1. Impossible, Imaginary, Useful Complex Numbers Ch. 17 Chris Conover & Holly Baust

2. SOLVE Solve the equation   x 2 +2x+7   Use the quadratic formula Solve on the calculator using a+bi mode

3. Overview Introduction Introduction Cardano Cardano Bombelli Bombelli De Moivre & Euler De Moivre & Euler Berkeley, Argand, and Gauss Berkeley, Argand, and Gauss Hamilton Hamilton Timeline Timeline

4. 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.” Example:

5. 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

6. BOMBELLI WARNING!!!   Not numbers   Used to simplify complicated expressions From previous example combined with the NEW language: WILD IDEA→

7. BOMBELLI Negative numbers can lead to real solutions so appearance can be tricky! USEFUL “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.”

8. 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

9. 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”

10. SIR WILLIAM ROWAN HAMILTON Interested in applying complex numbers to multi- dimensional geometry. Worked for 8 years to apply to the 3 rd dimension, only to realize that it only existed in the 4 th. 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

11. TIMELINE 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 e ix = 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

12. Works Cited Baez, John. Octonions. May 16, 2001. University of California. http://math.ucr.edu/home/baez/octonions. http://math.ucr.edu/home/baez/octonions 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.