1 Important Things to Review This Week

1. Absolute & Relative Error
2. Algebra
3. Limits
4. Derivatives
5. Integrals (anti-derivatives)
6. Series & Sums
7. Numeric Bases
   • Base 10 (decimal)
   • Base 2 (binary)
   • Base 8 (octal)
   • Base 16 (hexadecimal)

2 A Physics Example - Time Dilation

During my undergrad I minored in Physics. Quite a few of my stories are based on lessons I learned from Physics coursework. I have always been fascinated by science fiction (especially the Star Trek and Stargate franchises). Special Relativity comes up quite bit in science fiction.

One day I became curious… How much time dilation do I experience in my one hour commute? I ended up hacking together some quick-and-dirty C++ code. I ran into one of the first issues we will discuss this semester, finite precision, specifically rounding/truncation error.

However, finite precision is not our topic of interest here (not yet). Our interest is unfamiliar notation.

Lorentz Factor

$$ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}} } \rightarrow \gamma^{-1} = \alpha = \sqrt{1 - \frac{v^2}{c^2}} $$

Introduce More Notation

Let $\beta = \frac{v}{c} \rightarrow \gamma^{-1} = \alpha = \sqrt{1 - \beta^{2}}$

Much of what we will cover this semester will seem overwhelming, especially during the first 1 to 2 weeks. Do not psych yourself out. It is just notation.

Series Expansions

Eventually I ended up using series expansions to manipulate the $\gamma$ and time dilation equations.

Apply Taylor Series expansion for $\sqrt{1-x^2}$:

$$ 1 - \frac{x^2}{2} - \frac{x^4}{4} - \frac{x^6}{16} - \ldots $$

Or apply the Maclaurin Series expansion for

$$ \begin{array}{ll} \gamma &=& \frac{1}{\sqrt{1-\beta^2}} \\ & \approx & 1 + \frac{1}{2}\beta^{2} \end{array} $$

For $v \ll c$, this collapses to Newtonian Mechanics, i.e., $\gamma^{-1} = \sqrt{1-(\frac{v}{c})^{2} } \approx 1$.

3 What is the Point?

1. Finite Precision & Approximation

   • How is each number represented (i.e., represented in base-2)?
   • What is the error in representing a single number?

2. Error Propagation

   • How does error propagate through arithmetic operations?

3. Conditioning

   • How does a small change in input affect the result of a computation?
   • Can we trust the results of the computation?

4. Problem Domain

   • What does the problem represent?
   • Is there another method?

5. Notation

   • What does everything represent?

6. Know the Fundamentals

   • What math tricks can be used (e.g., change of variable and L’Hopital’s Rule)?