/sci/ - Science & Math

The main point is a little buried in a modern treatment. The point is that it is consistent to imagine little itty-bitty numbers, infinitesimals, adjoined to your conception of the real numbers, and these infinitesimals contain the idea of limit and asymptotics. So for example:

[math] (3 + dx)^2 = 9 + 6dx [/math]

where dx is an infinitesimal, so I dropped the dx^2, because the square of an infinitesimal is twice more infinitesimal than the infinitesimal and can be ignored. By definition, then, 6 is the derivative of squaring at 3. That means that

[math] 3.001^2 = 9.006 [/math]

up to certain negligible corrections. You can use this for party tricks:

[math] (1 + dx)^n = 1 + n dx [/math]

so that

[math] \sqrt{ 1.01} = 1.005 [/math]

You can use this to do arithmetic well, after you internalize the idea. You can also do calculations with trigonometry. Once you know enough, you see that

[math] \sin{dx} = dx [/math]

for infinitesimal dx (in radians) so that

[math] \sin{10 ^{\circ}} = 10 * 2\pi/ 360 [/math]

to a good approximation, because 10 degrees is small. It allows you to approximate quickly.

This infinitesimal idea is due to Cavalieri, it was developed by Leibnitz (Newton always thought in terms of limits), and it was given it's permanent final form inside modern mathematical logic by Abraham Robinson, after a century of suppression. It's a very exciting idea, it really is one of the greatest ideas humanity ever had.

>>11428737
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If Albert Einstein met Jesus what would be the first question he would ask?