几何与区间

Introduction: From Algebra Back to Geometry

The Three Elements of a Number Line and Constructing Rational Points

Which of the following statements about the three elements of a number line is correct?

Compass-and-Straightedge Construction of the Square Root of 2 and Postulate 1.9

Postulate 1.9 asserts: after fixing the origin, unit length, and positive direction, between the points on the line and the real numbers there is—

Types of Intervals

Which of the following numbers belongs to the open interval $(1,3)$?

Example: Marking the Square Root of 2 and the Square Root of 3 with Compass and Straightedge

How many integers are in the open interval $(1,3)$? Enter an integer.

Write the inequality $2\le x\le 5$ in interval notation. Enter the interval symbol.

What interval corresponds to the inequality $x<2$ or $x\ge 3$? Write it using the union symbol.

What do square brackets and parentheses denote in intervals?

In your own words, explain: why does the $\infty$ end of an interval always use a parenthesis rather than a square bracket? How is this related to Postulate 1.9?

Comprehensive Check