Problem 91 - Contemporary Precalculus

Question

Find the distance between the given numbers. \(\sqrt{2}\) and \(\sqrt{3}\)

Step-by-Step Solution

Answer

Answer: The distance between the numbers √2 and √3 is (√3 - √2).

Step 1: Calculate the difference

Find the difference between the two given numbers: \(\sqrt{3} - \sqrt{2}\)

Step 2: Obtain the absolute value of the difference

To find the distance between these numbers, we have to make sure to take the absolute value of the difference. The difference is already positive, so the absolute value is the same: \(|\sqrt{3} - \sqrt{2}| = \sqrt{3} - \sqrt{2}\)

Step 3: Present the final result

The distance between the numbers \(\sqrt{2}\) and \(\sqrt{3}\) is the absolute difference: \(distance = \sqrt{3} - \sqrt{2}\)

Key Concepts

Absolute Value

The absolute value of a number measures its distance from zero on the number line, without considering which direction from zero the number lies. It is always a non-negative number. To denote the absolute value of a number, we use two vertical bars, like this: \( |x| \). For example, \( |-5| = 5 \) and \( |5| = 5 \).

When it comes to the distance between two numbers, the absolute value becomes particularly useful. The distance between any two numbers, say \( a \) and \( b \), can be represented as the absolute value of their difference: \( |a - b| \). This formula is applicable for any real numbers, whether they are whole numbers, fractions, or even irrational numbers like square roots. Mathematically, we're saying that the distance between \( a \) and \( b \) is the same as the absolute value of \( a \) minus \( b \), or vice versa. As shown in the given exercise, \( |\sqrt{3} - \sqrt{2}| \) represents the distance between the two square roots.

Square Roots

Square roots fall under the umbrella of roots and radical expressions in mathematics. The square root of a number \( x \) is a number \( y \) that when multiplied by itself gives \( x \). This is written as \( y = \sqrt{x} \). Square roots are the opposite of squaring a number. For example, since \( 3^2 = 9 \), it follows that \( \sqrt{9} = 3 \).

When working with square roots, it's important to acknowledge that they represent the principal (or positive) root. In the context of finding distances in the exercise, we only consider the positive square root because distances are absolute values and cannot be negative. Additionally, operations involving square roots can sometimes result in irrational numbers, which do not have a precise decimal or fractional representation but can be approximated for practical uses.

Precalculus Arithmetic

Precalculus arithmetic encompasses the mathematical operations and concepts that prepare students for the study of calculus. These include properties of functions, complex numbers, and the manipulation of algebraic expressions. Among these concepts, understanding how to operate with roots, powers, and rational expressions is vital.

Learning precalculus arithmetic involves dealing with the addition, subtraction, multiplication, and division of real numbers, which can also include irrational numbers like \( \sqrt{2} \) and \( \sqrt{3} \). Having the skills to handle such arithmetic operations is crucial when you're calculating distances between numbers, as seen with the absolute value in our exercise example. Practicing these precalculus skills aids in developing strong foundations for more advanced mathematical concepts that are encountered in calculus and beyond.