Key Points

A number 'r' is called a rational number if it can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. The collection is denoted by $\mathbf{Q}$.

A number 's' is called an irrational number if it cannot be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Examples include $\sqrt{2}$, $\sqrt{3}$, and $\pi$.

Real numbers are the complete set of all rational and irrational numbers combined. Every point on the number line represents a unique real number, and this is why it is called the real number line.

The decimal expansion of a rational number is either terminating (e.g., $\frac{7}{8} = 0.875$) or non-terminating recurring (e.g., $\frac{1}{3} = 0.333... = 0.\overline{3}$).

The decimal expansion of an irrational number is always non-terminating and non-recurring. This means it goes on forever without a repeating block of digits.

To convert a recurring decimal like $x = 0.\overline{27}$ to a fraction, multiply by a power of 10 to shift the decimal. Here, $100x = 27.\overline{27}$, so $99x=27$, which gives $x = \frac{27}{99} = \frac{3}{11}$.

The sum or difference of a rational and an irrational number is always irrational. The product or quotient of a non-zero rational number and an irrational number is also irrational.

Rationalizing the denominator is the process of converting an irrational denominator to a rational one. For an expression like $\frac{1}{\sqrt{a}}$, we multiply by $\frac{\sqrt{a}}{\sqrt{a}}$ to get $\frac{\sqrt{a}}{a}$.

To rationalize a denominator of the form $\sqrt{a} + \sqrt{b}$, we multiply the numerator and denominator by its conjugate, $\sqrt{a} - \sqrt{b}$. The result uses the identity $(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a-b$.

For positive real numbers $a$ and $b$, the identity is $(\sqrt{a} + \sqrt{b})^2 = a + 2\sqrt{ab} + b$.

For positive real numbers $a$ and $b$, two key identities are $\sqrt{ab} = \sqrt{a}\sqrt{b}$ and $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

For any positive real number $a$ and rational exponents $p$ and $q$, we have $a^p \cdot a^q = a^{p+q}$. When multiplying powers with the same base, add the exponents.

For any positive real number $a$ and rational exponents $p$ and $q$, we have $(a^p)^q = a^{pq}$. To raise a power to another power, multiply the exponents.

For any positive real number $a$ and rational exponents $p$ and $q$, we have $\frac{a^p}{a^q} = a^{p-q}$. When dividing powers with the same base, subtract the exponents.

For positive real numbers $a, b$ and a rational exponent $p$, the rule is $a^p b^p = (ab)^p$.

For a positive real number $a$ and a rational exponent $\frac{m}{n}$ where $n > 0$, we define $a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$. For example, $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$.