The Look and Feel of Zero

The purpose of this exploration is to discover the various numerical mainfestations of what you would expect to be zero.

Try these exercises in both Claris Works and Excel. Compare and try to explain your results. If you have a calculator, also compare with that.

What is the largest number x for which 1 + x = 1?
What is the largest number x for which cos(x) = 1?
What is the largest number x for which exp(x) = 1?
What is the smallest positive number x that you can compute log(x)?
What is the largest number x for which sqrt(1 + x) = 1?
What is the largest number x for which 1 + x*x = 1?
Keep track of how many guesses it takes you to find the answers. Think about using a Bisection method for searching. Can you think of a better way to speed up your search for your numbers?
Can you think of other numerical measurements of zero? Try them!
What are the largest numbers for various situations the computer can compute?
Do your results from these investigations tell you anything about your ability to use a computer to model various systems? What precautions would you take? What approximations would you have to make?