Real Numbers Class 10 Mathematics Concept notes and Formula Sheet

 

Concept Notes

1. Definition of Real Numbers

• Real numbers include all the numbers that can be found on the number line. This set comprises both rational and irrational numbers. Real numbers can be positive, negative, or zero.

2. Types of Real Numbers

• Natural Numbers (N): Numbers used for counting, e.g., 1, 2, 3, …
• Whole Numbers (W): Natural numbers including zero, e.g., 0, 1, 2, 3, …
• Integers (Z): Whole numbers including negative numbers, e.g., -3, -2, -1, 0, 1, 2, 3, …
• Rational Numbers (Q): Numbers that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. Examples include $\frac{1}{2}, -\frac{3}{4}, 5$.
• Irrational Numbers: Numbers that cannot be expressed as a fraction, with non-repeating, non-terminating decimal expansions. Examples include $\sqrt{\mathrm{2, }}$$\pi$

3. Properties of Real Numbers

• Closure Property:
  • Addition: $a + b$ is a real number if $a$ and $b$ are real numbers.
  • Multiplication: $a \times b$ is a real number if $a$ and $b$ are real numbers.
• Commutative Property:
  • Addition: $a + b = b + a$
  • Multiplication: $a \times b = b \times a$
• Associative Property:
  • Addition: $(a + b) + c = a + (b + c)$
  • Multiplication: $(a \times b) \times c = a \times (b \times c)$
• Distributive Property:
  • $a \times (b + c) = (a \times b) + (a \times c)$
• Identity Elements:
  • Addition: The identity element is 0, i.e., $a + 0 = a$
  • Multiplication: The identity element is 1, i.e., $a \times 1 = a$
• Inverse Elements:
  • Addition: The additive inverse of $a$ is $-a$, i.e., $a + (-a) = 0$
  • Multiplication: The multiplicative inverse of $a$ (where $a \neq 0$) is $\frac{1}{a}$, i.e., $a \times \frac{1}{a} = 1$

4. Absolute Value

• The absolute value of a real number $a$ is denoted as $|a|$ and represents its distance from 0 on the number line.
  • For $a \geq 0$, $|a| = a$
  • For $a < 0$, $|a| = -a$

5. Real Number Line

• The real number line is a line on which every real number is represented by a unique point. It extends infinitely in both directions.

6. Decimal Representation

• Terminating Decimals: Decimal representations that come to an end, e.g., 0.5, 1.75.
• Non-Terminating, Repeating Decimals: Decimal representations that continue indefinitely with a repeating pattern, e.g., $\frac{1}{3} = 0.\overline{3}$.
• Non-Terminating, Non-Repeating Decimals: Decimal representations that continue indefinitely without repeating, e.g., $\sqrt{2}$, $\pi$.

7. Rationalization

• The process of converting an expression involving a square root in the denominator into an equivalent expression without a square root in the denominator.

Formula Sheet

1. Properties of Real Numbers:

   • Closure Property: $$a + b \in \mathbb{R}, \; a \times b \in \mathbb{R}$$
   • Commutative Property: $$a + b = b + a$$ $$a \times b = b \times a$$
   • Associative Property: $$(a + b) + c = a + (b + c)$$ $$(a \times b) \times c = a \times (b \times c)$$
   • Distributive Property: $$a \times (b + c) = (a \times b) + (a \times c)$$
   • Identity Elements:
     • Addition: $a + 0 = a$
     • Multiplication: $a \times 1 = a$
   • Inverse Elements:
     • Addition: $a + (-a) = 0$
     • Multiplication: $a \times \frac{1}{a} = 1$ (for $a \neq 0$)

2. Absolute Value:

   $$|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$$

3. Decimal Representations:

   • Terminating Decimal: Example: 0.25
   • Repeating Decimal: Example: $\frac{2}{3} = 0.\overline{6}$
   • Non-Repeating Decimal: Example: $\sqrt{2} \approx 1.414$

4. Rationalization:

   • ​$$\frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a}$$
   • ​$$\frac{1}{a + \sqrt{b}} \times \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b}$$

Examples for Practice

1. Identify whether $\frac{5}{7}$ and $\sqrt{3}$ are rational or irrational numbers.
2. Simplify the absolute value expressions $| -7 |$ and $| 3 - 8 |$.
3. Use the properties of real numbers to verify the following:
   • $3 \times (4 + 5) = (3 \times 4) + (3 \times 5)$
   • $(7 + 2) + 3 = 7 + (2 + 3)$
4. Rationalize the denominator of $\frac{5}{\sqrt{2}}$.