Uncertainty and Approximations

Estimations

• Most math problems you have seen have been carefully created for you to solve them exactly
• Most problems that need to be solved are not so tidy
• Physicist Enrico Fermi was famous for posing these types of questions
• Precise vs Approximate reasoning

We don’t always know quantities precisely. We also don’t always need to know the results of our estimations precisely. We will get comfortable with working with lots of uncertainty in our estimates.

Calculations and Approximations

• What is 1 + 1?
• What is 10 + 1?
• What is one thousand + 1?
• What is one million + 1?
• We see that adding one doesn’t matter in some cases.

Approximations

• For many problems, only the first digit and the number of zeros is important

Easier Multiplication

• Imagine we want to multiply several large numbers but only want a rough approximation.
• How can we do that?

Significant Figures

• Each significant figure that we add to the multiplication affects the accuracy less than the previous