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What inspired mathematicians invent imaginary numbers?

  1. Apr 26, 2013 #1
    Let me start by writing about the natural or counting numbers. The story of how, where and when we invented them is lost in the misty dawn of history. But perhaps our African ancestors, like our living simian cousins (and some other animals) evolved the ability to distinguish between few and many; in our case with articulated labels that gradually developed in parallel with language. Or perhaps numbers were invented in “... the age of stone when men stood up on their hind legs and began to count time by the sun”, as Lew Archer put it in Ross MacDonald’s story "The Moving Target".

    We really don’t know how all this happened, but the world-wide language of mathematics that is based on abstract numbers (algebra?) and perceived shapes (geometry?) is still evolving. Viva new inventions still to be made!

    Only a few millennia ago, in the Middle East, the counting numbers: 1,2,3.... etc., were up and running, as it were, serving the practical purpose of quantifying resources, such as territory, livestock and crops. But the counting numbers proved inadequate to satisfy our hard-wired-by-evolution need to talk about and describe everything we see or sense. Once resources have been quantified, they can be exchanged, lent and borrowed; practices that are lubricated by that abstract medium of exchange; money. Money has many representations; some quite real (hard or spot cash) and some not quite so real (binary 0's and 1's, easily exchanged but dangerously ephemeral).

    By classical times (I mean times that span the hinge between B.C. and A.D.) there already existed a venerable dichotomy between negative and positive numbers, perhaps dating also from the “age of stone”. I suspect that the invention of money and its social world of debits and credits, together with the ancient practice of stock theft, may have enhanced the need for the concept of negative numbers. Numbers that quantify debt are conveniently labelled negative. They can then be added to and algebraically cancel positive credit, so quantifying “flat broke” ith another invented number, zero. Margaret Atwood in her book "Payback", has argued persuasively that the ancient dichotomy of debtors and creditors reflects our innate sense of justice and fairness, hard-wired into us by Evolution.

    Also in Classical times, the gaps between the stepped sequences of counting numbers, both positive and negative, separated by the number zero, had been quantified and partly filled by fractions. Perhaps fractions were devised to quantify fair sharing of resources, like apple pie. The filling of gaps was completed when it was reluctantly recognised that there were numbers (like ∏) that weren’t fractions (a.k.a. rational numbers; the ratio of counting numbers). Such numbers, like ∏, are written with an endless, never-repeating sequence of digits, and are known as irrational numbers. Finally the continuous set of all numbers, negative, zero and positive, rational and irrational, including those used for counting (a.k.a. whole numbers) are the abstract concept on which much of mathematics (and hence physics, our quantified and predictive description of reality) is based. Mathematicians call this set of numbers simply “the Reals”; in the 17th century the mathematician John Wallis introduced the idea of an infinitely long number line as a useful geometrical analogue of the Reals. This concept still seems to be in use.

    Now consider how numbers can be used to quantify space or how we decorate space with imagined numerical position labels. As a start think first of a distant object in front of you, say a tree near your horizon. The natural but primitive way to quantify its distance from you might be to estimate how many steps it would take you to walk to it. For this purpose counting numbers suffice. You could construct a table of locations for such objects by associating with each its estimated count of steps and its bearing from some convenient direction, for instance North. Such a table would be a primitive set of polar coordinates R (say with the number of steps, ranging from 0 to about 10,000) and Φ (bearing, ranging from 0 to 2 ∏). I suggest that
    such a simple way of quantifying the two dimensions of space on an (assumed flat) Earth may have persisted from “the age of stone” until 1657, when the philosopher Réne Descartes put two and two together and made 3 (dimensions); replaced by 4 in the Annus Mirabilis of 1905.

    Descartes counted distances along oppositely directed lines with oppositely signed numbers, negative and positive, from an origin denoted zero. He did this along three mutually perpendicular lines that he called axes. Points in three dimensions thus acquired three numeric labels, now called the Cartesian coordinates of that point. Ever since this Cartesian system of coordinates was invented some 350 years ago, its adoption has greatly simplified calculations in physics, mainly because its axes are mutually orthogonal.

    But there is no such thing as a free lunch, and Cartesian coordinates that use negative numbers required a new algebraic invention to fully conform with long-established algebraic practice. In particular the algebraic operation of taking the square root of a number creates a problem with negative numbers. The problem was solved by the Gordian knot method: the label “imaginary” (denoted by i) is substituted for the square root of minus one, and the set of real numbers is replaced by a set of “complex” numbers for purposes of calculation. Complex numbers are the sum of two entities; a real number and a real number multiplied by i. This arcane construct, serving as “payment for lunch” --- lunch being algebraic consistency --- has
    enriched mathematics and physics in a way that Roger Penrose, in his tome Road to Reality (A complete Guide to the Laws of the Universe) describes as "magical".

    The introduction of complex numbers seems to me to have deep connections with one of the two symmetries of space, namely that of rotation symmetry. (The other is translation symmetry, whether it be in space or in time). The workings of the universe can be described in the same way by all observers who perceive it from perspectives distinguished only by their orientation with respect to any inertial (freely falling) reference frame. For instance i may be regarded as an operator active in a plane perpendicular to any potential rotation axis, in which distance along an axis (the real axis, say x) is proportional to a positive real number. (The number line) Suppose this plane is regarded as the complex plane x + iy (often called an Argand diagram) --- originally by Casper Wessel who first invented this representation of complex numbers in 1797 (Jean Argand also did in 1806). Then multiplication of z = x + iy twice by i rotates the real axis by ∏ and transforms the positive real axis into its collinear negative self, completing the Cartesian form of this axis.

