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### Course: Breakthrough Junior Challenge>Unit 1

Lesson 1: 2015 Challenge — Mathematics

# Infinitesimals and non-standard analysis

By Shaun Regenbaum

This video intuitively explains infinitesimals and the basics of Non-Standard Analysis. There are simplifications of advanced mathematics, just beware. Thanks to DM Ashura (Bill Shillito) for his awesome music and his constant support for my love of math and science. He is truly an inspiration.

## Want to join the conversation?

• When he puts the triangles into the circle, there is still a space between the base of each triangle and the outside of the circle (which he points out). Then, he bases the whole equation on the fact that the base of the triangle is the circumference of the circle. But they are NOT equal! The base of the triangle should be less than the circumference. What am I missing? (around )
• They are not equal. The height of the triangle is slightly less than R due to the missing section and you can't stack the triangle up next to each other to form a bigger triangle, that assumption doesn't work . In his example the hypotenuse is longer than R so it can't be one of the triangles from the circle.
• At around , I don't understand how the base of the final triangle would be the circumference of the circle. I mean, the base of each small triangle is not equal to the curvy bit underneath it (since a curved line reaching from point a to be would be bigger than a straight one going from a to b), meaning that if you add all the triangle bases together, they would not equal the circumference. I'm probably wrong but I just don't get it. Can anyone explain it?
• What is the Zorn's Lemma and what are ultrafilters
• Wait earlier in the video, didn't he say that 0.9 recurring and 1 were equal on the real numberline, because there wasn't any real number to represent the change difference(which was a infinitesimal), but 0.9 repeating and 1 were actually different and unequal. He then proceeds to say, later in the video, infinitesimals explain why 0.9 repeating equals 1...