CBSE Class 10 Maths Notes: Real Numbers

Definitions

• Natural Numbers (N): The counting numbers: {1, 2, 3, 4, …}.
• Whole Numbers (W): Includes natural numbers and zero: {0, 1, 2, 3, …}.
• Integers (Z): Include whole numbers and their negatives: {…, -3, -2, -1, 0, 1, 2, 3, …}.
• Rational Numbers (Q): Numbers that can be expressed in the form $p/q$, where $p$ and $q$ are integers, and $q \ne 0$. They can be represented as terminating or repeating decimals.
• Irrational Numbers (I): Numbers that cannot be expressed in the form $p/q$. Their decimal representations are non-terminating and non-repeating (e.g., $\sqrt{2}$, $\pi$).
• Real Numbers (R): The set of all rational and irrational numbers.

Core Principle

Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

Examples

* 12 = 2 x 2 x 3 = $2^2 \times 3$

* 35 = 5 x 7

* 100 = 2 x 2 x 5 x 5 = $2^2 \times 5^2$

Formulaes

• $a^m \times a^n = a^{m+n}$ (Product of powers)
• $\frac{a^m}{a^n} = a^{m-n}$ (Quotient of powers)
• $(a^m)^n = a^{m \times n}$ (Power of a power)
• $a^0 = 1$ (Zero exponent)
• $a^{-n} = \frac{1}{a^n}$ (Negative exponent)
• $\sqrt[n]{a} = a^{\frac{1}{n}}$ (Radical as a fractional exponent)

Basic Operations and Simplification

* Addition and Subtraction: Simplify radicals before adding or subtracting (e.g., $2\sqrt{3} + 4\sqrt{3} = 6\sqrt{3}$).

* Multiplication: Multiply the coefficients and the radicands (e.g., $\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4$).

* Division: Rationalize the denominator if needed (e.g., $\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$).

Algebraic Proofs (Example: $\sqrt{2}$ is irrational)

1. Assume the opposite: Assume $\sqrt{2}$ is rational. This means we can write $\sqrt{2} = \frac{p}{q}$, where $p$ and $q$ are integers, and $q \ne 0$, and $\frac{p}{q}$ is in its simplest form (i.e., $p$ and $q$ have no common factors other than 1).
2. Square both sides: $(\sqrt{2})^2 = (\frac{p}{q})^2$, which simplifies to $2 = \frac{p^2}{q^2}$.
3. Rearrange: Multiply both sides by $q^2$: $2q^2 = p^2$. This tells us that $p^2$ is an even number (since it’s a multiple of 2).
4. If $p^2$ is even, then $p$ is even: (Proof by contradiction: If $p$ were odd, $p^2$ would also be odd). Therefore, we can write $p = 2k$ for some integer $k$.
5. Substitute: Substitute $p = 2k$ into the equation $2q^2 = p^2$: $2q^2 = (2k)^2 = 4k^2$.
6. Simplify: Divide both sides by 2: $q^2 = 2k^2$. This means $q^2$ is also even, and consequently, $q$ is also even.
7. Contradiction: We initially assumed that $p$ and $q$ have no common factors other than 1. However, we have now shown that both $p$ and $q$ are even, meaning they share a common factor of 2. This contradicts our initial assumption.
8. Conclusion: Since our initial assumption led to a contradiction, the assumption must be false. Therefore, $\sqrt{2}$ is not rational; it is irrational.

Examples

* Finding the HCF and LCM: Using prime factorization to determine the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two or more numbers. For example, find the HCF and LCM of 12 and 18.

* Word Problems: Solving problems related to real-life scenarios that involve prime factorization, HCF, or LCM. For instance: “Find the largest number that divides 24 and 36 without leaving a remainder.”

* Proving Properties: Applying the Fundamental Theorem of Arithmetic and properties of real numbers to solve and prove properties of numbers.