The tidiness of elementary mathematics: Unterschied zwischen den Versionen

Aus DMUW-Wiki
Wechseln zu: Navigation, Suche
(→‎Applets)
(→‎Applets)
Zeile 13: Zeile 13:
  
 
<ggb_applet width="634" height="463"  version="4.0" ggbBase64="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" framePossible = "false" showResetIcon = "true" showAnimationButton = "true" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
 
<ggb_applet width="634" height="463"  version="4.0" ggbBase64="UEsDBBQACAAIAPBzaD8AAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT/LP88zLLNHQVKiuBQBQSwcI1je9uRkAAAAXAAAAUEsDBBQACAAIAPBzaD8AAAAAAAAAAAAAAAAMAAAAZ2VvZ2VicmEueG1szVh7j9M4EP8bPsUoQicQ2zbOs+Va0IGEDmnhkJY7odMJyU3c1DSJQ+y0XR7f/cZ2kqa7sCwsOm61W7/G8/yNZ7rzR/sihy2rJRflwiFj1wFWJiLlZbZwGrUaTZ1HD2/PMyYytqwprERdULVwAk3J04XDYhLT6YqMZilZjoJwSUeUpNORN40jL8WZF64cgL3kD0rxghZMVjRhZ8maFfRUJFQZwWulqgeTyW63G3eixqLOJlm2HO9l6gCqWcqF004eILujSzvfkHuuSyavn59a9iNeSkXLhDmgTWj4w9u35jtepmIHO56q9cKJQjRjzXi2RpsiFxcTTVShQyqWKL5lEq8OlsZmVVSOIaOlPr9lZ5D35jiQ8i1PWb1w3LHvxtHM6z9jB0TNWalaWtLKnHTc5lvOdpatnhmJgQNKiHxJNUf4+BE813PhRA/EDh4OUWSPXLvn+nbw7BDYIbQ0gb0eWNLA0gSWJvAd2HLJlzlbOCuaS/QgL1c1Rq9fS3WeM6NPu3GwnpygTZK/R2Jfu9S6HPdd90T/oZ9Pgs7XAyPJQKqqm28U2omM/OD6Ir0bGep3Mr3PmemFXzAzukKotfs6dpJwIBNFmV/zd0mif5WZFyXa9c0ERsF/YuJ80qXKvM0OkGtN26JHsULqfPFnEM407AmEmBtRjCgPgcxwiD3AbAASQhDikkwh0mMMfowHAfgwBU1HfDDJEU7xI4gNswhCZKZ3Y8xJICgogNAHYnIqAMwkMHmJOer5SBGGEOIlLZ54moUfQRDhyp9CgDrqlIwJEvp4Edco3gOfgK8vkxi8CCLNjwQ61aOpVh1ZehC5EBHNELMaM9pmM9JPwdfWRK27eFk16shFSZF2UyWqPhZIje/R4dWz79PRo3hrntMly7FOnOlIAmxprjPCCFqJUkEXRM/uZTWt1jyRZ0wpvCXhLd3SU6rY/ilSy062oU1EKV/WQj0ReVOUEiARudvrLHIymHu91rjwBwfB8CAcHESDefxZuQJPoJEM5YtaduQ0TZ9pisPTgJ78o8zPH9eMbirBj82YT0zJmbMmyXnKafkXglVL0X6BvgLp56qrQIEfdoqIOj07l4hg2P/NaqH9GI5nRz8OnNsjbxaP3eEPBlwmVCdfMDu+NMVL7VHoHl8iVjTb9hGie9Ybn9U6sVvD9eKZfCzyw5Yx/wmtVFOb3gHfxlob9VuZ5cxAxCQ2FuZksxT7M4sN3/J6dV7hyrUKLDPjdsCnwQtDJGjHpR0Njdasp3INjWso3A5sPO3PycwzFGZc2tFQIXqtaq2lpDOTuJ0YLs2D5jpHaWOgr8t8U3J12i0UTzYHSzX9i6ZYsh5AxyzJD2I5n1wA2HzD6pLlLZ4xko1opE3PAdRTlvACl/agdQjVwfoTFbC7Kctq1umdm67MusucukOoXto2rJ7WonhWbl8hEi4oMJ90Ws5lUvNKAw6WWAM27ICplEuKJSQd3tMJiKYnulSge5R2DaZmo9aiNo0Xvig46rzLWYFtFigDrrIpWM2T3tEb08GhUk2n9zi2mmsvg1i+xafuQnAOUcTjL8APaF6tqWbX5lNOz1l95BrD7blIW8Etncx1wwgFxwo4QrgXdI85ivyWEl