physicists began to figure out how carefully constructed artificial “metamaterials” actually could bend light the “wrong” way, leading to current research on cloaking devices.

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**9. Negative electric charge: Benjamin Franklin, 1747**

Franklin figured out that electric charge comes in positive and negative forms; he just guessed wrong about which was which, which is why electrons have negative charge even though they’re carriers of electric current.

**8. Negative mass (or negative weight): Friedrich Albrecht Carl Gren, 1786**

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OK, this one is tricky. Around 1700 the German physician Georg Stahl articulated the phlogiston theory (based on an idea of Johann Becher), an elaborate explanation for why things burn. Supposedly they contained a flammable substance (phlogiston) that disappeared into the air during combustion. It’s often asserted that Stahl’s phlogiston had negative weight, but that idea appeared only much later, when experiments showed that sometimes the combustion products (ashes) weighed more than the original burned substance. Gren, a German chemist, suggested that negative mass could account for the discrepancy. Both Stahl and Gren were wrong, by the way.

**7. Negative Energy: Hendrik Casimir, 1948**

Paul Dirac imagined a sea of negative energy electrons in the late 1920s during his work on quantum mechanics that led to the prediction of the existence of antimatter. But let’s give this prize to Casimir, who figured out how to create negative energy in a physical apparatus. You just have to put two mirrors, or shiny metal plates, very close to each other. Since the amount of energy in empty space is set at zero, the plates should just sit wherever you put them. But in fact, they are slightly attracted to each other (the Casimir effect). That’s because empty space actually isn’t empty, but has a bunch of quantum particles popping into and out of existence. Being quantum particles they behave like waves. When the plates are close enough together, the in-between space isn’t big enough for some of the waves. So there are fewer particles in the gap than there should be, hence less than zero energy. Really.

**6. Negative pressure: Saul Perlmutter et al, Brian Schmidt et al, 1998**

We’re not talking about vacuum pumps here, but rather cosmological negative pressure, which requires the universe’s expansion to accelerate. That’s what the two teams led by Perlmutter and Schmidt discovered when they measured the brightness of distant supernovas — evidence that the universe has, for the last few billion years, been expanding at an ever increasing rate. Because the universe is expanding faster and faster, some force other than ordinary gravity must be at work, because gravity would slow the expansion rate down. That force must exert negative pressure, because ordinary pressure would compress space; negative pressure expands it.

**5. Negative temperature: Robert Pound, Norman Ramsey, 1951**

We’re not talking about Antarctica here, but rather negative temperature on the absolute scale, where absolute zero represents the complete absence of heat, and hence supposedly the coldest temperature possible. Which it is. But it turns out that mathematically, coldest is not the same as lowest. On the absolute scale, temperature and entropy are related in such a way that in all ordinary circumstances the temperature is positive. Temperature is related to the average velocity (or energy) possessed by molecules, and most of the molecules won’t be as energetic (fast) as the very fastest. If they were, the fastest would just go even faster. But if you put an upper limit on how fast the molecules can go, then they all could be as fast as the fastest. In this case, when the majority of molecules are at the maximum energy, the ordinary formula for temperature is turned upside down, and that makes the temperature negative. Even though the temperature is negative, most of the atoms are very energetic, so the system is technically hotter than any system with a positive temperature (heat would always flow from a negative temperature system to a positive temperature system, which by definition makes the positive system colder).

**4. Negative probabilities: Paul Dirac, 1920s**

In his work leading to the prediction of antimatter, Dirac found not only that negative energies entered into the equations, but also so did negative probabilities. Ordinarily, the chance of something happening (its probability) is regarded as somewhere between 0 (no chance at all, like the Cubs winning the World Series) and 1 (absolutely certain, like A-Rod guilty of using PEDs). Having a less-than-zero chance of happening seems meaningless. But Dirac showed that in some situations negative probabilities at intermediate steps in quantum calculations could be useful, a point later discussed by Richard Feynman. Recently the mathematician John Baez has blogged in detail about the whole negative probability business.

**3. Negative curvature: Carl Friedrich Gauss, 1824**

Except maybe for Newton, Gauss was the greatest mathematician of his millennium. He figured out that it would be possible to devise a geometry in which the sum of a triangle’s angles was less than 180 degrees, which means the curvature of such a space would be negative. He usually doesn’t get credit for inventing non-Euclidean geometry, though, because he didn’t publish that work. He was a perfectionist and wouldn’t publish anything until he had everything worked out so well that nobody could find any way to criticize it. (In other words, if Gauss had written this blog it would never have been posted.)

**2. Negative numbers: Brahmagupta, seventh century**

There is some evidence that the ancient Chinese possessed the concept of negative numbers, but Brahmagupta, a Hindu astronomer, gets credit for explicitly articulating their status as actual numbers (and not “absurd impossibilities” as some of the Greeks thought). Brahmagupta called negative numbers “debts” (positive numbers were “fortunes”) and he outlined the arithmetical rules governing them. For instance: “The product of two debts … is one fortune.” Thus Brahmagupta anticipated by more than 13 centuries the mantra of Edward James Olmos in the movie *Stand and Deliver*: “A negative times a negative equals a positive.”

**1. Square roots of negative numbers: John Wallis, 1673**

Like negative numbers, the idea of the square root of a negative number was initially regarded as an impossibility, as negative numbers are not the square of anything. But Wallis, an English mathematician, argued otherwise; as Paul Nahin says in his book on the subject, Wallis “made the first rational attempt to attach physical significance” to the square root of –1. Wallis pointed out that negative numbers are not hard to visualize — they’re just the numbers to the left of zero on a number line. But if you add another axis to the number line (pointing straight up, from zero) you then have a whole plane to the left of zero. “Now what is admitted in lines must, on the same Reason, be allowed in Plains also” (he meant “planes”), Wallis wrote. And since you can draw a square in a plane, a side of a square on the negative side of zero would correspond to the square root of the negative number. Far from being physically meaningless, roots of negative numbers turn out to be necessary ingredients in the equations of quantum mechanics.