A teacher asks his students to write the numbers shown in order from least to greatest. Paul thinks the numbers are already in order. Sandra thinks the order should be reversed. Who is right?

Math History

There is a famous irrational number called Euler’s number, symbolized with an \(e\). Like \(\pi\), its decimal form never ends or repeats. The first few digits of \(e\)  are 2.7182818284.

a. Between which two square roots of integers could you find this number?

b. Between which two square roots of integers can you find \( \pi\)?

Analyze Relationships 

There are several approximations used for \(\pi \), including 3.14 and \(\frac{22}{7}\). \(\pi \) is approximately 3.14159265358979 . . .

a. Label \(\pi \) and the two approximations on the number line.

b. Which of the two approximations is a better estimate for \( \pi\)? Explain.

c. Find a whole number  \(x\) so that the ratio \(\large\frac{x}{113}\) is a better estimate for \( \pi \) than the two given approximations.

Communicate Mathematical Ideas

What is the fewest number of distinct points that must be graphed on a number line, in order to represent natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers? Explain.

Critique Reasoning

Jill says that \(12.\overline{6}\) is less than 12.63. Explain her error.