A number r is rational if it can be written as (p)/(q), where p and q are integers and q ≠ 0. An irrational number cannot be expressed in the form (p)/(q). The collection of all rational numbers and irrational numbers together forms the set of real numbers R.

A number r is rational if it can be written as (p)/(q), where p and q are integers and q ≠ 0. An irrational number cannot be expressed in the form (p)/(q). The collection of all rational numbers and irrational numbers together forms the set of real numbers R.

Chapter 1 number system give me all formulas

Answer: Summary of NCERT Class 9 Chapter 1 (Number Systems) Formulas: 1. Real Numbers: $\mathbb{R} = \text{Rational Numbers } (\mathbb{Q}) \cup \text{Irrational Numbers}$ . 2. Rational: $\frac{p}{q}$ ( $q \neq 0$ , $p, q \in \mathbb{Z}$ ), decimal is terminating or non-terminating repeating. 3. Irrational: Non-terminating non-repeating decimal expansion. 4. Surd Identities ( $a, b > 0$ ): - $\sqrt{ab} = \sqrt{a}\sqrt{b}$ - $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ - $(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a-b$ - $(a+\sqrt{b})(a-\sqrt{b}) = a^2-b$ - $(\sqrt{a}+\sqrt{b})^2 = a+2\sqrt{ab}+b$ 5. Rationalisation: $\frac{1}{\sqrt{a}+\sqrt{b}} = \frac{\sqrt{a}-\sqrt{b}}{a-b}$ 6. Laws of Exponents ( $a, b > 0$ ): - $a^m \cdot a^n = a^{m+n}$ - $(a^m)^n = a^{mn}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $a^m b^m = (ab)^m$ - $a^0 = 1$ , $a^{-n} = \frac{1}{a^n}$ , $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$

Step-by-step solution

Step 1: Classification of Numbers

A number $r$ is rational if it can be written as $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ . An irrational number cannot be expressed in the form $\frac{p}{q}$ . The collection of all rational numbers and irrational numbers together forms the set of real numbers $\mathbb{R}$ .

Step 2: Decimal Expansions of Real Numbers

The decimal expansion of a rational number is either terminating or non-terminating recurring (repeating). Conversely, the decimal expansion of an irrational number is non-terminating and non-recurring.

Step 3: Identities for Positive Real Numbers (Square Roots)

For any positive real numbers $a$ and $b$ , these algebraic identities help simplify surds and square root expressions. They directly mirror the standard algebraic identities $(x+y)(x-y) = x^2 - y^2$ and $(x+y)^2 = x^2 + 2xy + y^2$ .

Step 4: Rationalising the Denominator

When the denominator of an expression contains a term with a square root, multiplying both numerator and denominator by its conjugate rationalises the denominator, turning it into a rational number.

Step 5: Laws of Exponents for Real Numbers

Let $a > 0$ and $b > 0$ be real numbers, and let $m$ and $n$ be rational numbers. These fundamental exponent laws govern multiplication, division, and powers of real numbers.

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