Problem 20 - 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. $$2.8$$

Step-by-Step Solution

Answer

The number 2.8 belongs to the rational numbers (\(\mathbf{Q}\)).

Step 1: Identify the Nature of the Real Number

The number is 2.8, which is a decimal. A decimal number cannot be a natural number (\(\mathbf{N}\)), as natural numbers do not include decimals or fractions. Similarly, it cannot be a whole number (\(\mathbf{W}\)), which includes 0 and positive integers without decimals.

Step 2: Determine if it is an Integer

Integers (\(\mathbf{Z}\)) are whole numbers including zero and negatives without fractions or decimals. Thus, 2.8, having a decimal part, cannot be an integer.

Step 3: Assess if it is Rational

Rational numbers (\(\mathbf{Q}\)) are numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. The number 2.8 can be expressed as \(\frac{28}{10}\) or further simplified to \(\frac{14}{5}\), making it a rational number.

Step 4: Examine for Irrationality

Irrational numbers (\(I\)) cannot be expressed as a fraction of two integers. Since 2.8 can be written as a fraction, it is not an irrational number.

Key Concepts

Number Sets

Number sets are groups of numbers defined by shared properties. They help us categorize numbers based on their characteristics.

Some common number sets you should know about include:

• Natural numbers ( 13;
  78;): Whole numbers starting from 1, used in counting.
• Whole numbers ( 13;
  78;): Natural numbers plus zero.
• Integers ( 13;
  78;): Whole numbers that also include negative numbers.
• Rational numbers ( 13;
  78;): Numbers that can be expressed as fractions.
• Irrational numbers ( 13;
  78;): Numbers that cannot be expressed as simple fractions.
• Real numbers: Includes both rational and irrational numbers, covering all possible numbers we encounter in everyday calculations.

Identifying the correct set helps in understanding the number's properties.

Real Numbers

Real numbers are a vast set comprising all possible numbers that can be plotted on a continuous line.

Real numbers include:

• Positive numbers
• Negative numbers
• Zero
• Both rational and irrational numbers

The unique thing about real numbers is that they fill up every single point on the number line. This includes perfect squares like 4, as well as decimals like 2.8.

Integers

Integers are an interesting group. They include all whole numbers plus their negative counterparts.

Key characteristics of integers:

• Include numbers like -3, -2, -1, 0, 1, 2, 3, etc.
• Excludes fractions and decimals.
• Can be positive or negative.

In the number line, integers are represented by evenly spaced points.
Though infinities exist, each integer is separate and distinct.

Natural Numbers

Natural numbers are those you first learned when you began counting.

These numbers start at 1 and climb up forever like 1, 2, 3, and so on.

• First numbers children learn.
• Used for counting everyday objects.

These do not include zero, negatives or decimals.
Natural numbers are foundational in mathematics and are the building blocks for other sets you'll learn about.

Whole Numbers

Whole numbers resemble natural numbers but with a small twist.

They include all natural numbers and add the number zero into the mix.

• Starts with 0, then continues: 1, 2, 3, ...
• These numbers are non-negative.
• Include all the figures you might use for counting items including none.

This small inclusion of zero makes whole numbers very important in mathematical operations, filling in the gap left by natural numbers.

Irrational Numbers

Irrational numbers are fascinating because they can't be expressed as simple fractions.

You'll often encounter them in various situations, notably:

• The number 13;
  7c;, an infinitely non-repeating decimal.
• The square root of numbers like 13;
  78;(2) which cannot be simplified into a fraction.

This means they can be written as decimals, but these decimals neither repeat nor terminate.
Irrational numbers fill in the gaps on the number line between the rationals, creating a complete picture of real numbers.