edu:topics:complex:start
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edu:topics:complex:start [2017/11/10 15:05] simon [more] |
edu:topics:complex:start [2018/02/21 23:51] (current) simon [notes] |
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| But for others there needs to be something else. | But for others there needs to be something else. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | <quote></quote>http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf<quote> | ||
| + | According to [B. L. van der Waerden, A History of Algebra, Springer Verlag, NY 1985.], | ||
| + | |||
| + | "Cardano was the first to introduce complex numbers a + √−b into algebra, but had misgivings about it." | ||
| + | |||
| + | <quote></quote>In Chapter 37 of Ars Magna the following problem is posed:<quote> | ||
| + | "To divide 10 in two parts, the product of which is 40”. | ||
| + | |||
| + | It is clear that this case is impossible. Nevertheless, we shall work thus: | ||
| + | |||
| + | We divide 10 into two equal parts, making each 5. | ||
| + | |||
| + | These we square, making 25. | ||
| + | |||
| + | Subtract 40, if you will, from the 25 thus produced, as I showed you in the chapter on operations in the sixth book leaving a remainder of -15, | ||
| + | |||
| + | the square root of which added to or subtracted from 5 gives parts the product of which is 40. | ||
| + | |||
| + | These will be 5 + √−15 and 5 − √−15. | ||
| + | |||
| + | Putting aside the mental tortures involved, multiply 5 + √−15 and 5 −√−15 | ||
| + | |||
| + | making 25 − (−15) which is +15. Hence this product is 40. | ||
| + | </quote></quote> | ||
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| ---- | ---- | ||
| + | just like calculus was re-defined in terms of limits to eliminate "impossible" infinities, so | ||
| + | complex numbers are redefined considering groups and sets ... but this requires a great deal more theory: | ||
| <quote></quote>http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf<quote> | <quote></quote>http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf<quote> | ||
| - | According to [B. L. van der Waerden, A History of Algebra, Springer Verlag, NY 1985.], | + | Augustin-Louis Cauchy (1789-1857) initiated complex function theory in an 1814 memoir |
| - | + | submitted to the French Acad´emie des Sciences. The term analytic function was not | |
| - | <quote>Cardano was the first to introduce complex numbers a + | + | mentioned in his memoir, but the concept is there. The memoir was published in 1825. |
| - | √ | + | Contour integrals appear in the memoir, but this is not a first, apparently Poisson had |
| - | −b into | + | a 1820 paper with a path not on the real line. Cauchy constructed the set of complex |
| - | algebra, but had misgivings about it.” In Chapter 37 of Ars Magna the following problem | + | numbers in 1847 as R[x]/(x2 + 1) |
| - | is posed: “To divide 10 in two parts, the product of which is 40”. | + | <quote>We completely repudiate the symbol √ |
| - | It is clear that this case is impossible. Nevertheless, we shall work thus: We | + | −1, abandoning it without regret because |
| - | divide 10 into two equal parts, making each 5. These we square, making 25. | + | we do not know what this alleged symbolism signifies nor what meaning |
| - | Subtract 40, if you will, from the 25 thus produced, as I showed you in the | + | to give to it. |
| - | chapter on operations in the sixth book leaving a remainder of -15, the square | + | |
| - | 2 | + | |
| - | root of which added to or subtracted from 5 gives parts the product of which is | + | |
| - | 40. These will be 5 + √ | + | |
| - | −15 and 5 − | + | |
| - | √ | + | |
| - | −15. | + | |
| - | Putting aside the mental tortures involved, multiply 5 + √ | + | |
| - | −15 and 5 − | + | |
| - | √ | + | |
| - | −15 | + | |
| - | making 25 − (−15) which is +15. Hence this product is 40. | + | |
| </quote></quote> | </quote></quote> | ||
| + | |||
| + | ---- | ||
| ... from pp2 to top of numbers ... that the ancient pyth, cubis and quadratics are linked here with complex | ... from pp2 to top of numbers ... that the ancient pyth, cubis and quadratics are linked here with complex | ||
| Line 199: | Line 217: | ||
| ====notes==== | ====notes==== | ||
| - | [[edu:resources:schrodinger-life]] | [[edu:resources:gleick-chaos]] | galileo | | + | [[books:schrodinger]] | [[edu:resources:gleick-chaos]] | galileo | |
| copernicus | [[edu:resources:penrose-physics]] | copernicus | [[edu:resources:penrose-physics]] | ||
edu/topics/complex/start.1510286747.txt.gz · Last modified: 2017/11/10 15:05 by simon
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