Impossible, Imaginary, Useful Complex Numbers

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

2. SOLVE • Solve the equation • x2+2x+7 • Use the quadratic formula Solve on the calculator using a+bi mode

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

4. GIROLAMO CARDANO • 1545 • Published The Great Art • Formula Works for many cubics….but WAIT! Example: The process of dealing with the square root of negative one is “as refined as it is useless.”

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 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

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 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

12. Works Cited • Baez, John. Octonions. May 16, 2001. University of California. 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.