Dimensions | Chapter 5

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Dimensions | Chapter 5
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École Normale Supérieure de Lyon (ENS-Lyon)

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e parliamentary Dougherty my
time I work in Ms. centered on the complex numbers my contributions helped to advance buzz algebraic geometry and the theory of dynamical systems complex numbers have a long history you see here on Monday but and cut down mathematical pioneers who lived during the Renaissance on the right Koshy camps consolidated the theory during the 19th century complex numbers are not previous complicated is the name might want at 1st they pulled impossible even today they still sometimes called imagine well it's true it does take a little imagination yet today these numbers are everywhere in science and I'm not really mysterious anymore In particular thanks to them 1 could construct beautiful battles something on worked on not I produced the film the dynamics of the rat it was 1 of the best animated films in mathematics let me begin by explaining the complex numbers on the right mathematicians just laugh writing you Lucy in a minute that my real real these squarely track behave robber what we sometimes that's a graduated line on the 1 of the most beautiful ideas in mathematics is the geometry algebra this is the starting point the algebraic geometry just as we can add numbers we can add points here is a break point on the line and another Ray 1 let's at these 2 points we get the green . 1 plus 2 equals 3 when the red and points of the Green Point which is the source must move more interesting still is modification of points let's look of modification by minors take for instance it transforms the . 1 . 1 2 calls if you modify
again by you have to to do the same thing change sides with respect to the origin and double the distance from the origin you get phone calls if we multiplied twice a month we multiplied by multiplying by minus 1 histories each point is sent to the symmetrical point with respect to the origins In other words your half the rotation by 180 degrees when we not apply number by itself the
result is always positive for instance if we multiplied by more than this 1 we make off so that if we did it 1 time well we come back to the initial point it is what minus 1 times minus 1 easy puts it plus 1 enough you see for instance the modification by minus 1 send to commanders and that if you multiply 1 more time on this 1 you come back 2 obvious Is therefore there is no number multiplied by them minus 1 another way of saying This is the minus 1 has no square he said but of course we are underestimating the inventiveness of petition at the beginning of the 19th century Robert Our goal had a really great idea he said to himself since multiplied by minus 1 is a 180 degree rotation its square root is rotation by one-half of 190 90 degrees if I due to quarter and 1 after another I end up doing a half the square of the quarter is a hot and wonders what it's
easy when you know on the other side is therefore that the square root of minus 1 is represented by the point which is the image of 1 by 90 degree rotation but of course these forces us to be a horizontal straight line since we had just agreed to associate the number 12 point the plane which is not on the as this construction is a bit strange we say that this point the square root of minus 1 is an imaginary number and mathematicians denoted by but once we have the courage to Everything else is easy we can represent 2 3 of and so on each point in the plane represents a complex number and conversely each complex number defines a point the fight points in the plane become numbers in their own right these numbers to be just like usual look at the break point which is the point 1 us to move on let's and 3 plus which is the point well he adds interest schoolchildren that uses plus 3 part geometrically This is just additional factors he see that it's no problem to a complex numbers much more interesting these complex numbers could also be multiplied just 1 real
numbers next season we know multiply complex number by 2 different since 2 times 1 plus 2 I'm here too fast for geometrically multiplied by 2 easy he's just scaling up by a factor of 2 if we double the rate we get the Greenpoint my going on is not difficult since we know that corresponds to a quarter In order to
multiply 3 I thought we just have to rotate quarter we get minus 1 last 3 not so complicated these complex numbers he and finally we can multiply any too complex numbers with no problem look silly for instance let's try to multiply 2 plus 1 . 5 9 and 1 class 2 . 4 all right we proceed as usual we 1st multiplied by 2 and then by 1 . 5 and entry at the results therefore we get
which is minus 2 plus 4 . 8 part minus 1 . 5 3 . 6 20 by square minus 1 since we invented on so we get minus 2 4 . 8 minus 1 . 5 9 minus 3 . 6 and tidying up that gives miners to minus 3 . 6 plus 4 . 8 9 Monday's 2 . 5 giving us in all minus 5 . 6 plus 3 . 3 there you are we know how to multiply complex numbers In other words we can multiply points June 20 that's amazing we thought that the plane was dimension to since 2 numbers it necessary to make a point and now I'm telling you that what numbers enough of course we change our numbers and now we're dealing with complex numbers it seems the right time to define to the modulus and they argument complex number the marchers the complex numbers said just that distance from the origin the point represents it is in the frame let's use the rumored to determine the modulus the right point we choose 2 plus 1 . 5 on the sea Major 2 . the modulus of 2 cost 1 . 1 on is 2 . for the point I get 2 . 6 and the Green Point which is the product of the 2 points on 6 . behind as a rule the modulus of a product of 2 complex numbers he's just the product Of the much the 2
argument complex number is measured by the angle between Texas and the straight line joining the origin of the point here for instance argument of the red complex number is 36 . 8 degrees the argument of the real point is 112 . 6 degrees and the product the Greenpoint we get the 149 . 4 degrees some of the arguments of the 2 numbers multiplied too complex numbers marginal and multiply arguments but let's finish up office encounter with complex numbers the HISTORY Graphic projections considers is tangent origins using xerographic graphic projections each point on that each complex cars on the point on his face under the North Pole is on the pole from which I reject as the To we say that it corresponds to therefore modifications say sphere is of complex projective line but why not because 1 needs anyone on describe its points why complex both because this numbers complex 1 rejected because we added appointed infinity using the projection mathematicians strange when they tried to tell us that the sphere is a straight line we need to do a more I
mean you you know whom he uses only E a behavior a a lot