9BhS0zxqI8tMxWs/YVwQZEN+R7XTNjPTvH1Dfd/IrvBx5FJ/H3CAp6ZM0hDxS+0BtsQ6VpjVSblmbyO09TVvbq0hLhY4KAj1Sl7XU1QivGLLb7uxU6wLwIg9i3odFB2lc1itN8WifjtwXcxPHu/h4sYP/m7gbuA7kHE2hntj4dh3fVlAYSzoHN1bEcpMIwmN2Lczmc7jXD6X6vT692C9ZJRVrX/PKuEerXO/APdlTJh/2bD5v75NMn8wl37KFz2UWag3PM7ue6iMtT+oq9tnsHNJrmQWKyrvpe0RQLt20a+/v45a5WL3XdAY18/KYVXyqQ+pvq9EKZg/f2m+5XfZ61/s4sFO8esDj6dkBm3+ft4etiPP9/AqR3FSBH5Nsh6f1kJ/1gSJJx4B2BL+ggeV0QwsJZt05eWxTmQsPxM868hLj1zZ15g+y+XCCuWx++hlXyNbBOhk2CacTb/yk9/BdQSwcINd+bIxoGAADwEgAAUEsBAhQAFAAIAAgA8HNoP9Y3vbkZAAAAFwAAABYAAAAAAAAAAAAAAAAAAAAAAGdlb2dlYnJhX2phdmFzY3JpcHQuanNQSwECFAAUAAgACADwc2g/Nd+bIxoGAADwEgAADAAAAAAAAAAAAAAAAABdAAAAZ2VvZ2VicmEueG1sUEsFBgAAAAACAAIAfgAAALEGAAAAAA==" framePossible = "false" showResetIcon = "true" showAnimationButton = "true" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />
 +
 +
 +
 +
<ggb_applet width="634" height="463"  version="4.0" ggbBase64="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" framePossible = "false" showResetIcon = "true" showAnimationButton = "true" enableRightClick = "false" errorDialogsActive = "true" enableLabelDrags = "false" showMenuBar = "false" showToolBar = "false" showToolBarHelp = "false" showAlgebraInput = "false" useBrowserForJS = "true" allowRescaling = "true" />

Version vom 8. November 2011, 23:34 Uhr

We all expect a certain tidiness to mathematics, and for many this provides a gratifying aesthetic pleasure. However, there are some wrinkles. One of these arises in the integral of x^k. Calculus provides us with the formula \int x^k\mathrm{d}x = \frac{x^{k+1}}{k+1}. However, this equation is only correct for k\neq -1. If we try to take k=-1 the right hand side is meaningless because we have a zero on the denominator of the fraction, i.e. \frac{1}{0}. But in this case a separate argument gives the answer \int x^{-1}\mathrm{d}x = \int \frac{1}{x}\mathrm{d}x = \ln(x).

Our expectation is that these two formulae should be reconciled. Indeed, if we let k approach -1 in the first we should end up with the second, but that fails to happen. For each x, \lim_{k\rightarrow -1} \frac{x^{k+1}}{k+1} is undefined.

This vingette explores this issue.

Main article

Applets


Von „http://dmuw.zum.de/index.php?title=The_tidiness_of_elementary_mathematics&oldid=30223“