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Rational and Irrational Numbers

1. Introduction to the Number System

The complete number system is divided into two main categories: Imaginary numbers (e.g., the square root of negative numbers) and Real numbers.
Real Numbers consist of Rational Numbers and Irrational Numbers.
Rational numbers further branch into Integers (Positive, Negative, and Zero) and Non-Integral Rationals (fractions).

A rational number is any number that can be expressed in the form a/b, where 'a' and 'b' are integers and 'b' is not equal to zero.
Every integer and every decimal number is a rational number.
For standard rational numbers, 'a' and 'b' are co-primes (no common factor other than 1), and 'b' is usually positive.
Two rational numbers a/b and c/d are equal if and only if a × d = b × c.
There are infinitely many rational numbers between any pair of rational numbers. A simple way to find one rational number between 'a' and 'b' is to use the formula (a + b) / 2.

To find 'n' rational numbers between 'x' and 'y' (where x < y), first calculate a common difference d = (y - x) / (n + 1).
The required rational numbers will then be x + d, x + 2d, x + 3d, ..., x + nd.
Alternative Method: Make the denominators of the given fractions equal by finding the L.C.M., and then multiply both the numerator and denominator by (n + 1) to create enough spacing to easily pick numbers between them.

The sum, difference, and product of two or more rational numbers is always a rational number (This is known as the closure property).
The division of a rational number by a non-zero rational number is always a rational number.

Every rational number can be expressed either as a terminating decimal or a non-terminating recurring decimal.
Terminating Decimals: Occur when the division is exact (no remainder). A rational number can be expressed as a terminating decimal if its denominator (in lowest form) can be prime factored strictly into powers of 2 and 5 (i.e., 2^m × 5ⁿ).
Non-Terminating Recurring Decimals: Occur when the division never ends, but a digit or set of digits repeats continuously (the "period"). Indicated by a dot or bar over the repeating digits.
Conversion: To convert a recurring decimal back to a fraction, set it to an equation (let x = ...), multiply by powers of 10 to shift the decimal point past the repeating parts, and subtract the equations to solve for x.

Irrational numbers include the square roots, cube roots, etc., of natural numbers whose exact values cannot be obtained (e.g., √2, √3).
They are represented by non-terminating and non-recurring decimals.
The mathematical constant π is an irrational number (22/7 is merely an approximate rational value used for calculation).
To find an irrational number between two positive numbers 'a' and 'b' (where their product is not a perfect square), use the formula √(ab).
The sum, difference, or product of two irrational numbers may or may not be irrational. However, the product of a non-zero rational number and an irrational number is always irrational.

The Real Numbers (R) system is simply the union of the set of rational numbers and the set of irrational numbers.

If 'x' is a positive rational number and 'n' is a positive integer greater than 1, such that the n^th root of x (ⁿ√x) is irrational, it is called a surd or radical of order n.
Key Rule: Every surd is an irrational number, but every irrational number is not necessarily a surd (for example, π is irrational but not a surd).

Concept: When two surds are multiplied together to produce a rational number, they are called rationalising factors of each other.
Rationalising the Denominator: To simplify an expression with a surd in the denominator, you multiply both the numerator and the denominator by the lowest rationalising factor of the denominator. (e.g., multiply by √2 / √2 to remove a √2 from the bottom).

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