# Did chaos cause mayhem in Jurassic Park?

I’ll happily link to this very readable introduction to chaos theory and the butterfly effect: http://plus.maths.org/content/did-chaos-cause-mayhem-jurassic-park

A sampling:

Suppose that we want to predict the future state of a system — the weather, for example — that is sensitive to initial conditions. We could measure its current state, and then iterate the system’s governing function on that seed value. This would yield an answer, but if our measurement of the system’s current state had been slightly imprecise, then the true result after a few iterations might be wildly different. Since empirical measurement with one hundred percent precision is not possible, this makes the predictive power of the model more than a few time-steps into the future essentially worthless.

The popular buzz-word for this phenonemon is the butterfly effect, a phrase inspired by a 1972 paper by the chaos theory pioneer Edward Lorenz. The astounding thing is that the unpredictability arises from a deterministic system: the function that describes the system tells you exactly what its next value will be. Nothing is left to randomness or chance, and yet accurate prediction is still impossible. To describe this strange state of affairs, Lorenz reportedly used the slogan

Chaos: When the present determines the future, but the approximate present does not determine the approximate future.

Chaotic dynamics have been observed in a wide range of phenomena, from the motion of fluids to insect populations and even the paths of planets in our solar system.

# Different definitions of e: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on the different definitions of $e$ that appear in Precalculus and Calculus.

Part 1: Justification for the formula for discrete compound interest

Part 2: Pedagogical thoughts on justifying the discrete compound interest formula for students.

Part 3: Application of the discrete compound interest formula as compounding becomes more frequent.

Part 4: Informal definition of $e$ based on a limit of the compound interest formula.

Part 5: Justification for the formula for continuous compound interest.

Part 6: A second derivation of the formula for continuous compound interest by solving a differential equation.

Part 7: A formal justification of the formula from Part 4 using the definition of a derivative.

Part 8: A formal justification of the formula from Part 4 using L’Hopital’s Rule.

Part 9: A formal justification of the continuous compound interest formula as a limit of the discrete compound interest formula.

Part 10: A second formal justification of the continuous compound interest formula as a limit of the discrete compound interest formula.

Part 11: Numerical computation of $e$ using Riemann sums and the Trapezoid Rule to approximate areas under $y = 1/x$.

Part 12: Numerical computation of $e$ using $\displaystyle \left(1 + \frac{1}{n} \right)^{1/n}$ and also Taylor series.

# Different definitions of logarithm: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on how the different definitions of logarithm are in fact equivalent.

Part 1: Introduction to the two definitions: an antiderivative and an inverse function.

Part 2: The main theorem: four statements only satisfied by the logarithmic function.

Part 3: Case 1 of the proof: positive integers.

Part 4: Case 2 of the proof: positive rational numbers.

Part 5: Case 3 of the proof: negative rational numbers.

Part 6: Case 4 of the proof: irrational numbers.

Part 7: Showing that the function $f(x) = \displaystyle \int_1^x \frac{dt}{t}$ satisfies the four statements.

Part 8: Computation of standard integrals and derivatives involving logarithmic and exponential functions.

# Improvisation in the Mathematics Classroom

Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight “Improvisation in the Mathematics Classroom” by Andrea Young. Here’s the abstract:

This article discusses ways in which improvisational comedy games and exercises can be used in college mathematics classrooms to obtain a democratic and supportive environment for students. Using improv can help students learn to think creatively, take risks, support classmates, and solve problems. Both theoretical and practical applications are presented.

Full reference: Andrea Young (2013) Improvisation in the Mathematics Classroom, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 23:5, 467-476, DOI: 10.1080/10511970.2012.754809

# What to Expect When You’re Expecting to Win the Lottery

I can’t think of a better way to tease this video than its YouTube description: