edu:topics:complex:start
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edu:topics:complex:start [2017/11/10 15:02] simon [more] |
edu:topics:complex:start [2018/02/21 23:51] (current) simon [notes] |
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| - | perhaps re the lesson connecting to Argand Plane, use: | + | <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.], | ||
| - | <quote> </quote> | + | "Cardano was the first to introduce complex numbers a + √−b into algebra, but had misgivings about it." |
| - | <quote>http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf</quote> | + | <quote></quote>In Chapter 37 of Ars Magna the following problem is posed:<quote> |
| - | There are indications that Gauss had been in possession of the geometric representation of complex numbers since 1796, but it went unpublished until 1831, when he submitted his ideas to the Royal Society of Gottingen. Gauss introduced the term complex number | + | "To divide 10 in two parts, the product of which is 40”. |
| - | “If this subjet has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1 and | + | It is clear that this case is impossible. Nevertheless, we shall work thus: |
| - | √ −1, instead of being | + | |
| - | called positive, negative and imaginary (or worse still, impossible) unity, been given the names say,of direct, inverse and lateral unity, there would hardly have been any scope for such obscurity.” | + | |
| - | </quote> | + | |
| - | ---- | + | We divide 10 into two equal parts, making each 5. |
| - | <quote></quote><quote>http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf</quote> | + | These we square, making 25. |
| - | According to [B. L. van der Waerden, A History of Algebra, Springer Verlag, NY 1985.], | + | |
| + | 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 | ||
| - | <quote>Cardano was the first to introduce complex numbers a + | ||
| - | √ | ||
| - | −b into | ||
| - | algebra, but had misgivings about it.” In Chapter 37 of Ars Magna the following problem | ||
| - | is posed: “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 | ||
| - | 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. | making 25 − (−15) which is +15. Hence this product is 40. | ||
| </quote></quote> | </quote></quote> | ||
| + | |||
| + | ---- | ||
| + | |||
| + | perhaps re the lesson connecting to Argand Plane, use: | ||
| + | |||
| + | <quote></quote>http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf<quote> | ||
| + | There are indications that Gauss had been in possession of the geometric representation of complex numbers since 1796, but it went unpublished until 1831, when he submitted his ideas to the Royal Society of Gottingen. Gauss introduced the term complex number | ||
| + | |||
| + | <quote>If this subjet has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1 and | ||
| + | √ −1, instead of being | ||
| + | called positive, negative and imaginary (or worse still, impossible) unity, been given the names say,of direct, inverse and lateral unity, there would hardly have been any scope for such obscurity. | ||
| + | </quote></quote> | ||
| + | |||
| + | ---- | ||
| + | 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> | ||
| + | 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 | ||
| + | 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 | ||
| + | a 1820 paper with a path not on the real line. Cauchy constructed the set of complex | ||
| + | numbers in 1847 as R[x]/(x2 + 1) | ||
| + | <quote>We completely repudiate the symbol √ | ||
| + | −1, abandoning it without regret because | ||
| + | we do not know what this alleged symbolism signifies nor what meaning | ||
| + | to give to it. | ||
| + | </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 201: | 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.1510286573.txt.gz · Last modified: 2017/11/10 15:02 by simon
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