Problem 15 - Beginning and Intermediate Algebra

Question

Give a number that satisfies the given condition. An irrational number that is between \(\sqrt{12}\) and \(\sqrt{14}\)

Step-by-Step Solution

Answer

An example is \(\sqrt{13}\) or the number \(\pi\).

Step 1: Understand the Condition

Identify the given condition in the problem: find an irrational number between \(\sqrt{12}\) and \(\sqrt{14}\). Irrational numbers are numbers that cannot be expressed as a fraction where both numerator and denominator are integers.

Step 2: Approximate the Bounds

Calculate approximate values for the bounds. \(\sqrt{12}\) is approximately 3.464 and \(\sqrt{14}\) is approximately 3.742.

Step 3: Identify an Irrational Number

Choose an irrational number between these two approximate values. One easy option is \(\pi - 3\), which is approximately 3.14159 - 3 = 0.14159. Scale this by adding 3 to both sides to get 3.14159.

Step 4: Verification

Verify that the chosen number, approximately 3.14159, lies between 3.464 and 3.742. Since \(\pi\) is known to be irrational, \(\pi\) (which is 3.14159...) clearly satisfies the given condition.

Key Concepts

Square Roots

Square roots are fundamental in mathematics. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9.

There's an essential distinction between perfect squares (like 4, 9, and 16) and non-perfect squares (like 12 and 14). When you take the square root of a perfect square, you get a whole number. But for non-perfect squares, the result is not a whole number. This leads us into the realm of irrational numbers.

In our exercise, \( \sqrt{12} \ \) and \( \sqrt{14} \ \) are non-perfect square roots, meaning they are irrational and not exact. They're approximately 3.464 and 3.742 respectively.

Approximations

When dealing with square roots of non-perfect squares, we often use approximations to make calculations easier. Approximations are rough estimates of a number.

For instance:

• \( \sqrt{12} \ \) is approximately 3.464
• \( \sqrt{14} \ \) is approximately 3.742

Approximations help us understand where an irrational number falls on the number line. These rounded figures make it easier to identify a number between them.

In this context, \( \pi \ - 3 \ \) is chosen because approximated to 3.14159, it lies between the bounds.

Irrational Number Properties

Irrational numbers cannot be written as simple fractions. They have non-repeating, non-terminating decimal parts. For instance, \( \pi \ \) (3.14159...) is irrational because its decimal representation doesn't end or repeat.

Other famous examples include \( \sqrt{2} \ \) and Euler's number \( \ e \ \). These numbers are critical in various branches of mathematics.

In the given problem, we needed an irrational number between \( \sqrt{12} \ \) and \( \sqrt{14} \ \). By selecting \( \pi - 3 \ \) and verifying its properties, we confirmed \( \pi - 3 + 3 = \pi \ \) lies comfortably between 3.464 and 3.742.

This concludes that \( \pi \ \) satisfies the conditions of the exercise, highlighting the distinct characteristics of irrational numbers.