Problem 4 - Prealgebra

Question

Name all of the sets of numbers to which each real number belongs. Let \(\mathbf{N}=\) natural numbers, \(\mathbf{W}=\) whole numbers, \(\mathbf{Z}=\) integers, \(\mathbf{Q}=\) rational numbers, and I = irrational numbers. $$\sqrt{12}$$

Step-by-Step Solution

Answer

\( \sqrt{12} \) is an irrational number.

Step 1: Simplify the Square Root

First, try to simplify the square root if possible. \( \sqrt{12} \) can be simplified as \( \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \).

Step 2: Recognize Type of Number

Since \( \sqrt{3} \) cannot be simplified to a rational number and stays in the form of an irrational number, \( 2\sqrt{3} \) is also an irrational number. Thus, \( \sqrt{12} \) belongs to the set of irrational numbers \( I \).

Step 3: Conclusion of Number Set

As \( \sqrt{12} \) was simplified to an irrational form \( 2\sqrt{3} \), it only belongs to the set of irrational numbers \( I \). It does not belong to \( \mathbf{N} \), \( \mathbf{W} \), \( \mathbf{Z} \), or \( \mathbf{Q} \) because it is not a whole, natural, integer, or rational number.

Key Concepts

Irrational Numbers

Irrational numbers are those that cannot be expressed as a simple fraction or ratio of two integers. This means if a number cannot be accurately expressed in the form \( \frac{a}{b} \), where both \(a\) and \(b\) are integers and \(b eq 0\), it is considered irrational. Some common examples of irrational numbers include \(\pi\), \(e\), and square roots of non-perfect squares like \(\sqrt{2}\) and \(\sqrt{3}\).

• They have non-repeating, non-terminating decimal expansions.
• Irrational numbers are represented with the symbol \(I\).
• The sum or product of a rational and an irrational number is always irrational (as long as the rational part is not zero in multiplication).

For instance, in the original exercise, simplifying \( \sqrt{12} \) results in \( 2\sqrt{3} \), where \( \sqrt{3} \) cannot be simplified to a rational number, confirming that \( 2\sqrt{3} \) is irrational.

Square Root Simplification

Square root simplification helps in expressing a square root as simply as possible by factoring out perfect squares. To simplify a number under the square root, you look for factors that are perfect squares (like 4, 9, 16, etc.) and factor them out.

• Find factors of the number inside the square root that are perfect squares.
• Use the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) for simplification.
• Replace perfect squares with their root values.

As in our example, \( \sqrt{12} \) can be broken down into \( \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \). This method shows that after simplification, we get an expression that is often easier to work with, especially if further calculations are needed.

Number Sets

Number sets are categories used to classify numbers based on their properties, helping in understanding their nature and usage in mathematics.

Natural Numbers (\(\mathbf{N}\))

Natural numbers are positive integers starting from 1, like 1, 2, 3, and so on. These do not include zero or any fractions/decimals.

Whole Numbers (\(\mathbf{W}\))

Whole numbers include all natural numbers plus zero. They are non-negative integers.

Integers (\(\mathbf{Z}\))

Integers consist of all whole numbers and their negative counterparts, including zero. This means numbers like -3, 0, and 4 are all integers.

Rational Numbers (\(\mathbf{Q}\))

Rational numbers can be expressed as a fraction of two integers. For example, \( \frac{1}{2} \), 0.75, and -2 all fit into this set because they can be rewritten as fractions.

Irrational Numbers (\(I\))

As stated earlier, irrational numbers cannot be written as simple fractions. Numbers like \(\sqrt{12}\), after simplification to \(2\sqrt{3}\), remain irrational as they cannot be expressed as fractions.By understanding these sets, we can identify the group any given number belongs to, aiding in mathematical problem-solving processes.