1. The Number System: A Quick Overview

To understand Real Numbers, we first need to look at the building blocks of the number system:

1. Natural Numbers ($\mathbb{N}$): These are the basic counting numbers starting from 1.

   • Example: $\{1, 2, 3, 4, \dots\}$

2. Whole Numbers ($\mathbb{W}$): These include all natural numbers plus zero.

   • Example: $\{0, 1, 2, 3, 4, \dots\}$

3. Integers ($\mathbb{Z}$ or $\mathbb{I}$): This set includes zero, positive whole numbers, and their negative counterparts.

   • Example: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$

4. Rational Numbers ($\mathbb{Q}$): Numbers that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

5. Irrational Numbers ($\mathbb{Q}^*$): These are numbers that cannot be represented in the $\frac{p}{q}$ form. They have non-repeating, non-terminating decimals.

2. What are Real Numbers ($\mathbb{R}$)?

Real Numbers are the set of all rational and irrational numbers combined. If you can place it on a continuous number line, it is a real number.

Formula: $\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}^*$

Examples of Real Numbers:

• Fractions & Decimals: $\frac{1}{2}, -\frac{3}{5}, 0.1, \frac{3}{5}$

• Roots (Radicals): $\sqrt{2}, \sqrt{5}, 2\sqrt{2}, 2+\sqrt{2}, \sqrt{5}+\sqrt{2}$

• Mathematical Constants:

  • $\pi \approx 3.14 \dots \approx \frac{22}{7}$

  • $e \approx 2.71 \dots$

Notation Tip: The symbol $\in$ means “belongs to.” So, $\sqrt{5} \in \mathbb{R}$ is read as “$\sqrt{5}$ belongs to the set of Real Numbers.”

3. Additions and Subtractions of Real Numbers

When adding or subtracting real numbers, especially those involving roots (irrationals), you treat the roots like variables. You can only add or subtract “like terms” (numbers with the same root).

Example 1: Basic Addition

Problem: Add $(2 + \sqrt{3})$ and $(3 + 2\sqrt{3})$

Solution:

1. Write the expression: $(2 + \sqrt{3}) + (3 + 2\sqrt{3})$

2. Remove parentheses: $2 + \sqrt{3} + 3 + 2\sqrt{3}$

3. Group whole numbers and like roots: $(2 + 3) + (\sqrt{3} + 2\sqrt{3})$

4. Final Result: $5 + 3\sqrt{3}$

Example 2: Working with Different Roots

Problem: Add $(2\sqrt{2} + 5\sqrt{3})$ and $(\sqrt{2} – 7\sqrt{3})$

Solution:

1. Write the expression: $(2\sqrt{2} + 5\sqrt{3}) + (\sqrt{2} – 7\sqrt{3})$

2. Remove parentheses: $2\sqrt{2} + 5\sqrt{3} + \sqrt{2} – 7\sqrt{3}$

3. Group the $\sqrt{2}$ terms and the $\sqrt{3}$ terms:

   • $(2\sqrt{2} + \sqrt{2}) = 3\sqrt{2}$

   • $(5\sqrt{3} – 7\sqrt{3}) = -2\sqrt{3}$

4. Final Result: $3\sqrt{2} – 2\sqrt{3}$

Example 3: Subtracting Expressions with Radicals

When subtracting one real number expression from another, it is crucial to distribute the negative sign to every term inside the parentheses of the expression being subtracted.

Problem:

Subtract $3 + \sqrt{5}$ from $7 – 6\sqrt{5}$.

Solution:

1. Set up the expression: Place the term you are subtracting from first, and put the term being subtracted in parentheses.
   $$(7 – 6\sqrt{5}) – (3 + \sqrt{5})$$
2. Distribute the negative sign: Remove the parentheses by changing the signs of the terms inside.
   $$7 – 6\sqrt{5} – 3 – \sqrt{5}$$

3. Group and combine like terms:

   • Combine the whole numbers: $7 – 3 = 4$.

   • Combine the radical terms: $-6\sqrt{5} – \sqrt{5} = -7\sqrt{5}$.

4. Final Result:
   $$4 – 7\sqrt{5}$$