Problem 120 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a calculator to find a decimal approximation for each irrational number, correct to three decimal places. Between which two integers should you graph each of these numbers on the number line? $$\sqrt{3}$$

Step-by-Step Solution

Answer

The decimal approximation for \( \sqrt{3} \) is 1.732. Therefore, it is between the integers 1 and 2 on the number line.

Step 1: Calculate Decimal Approximation

Using a calculator, find a decimal approximation for \( \sqrt{3} \) correct to three decimal places.

Step 2: Determine the Two Integers

It is then necessary to analyze the calculated number to find which two integers these numbers fall in between of.

Key Concepts

Understanding Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as simple fractions. That means they can't be represented exactly by a simple ratio of two integers. An intriguing characteristic of an irrational number is that its decimal representation is non-terminating and non-repeating. Examples include \( \pi \) and \( \sqrt{2} \) among others.

These numbers frequently occur in various mathematical and real-world contexts, ranging from geometry to the growth patterns of natural phenomena. It is often necessary for students to approximate these irrational numbers to make them easier to work with, usually by rounding them to a certain number of decimal places as we do in calculators or when comparing to rational numbers.

The Mystery of Square Roots

Square roots form a critical part of our discussion on irrational numbers. To understand what a square root is, picture a square with an area of \( x \) square units. The side of this square would be \( \sqrt{x} \) units long.

The square root of a number is a value that, when multiplied by itself, gives the original number. While some numbers have square roots that are themselves whole numbers—like \( \sqrt{4} \) being \( 2 \)—others, like \( \sqrt{3} \) in our exercise, do not. In fact, square roots of non-perfect squares are irrational. This is why we need to approach them through approximation to make them manageable.

Graphing on the Number Line

Finding a Home on the Number Line

After working out a decimal approximation for an irrational number, the next challenge is determining where it belongs on the number line. In a mathematical and graphical context, a number line allows us to visually represent numbers and their relationships to each other.

Each point on the number line corresponds to a number, and each number corresponds to a unique point on the line. This rule holds true for irrational numbers, too, even though we never find them occupying a precise 'tick' on the line because of their non-repeating, non-terminating decimal nature. Instead, we identify the integers between which these approximated irrational numbers would fall, which helps in graphing and understanding their approximate value relative to known quantities.