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Impossible, Imaginary, Useful Complex Numbers. Ch. 17 Chris Conover & Holly Baust. SOLVE. Solve the equation x 2 +2x+7 Use the quadratic formula. Solve on the calculator using a+bi mode. Overview. Introduction Cardano Bombelli De Moivre & Euler Berkeley, Argand, and Gauss Hamilton

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impossible imaginary useful complex numbers

Impossible, Imaginary, Useful Complex Numbers

Ch. 17

Chris Conover & Holly Baust

  • Solve the equation
    • x2+2x+7
    • Use the quadratic formula

Solve on the calculator using a+bi mode

  • Introduction
  • Cardano
  • Bombelli
  • De Moivre & Euler
  • Berkeley, Argand, and Gauss
  • Hamilton
  • Timeline
girolamo cardano
  • 1545
    • Published The Great Art
  • Formula

Works for many cubics….but WAIT!


The process of dealing with the square root of negative one is “as refined as it is useless.”

rafael bombelli
  • 1560s
    • Operating with the “new kind of radical”
    • Invented NEW LANGUAGE
  • Old language
    • “two plus square root of minus 121”
  • New Language
    • “two plus of minus square root of 121”
    • “plus of minus” became code
  • Explained the rules of operation
  • WARNING!!!
    • Not numbers
    • Used to simplify complicated expressions
  • From previous example combined with the NEW language:


  • Negative numbers can lead to real solutions so appearance can be tricky!

“And although to many this will appear an extravagant thing, because even I held this opinion some time ago, since it appeared to me more sophistic than true, nevertheless I searched hard and found the demonstration, which will be noted below. ... But let the reader apply all his strength of mind, for [otherwise] even he will find himself deceived.”

de moivre euler
  • De Moivre
    • At this time mathematicians knew that:
      • (a+bi)(c+di) = (ac-bd) + i(bc+da)
    • If you think of this in the right frame of mind you can see the similarities in the REAL parts in the formula:

cos(x+y) = cos(x)cos(y)-sin(x)sin(y)

    • Similarly, you can notice the relationship between imaginary parts of formula: sin(x+y) = sin(x)cos(y) + sin(y)cos(x)
    • From here it is not hard to see De Moivre’s formula:

(cos(x)+isin(x))n = cos(nx)+isin(nx)

  • Euler
berkeley argand and gauss
  • Bishop George Berkeley
    • Would say that all numbers were useful functions
  • J.R. Argand
    • First to suggest the mystery of these “fictitious” or “monstrous” imaginary numbers could be eliminated by geometrically representing them on a plane
    • Published booklet in 1806
    • Points
    • Results ignored until Gauss suggested a similar idea
  • Gauss
    • Proposed similar idea and showed it could be useful mathematically in 1831
    • Coined the term “Complex number”
sir william rowan hamilton
  • Interested in applying complex numbers to multi-dimensional geometry.
  • Worked for 8 years to apply to the 3rd dimension, only to realize that it only existed in the 4th.
  • Quaternions

q = w+xi+yj+zk, where i, j, and k are all different square roots of -1 and w, x, y, and z are real numbers

  • 1545: Cardano’s The Great Art
  • 1560: Bombelli’s new language
  • 1629: Girard assumption of roots and coefficients
  • 1637: René Decartes coined the term “imaginary”
  • 1730: De Moivre’s formula (cos(x)+isin(x))n = cos(nx)+isin(nx)
  • 1748: Euler’s formula eix = cos(x)+isin(x)
  • 1806: Argand’s booklet on graphing imaginary numbers
  • 1831: Gauss coined the term “complex number”
  • 1831: Gauss found complex numbers useful in mathematics
  • 1843: Hamilton discovered quaternions
works cited
Works Cited
  • Baez, John. Octonions. May 16, 2001. University of California.
  • Berlinghoff, William P., and Fernando Q. Gouvêa. Math Through the Ages: a Gentle History for Teachers and Others. Farmington: Oxton House, 2002. 141-146.
  • Hahn, Liang-Shin. Complex Numbers & Geometry. Washington, DC: The Mathematical Association of America, 1994.
  • Hawkins, F M., and J Q. Hawkins. Complex Numbers & Elementary Complex Functions. New York: Gordon and Breach Science, 1968.
  • Lewis, Albert C. "Complex Numbers and Vector Algebra." Campanion Encyclopedia of the History and Philosophy of the Mathematical Sciences. 2 vols. New York: Routledge, 1994.