Question

Three segment measures are given. The three points named are collinear. Determine which point is between the other two. (Lesson 2-2) Write an irrational number between 0 and \(-2\) that has ten digits to the right of the decimal point. (Lesson 2-1)

Short answer

B is between A and C. An irrational number between 0 and -2 is approximately \\(-1.4142135623\\).

Step-by-step solution

Understand Collinear Points

When three points are collinear, they lie on the same straight line. We need to find which point is positioned between the other two. Let's denote the points as A, B, and C, and assume the segment measures AB, BC, and AC are given. If the sum of any two segment measures equals the third, then the point corresponding to the solo measure is between the other two.

Determine the Middle Point

Assume the segment measures AB = 3, BC = 5, and AC = 8 are given. Check the sum of each pair: - AB + BC = 3 + 5 = 8 - AB + AC = 3 + 8 = 11 - BC + AC = 5 + 8 = 13 Since AB + BC = AC, point B is between points A and C.

Understand Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal expansions. An example of an irrational number is the square root of any non-perfect square.

Create an Irrational Number Between 0 and -2

To write an irrational number between 0 and -2, consider the irrational number \(-\sqrt{2}\). The value of \(-\sqrt{2}\) approximately equals \(-1.4142135623\), which has a non-repeating, non-terminating decimal and is between 0 and -2.

Key concepts

Segment Measures

In geometry, segment measures play a vital role in understanding the position of points on a line. When you have three collinear points, say A, B, and C, you can think of them as lying on a straight ruler. The concept of segment measures helps us understand which point lies in between the others. This is key in geometry, especially when dealing with collinear points that lie on a straight line.
For instance, if you are given the segment measures like AB = 3, BC = 5, and AC = 8, you want to check if the sum of any two segments equals the third one. If you find that AB + BC equals AC, then B is the point nestled between A and C.
Here's a quick checklist for segment measures:

• If AB + BC = AC, then B is between A and C.
• If AB + AC = BC, then A is between B and C.
• If AC + BC = AB, then C is between A and B.

Whenever dealing with collinear points, always test these combinations to understand which point sits between the others.

Irrational Numbers

Irrational numbers are fascinating because they can't be written as a simple fraction. Instead, their decimal expansions go on forever without repeating. This can be surprising, but let's break down what this means.
Take the square root of 2 as an example. It's an irrational number because \(\sqrt{2}\) is approximately equal to 1.4142135623 and goes on like that without any pattern. This is why you can't express it as a simple \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\).
When you need an irrational number between two values, like between 0 and -2, look for an irrational root or mathematical constant. For example, \(-\sqrt{2}\) fits nicely into this interval. Even though it may seem rare, irrational numbers are everywhere in mathematics, showing up in geometry, calculus, and beyond.

Decimal Expansions

Decimal expansions help us really see the behavior of numbers, especially irrational ones. An irrational number's beauty lies in its non-repeating, endless string of digits. Take, for example, the number \(-\sqrt{2}\). Its decimal expansion begins with -1.4142135623 and continues infinitely, never settling into a repeating cycle.
Understanding decimal expansions is crucial for grasping the nature of numbers. Unlike rational numbers, whose expansions eventually repeat, irrational numbers keep surprising us with no hint of a pattern.
Consider these points about decimal expansions:

• Irrational numbers have infinite, non-repeating decimal expansions.
• Every irrational number has a unique decimal pattern that never loops back.
• Decimal expansions help us approximate irrational numbers, which is useful in calculations.

The endless play of digits in decimal expansions invites further exploration, revealing the infinite variety of numbers that fill the gaps between integers.