Problem Solving

I want to share my approach to solving problems. These techniques are especially suited for math problems, but they work for most other problems I've solved. I believe that a lot of problem solving is organizing your work and information. When a problem is confusing, it is probably because it hasn't been drawn out in an organized and systematic way. I typically separate my work into a few sections:

• Restatement of the problem, both written and verbal.
• A diagram or drawing of the problem to visualize the symbols. This is probably the most important process for me because my ability to see visual patterns and transformations is strong. This and the previous step will help you think about and define symbols that you introduce in the next step.
• Table of symbols, with column headers for values (initial or other), the value's measurement unit, and description of what the symbol represents. This table should also include conditions on the variables you've introduced e.g. statements like "x is positive" or "V not equal to 0".
• A written statement of approach that describes how the problem should generally be solved. This statement is a prediction of the necessary steps to get an answer. When solving programming problems, this is usually done with pseudocode.
• Manipulation and reconfiguration of equations or functions e.g. differentiation, integration, refactoring. It  is always good to state what facts you use to make jumps in logic for your own reference and for others to follow along. The more you explain yourself in writing or another durable way, the better value your work has as a reference for yourself in the future and for others.
• Numerical calculations. This can be a subtle step that ruins your answer. Do calculations carefully, substitute the correct values by referring to symbol table and only after you've fully completed the last step. Be particularly wary of sign.
• Solution and concluding remarks.  View all your solutions as something you'll need to go back to in the future when you forget, so write down a little journal entry stating the process and how it differed from what you expected.

This is a short list, but it is very helpful for approaching many problems. Another great resource for problem solving is Stewart Calculus: Principles of Problem Solving.  If you want to master the problem, teach it and speak the solution to someone. 

Slide Rules

I spent a good portion of the day learning about slide rules because I received my Concise No. 28 Circular Slide rule yesterday and I want to start using it for calculations. I believe that it encourages good calculation habits because it demands efficient calculations. I looked for reference materials by searching through manuals at http://sliderule.ozmanor.com/man/man-download.html and I found that the Keuffel & Esser manual had the best information on the theory of slide rules. It quickly allowed me to grasp the ideas behind a slide rule and therefore its operation.

NB: I only understood the manual because I understand the properties of logarithms and exponents. Please read up on that before you try to understand the following breakdown of what I learned.

The section on slide rule theory starts by comparing logarithmic slide rule operation to linear slide ruler operation. With the linear slide ruler, one can perform addition and subtraction by setting units together side by side. Since we can perform addition and subtraction with rulers, we can do something more interesting -- add and subtract scaled units. It turns out that using logarithmic scaled units performs multiplication and division.

Most operations can be done with plain logarithmic scales. I won't go into the plain scales, but I will go into a cool variant called the "folded scale," which typically starts at Pi, but could start at any number. If the folded scale starts at Pi (in other words, if the index is Pi), when the slide is closed, Pi is above "1" on the regular scale and the indexes are lined up. So, by moving the cursor to any position on the regular scale, the number it lands on will be the multiplier of Pi. The product of Pi and this number will be under the cursor on the folded scale. The folded scale also goes through a change in distance equal to the change you make on the unfolded scale. Because of this, products that fall off the end of a linear slide rule scale can usually be picked up on the folded scale and read off. On my circular slide rule, I don't need to worry about ever falling off the edge because a circle has no edges.

The inverse scale is a variant of the original scale. Simply, it is the original logarithmic scale laid down from right to left instead of from left to right. It is laid down like this because the entire length of the original scale is log(10). When you set the cursor, you break it into two pieces. One piece is length log(N) and the other length log(10) - log(N) = log(10/N). Since log(10/N) is the same as the inverse with a different characteristic, if you measure it, you are measuring log(1/N). So, the cursor would be on 1/N if you lay a scale down from the right.

The greatest power of a slide rule is not in doing single multiplication and division problems. It's greatest power is to view scale (K x) and proportion (a/b = c/d)  relationships at a glance. By multiplying by a number, K, you can find all products of K by setting the cursor to your desired multiplicand. By setting two numbers on top of each other, you are in effect setting all other marks between the scales to reflect that proportion. To find an equivalent proportion, simply set the cursor. It's remarkable that we don't use slide rules still because of this amazing property.

After writing this, I realized that the slide rule is an example of the broader topic of transformations. Transformations in general are used to make operations easier or faster to compute. The most famous in electrical engineering are the Laplace, Fourier, and Phasor transforms. Where logarithms transform the operation of multiplication to the easier operation of addition, these transformations change functions to make differential equations easier to solve.

Why Blog?

This blog will serve as my lab notebook for any experiments or projects that I undertake. While I don't necessarily have grand hypotheses to test, I would like to at the very least record the smaller hypotheses I make on a daily basis and the processes I go through to test them as I build or repair things. My posts should share the most important things that I learn on any given day that help me achieve my project goals. I would like posts to follow this skeleton (unless I have something random to share)

1. A problem and my process, apparatus, and solution
2. One thing I learned
3. New capabilities I gained from what I learned
4. Mistakes I made while trying to solve the problem
5. Strategies for solving a given problem and next steps if no solution is found
6. Experiment log and results
7. Diagrams, pictures, and memory aids
8. Links to relevant resources

I hope that by having a good place to post my work, I will do better about recording my progress and create a good routine. The blog description is loose enough to allow for mathematical content and engineering content, so I'll do my best to post on both subjects. If I include any mathematical expressions, I will post them as images, but you can contact me to receive LaTeX source. Same goes for any programs I post about.