— Imag­i­nary num­bers

The gleam in the ‘ i’

Cosmos - - Contents -

Imag­i­nary num­bers make the uni­verse real

THE OLD APHO­RISM “my enemy’s enemy is my friend” has a math­e­mat­i­cal equiv­a­lent: mul­ti­ply­ing two neg­a­tives makes a pos­i­tive. In mon­e­tary terms, where a neg­a­tive num­ber is def­i­nitely the enemy, it is the same as say­ing that re­duc­ing a debt is equiv­a­lent to mak­ing a gain.

The sim­plest case of two neg­a­tives mak­ing a pos­i­tive is:

-1x -1 = 1 Mul­ti­ply­ing a num­ber by it­self is known as squar­ing the num­ber, so the square of -1 is 1, the square of -2 is 4, and the square of -3 is 9, etc. How­ever, the square of 1 is also 1, the square of 2 is 4 and so on. The square is the same whether the num­ber is neg­a­tive or pos­i­tive.

Go­ing back­wards, a pro­ce­dure called “tak­ing the square root” re­verses the re­sult. The square root of 9 is 3, but it is also -3; there are two so­lu­tions to any square root cal­cu­la­tion.

All this is well and good, and drummed into most of us in school, but what hap­pens if you try to take the square root of a neg­a­tive num­ber, such as -9? No or­di­nary num­ber, when mul­ti­plied by it­self, yields -9, so how can you do the re­verse?

This ques­tion stymied math­e­ma­ti­cians for years. What they fi­nally fig­ured out was the need for a new type of num­ber en­tirely. By the 18th cen­tury, they ex­tended the num­ber sys­tem to in­clude the square root of neg­a­tive num­bers. This is what they did:

ix i = -1 which, re­ar­ranged, reads:

i = √-1 The new species of num­ber here is sym­bol­ised by i be­cause in the early days it was con­sid­ered an “imag­i­nary” num­ber rather than a “real” num­ber.

Many math­e­ma­ti­cians were sus­pi­cious and even de­risory about it. The name has stuck, even though to­day we ac­cept imag­i­nary num­bers are just as real as real num­bers. You can get more imag­i­nary num­bers by mul­ti­ply­ing i by real num­bers – 2i, 3i, 4i and so on – and there is no prob­lem com­bin­ing real and imag­i­nary num­bers. For ex­am­ple, 5 + 3i is a per­fectly good num­ber. Such com­bi­na­tions are called ‘com­plex’ num­bers, though the rules for ma­nip­u­lat­ing them are very sim­ple.

What are imag­i­nary num­bers good for? It turns out that by em­brac­ing i, the scope and power of math­e­mat­i­cal ma­nip­u­la­tions is enor­mously broad­ened, open­ing the way to a plethora of new short­cuts and tricks.

Ex­po­nen­tial growth and de­cay (which I ex­plained in Cos­mos is­sue 71, p76) when raised to the power of i su­per­im­poses a wave-like os­cil­la­tion on to the pat­tern of growth or de­cay. The world is re­plete with quan­ti­ties that os­cil­late and ei­ther grow or de­cay at the same time. For an ex­am­ple of how this works, think of a swing­ing pendulum that grad­u­ally slows – it os­cil­lates as it “de­cays”. Us­ing i also greatly sim­pli­fies the math­e­mat­i­cal de­scrip­tion of sys­tems that use com­pli­cated os­cil­lat­ing wave­forms, such as acous­tic and elec­tronic sig­nals.

But imag­i­nary num­bers are not just a com­pu­ta­tional con­ve­nience. Mother Na­ture got there long be­fore math­e­ma­ti­cians. We have known since Ein­stein’s the­ory of rel­a­tiv­ity that space and time are not in­de­pen­dent but fun­da­men­tally tied to­gether by the speed of light into a uni­fied “space­time”.

Though re­lated, space and time are not the same: it is i that al­lows us to com­bine

them. To mea­sure the space­time in­ter­val be­tween two cos­mic events, for in­stance, you have to ex­press the time in­ter­val in the same units as spa­tial dis­tance – achieved by mul­ti­ply­ing by i. One can there­fore say that space is “imag­i­nary time” (in the tech­ni­cal sense of imag­i­nary num­bers), a term pop­u­larised by Stephen Hawk­ing in A Brief His­tory of Time. Hawk­ing dis­cusses a the­ory of the Big Bang in which the uni­verse started out with four space di­men­sions, so time was imag­i­nary (in the √ -1 sense) at the out­set.

Na­ture also uses com­plex num­bers in quan­tum me­chan­ics. If you were the Great Cos­mic De­signer and tried to come up with laws for atomic pro­cesses us­ing only real num­bers, the re­sult­ing prop­er­ties of atoms would be very dif­fer­ent from what we ob­serve.

There’s an­other rea­son to ap­pre­ci­ate the num­ber i: its el­e­gance. It won a pub­lic beauty con­test ear­lier this year when the BBC asked peo­ple to vote for the most el­e­gant math­e­mat­i­cal re­la­tion­ship of all time. The win­ner was de­clared to be e +1=0

i where e stands for ex­po­nen­tial. This for­mula was dis­cov­ered in 1748 by the bril­liant Leon­hard Euler, known as “the Mozart of math­e­mat­ics”. By in­vok­ing i, Euler was able to com­bine e with three of the most ba­sic el­e­ments of the en­tire num­ber sys­tem: 0, 1 and . Euler’s for­mula is a pro­found re­la­tion­ship that seems to be speak­ing to us from some sort of math­e­mat­i­cal nir­vana.

All of which raises the ques­tion of how much math­e­mat­i­cal beauty still re­mains hid­den from us be­cause of lim­i­ta­tions on our ex­ist­ing num­ber sys­tem. Is there a fu­ture Euler out there who (to bor­row an­other apho­rism) will help us to be­hold it?

PAUL DAVIES is a the­o­ret­i­cal physi­cist, cos­mol­o­gist, as­tro­bi­ol­o­gist and best-sell­ing au­thor.


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