Important equations and numbers to memorize

Many students in science and engineering today claim that the need to memorize certain numbers or equations is outdated with the advent of the internet and the ease with which we may find information. As an undergraduate, I often heard my engineering colleagues complain when they were not offered an equation sheet for a test because real-world engineers don't have to look up "well-known" equations when solving problems.

However, I contend that there are advantages to memorizing a few important equations and numbers. These reasons include the following:

1. You may work faster since you will not need to constantly reference those equations.
2. You may mentally check your work to ensure that the numbers are correct.
3. You may mentally approximate the solutions to certain calculations if you don't have access to reference material. I remember reading once how Feynman would challenge mathematicians by approximating powers of numbers in his head by knowing the natural logarithms of 2 and 10.
4. You may impress the ladies with your abilities (not guaranteed).

So what are some equations and constants I've found useful throughout the years? Here is my list:

1. The binomial expansion
2. The Taylor expansion
3. The power series representation of e 
4. The speed of light in vacuum and the impedance of free space (about 377 Ohms)
5. The refractive index of common glass (~1.5) and water (~1.33)
6. Boltzmann's constant and the thermal energy at 300 K (4.14 Joules) and thermal voltage at 300 K 0.026 eV)

Other important things to memorize, depending on your field, include:

1. Stirling's approximation
2. The Krebs cycle (a biologist once told me that many ideas in molecular biology come back to this)

Consider these as tools in your toolbox. They can make your job a lot easier.

Addendum: Not memorizing the basics is a mistake to avoid in physics according to this Back Page of APS News.