    Was it the introduction of Cartesian coordinates that provided the dominant motive behind the introduction of complex numbers? or was it the fascination mathematicians once had with the use of formulae to solve equations (e.g. quadratic equations) that popularised i? This seems to be the view taken in Mario Livio's book "The equation that couldn't be solved"?

    Can somone please unravel my muddled thinking?
    Last edited: Apr 26, 2013
  2. jcsd
  3. Apr 26, 2013 #2
    As far as I know, they just came up when the cubic formula was applied to certain cubic equations. They were ignored at first, or just treated as a curiosity. It was recognized that they gave correct results if treated in a purely formal way, but they weren't considered on par with real numbers until the geometric viewpoint caught on (maybe around 1750-1800?).

    An Imaginary Tale: The Story of √(-1) should fully answer your question, but that's the gist of it. At any rate, mathematicians did not invent i for the purpose of solving equations of the form x^2+1=0. That's the idea one gets when reading most textbooks.
  4. Apr 27, 2013 #3
    Thanks very much for steering me to my namesake Paul Nahin's book, which will certainly clarify much of my thinking. I've just skimmed through a web-available facsimile of it (cf. Google), and plan to buy a hard copy. What a stellar array of mathematical names seem to have been involved in trying to understand the nature of complex numbers! At the moment I'm still inclined to stay with my "no free lunch" take, but I guess that'll change.
    Last edited: Apr 27, 2013
  5. Apr 28, 2013 #4
  6. Apr 28, 2013 #5
    From my historic research, Cardano (1501-1576) had stumbled upon this concept when investigating the factorization of cubic equations.

    The motivation was probably a popular game among the noble classes in the 1500s. In the game, mathematicians would compete in solving polynomials for a group of spectators. Whoever solved the polynomial the fastest won. This inspired mathematicians to come up with novel techniques to solve factorization problems. But at the same time, it encouraged them to conceal their methods.

    The concept occurred just before Descartes.

    I'm still not certain if it is appropriate to say complex numbers were "invented" as opposed to "discovered."
  7. Apr 29, 2013 #6
    thelema418: Thanks; mathematics in the 1500's seems to have been a sponsored competitive sport! Great times to be had then. But I suppose success brings similar rewards even today; Field's medals to math folk, the gifts of Yuri Milner to physics folk and Nobel Prizes to a wide range of deserving people.

    Using the word "discovered" implies a belief that complex numbers existed (somehow) before say, Cardano, just as the continent of N. America existed (some believe) before say, Columbus. I think Roger Penrose would, together with many mathematicians, take this view about their beloved subject.

    But mathematical concepts remain abstract human constructs (even 1 + 1 = 2!), however effective they may prove to be for conserving resources or helping physics to describe the universe we find ourselves in. I think that talking of such marvellous inventions as "discovered" is, sadly, only anthro'centric vanity.
    Last edited: Apr 29, 2013
  8. Apr 29, 2013 #7


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    You feel that saying we "discovered" something is more "anthropocentric vanity" than saying we created it?

    In any case both "discovered" and "created" can be used for mathematical concepts. When we first state the defining properties of something like the complex numbers, we are creating[ them. When we derive other properties those, we are discovering those new properties that were created when we first defined the complex numbers.
  9. Apr 29, 2013 #8
  10. May 2, 2013 #9
    I apologise if I touched a nerve, Halls of Ivy. But I still think that "discovered " is a word that should be reserved for "finding out" that something (like a country, or a fact) already exists, rather than used for something that is the more mundane product of human ingenuity. I see älso that you prefer the word "created" and avoid the more pedestrian "invented". But I do think that a lot of what we do is anthro'centric vanity. We're full of it!
  11. May 2, 2013 #10


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    But you also wrote:

    Except all facts are human constructs.

    And your example of countries is an awful one. Do you know what is the requirement to be a country? That other countries recognise your sovereignty. Just having an explorer finding a previously unknown group of people does not mean he "discovered" a new country. Read about Terra Nullius and Australian Aborigines.
  12. May 2, 2013 #11
    I would say they were constructed to fulfill a certain requirement.
    As the real numbers were.
  13. May 2, 2013 #12

    I refer you to the O.E.D. In addition to being full of anthro'centric vanity, I'm not P.C., so the Abos' plight misses me.
  14. May 2, 2013 #13
    Solkar: I agree. In the case of complex numbers this was to allow all real numbers (negative as well as positive) to conform with existing algebraic practice, namely to allow the operation "take the square root". As I wrote, there is no such thing as a free lunch!
  15. May 2, 2013 #14


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    In order to demonstrate whether a statement is a fact, you need to be able to show that it corresponds with experience. That experience will be a human one. That's why it is a human construct.

    That has nothing to do with what I said. It is a counter-example to your use of "country" as "something that already exists".
  16. May 2, 2013 #15
    O.K. Substitute "planet" as an example of something that can be discovered, rather than "invented".
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