RATIONAL & IRRATIONAL NUMBERS

Rational numbers are a fundamental set of numbers in mathematics. They are defined as any number that can be expressed as a fraction where the numerator and denominator are both integers (whole numbers) and the denominator is not zero. Examples of rational numbers include 1/2, -3/4, 5, and 0.

Rational numbers form a subset of real numbers, meaning they exist on the number line along with integers, decimals, and irrational numbers

What is a Rational Number?

A rational number is any number that can be expressed as the quotient or fraction p​/q, where $p$ and $q$ are integers and $q$ is not equal to zero. In other words, a rational number is the result of dividing one integer by another, and it can be represented as a fraction where the numerator ($p$) and denominator ($q$) are integers.

Key characteristics of rational numbers:

Rational numbers can be represented as fractions, decimals that either terminate or repeat, and integers (since every integer can be expressed as a fraction with a denominator of 1)

Excluded Values: The denominator ($q$) in a rational number cannot be zero, as division by zero is undefined.

Closure under Operations: Rational numbers are closed under addition, subtraction, multiplication, and division, meaning that the result of these operations between two rational numbers is always another rational number.

Examples:1/2, -3/5, 2.5, and even 7 are all happy members of the rational number family

On the number line, rational numbers can be located at specific points, and they include integers as well. Rational numbers provide a way to represent quantities that can be expressed as a ratio of two integers, making them a fundamental concept in mathematics

How to identify rational numbers?

To identify rational numbers, you need to look for numbers that can be expressed as a fraction, where the numerator and denominator are integers, and the denominator is not zero.

Here's how you can identify rational numbers:

Types of Rational Numbers

Positive Rational Numbers:

These are rational numbers greater than zero.

Examples:

• 3/4
• 5/2
• 7/7

Negative Rational Numbers:

These are rational numbers less than zero.

Examples:

• -5/3
• -1/2
• -8/8

Whole Numbers:

Whole numbers are rational numbers with a denominator of 1.

Examples:

• 3
• 0
• -9

Integers:

Integers include positive and negative whole numbers as well as zero.

Examples:

• -5
• 2
• 0

Proper Fractions:

Proper fractions have numerators smaller than their denominators.

Examples:

• 1/3
• 5/7

Improper Fractions:

Improper fractions have numerators equal to or greater than their denominators.

Examples:

• 7/4
• 10/3

Mixed Numbers:

Mixed numbers are a combination of a whole number and a proper fraction.

Examples:

• 2 1/4
• -3 2/5

Terminating Decimals:

Decimals that have a finite number of digits after the decimal point.

Examples:

• 0.75
• -2.4

Repeating Decimals:

Decimals that have a repeating pattern of digits.

Examples:

• 0.333...
• 1.234‾

Reciprocal of Rational Numbers:

The reciprocal of a rational number is obtained by swapping the numerator and the denominator.

Example:

The reciprocal of 3/5 is 5/3.

Image Source: Geeks for Geeks

Arithmetic Operations on Rational Numbers

Rational numbers, the ones expressed as fractions (p/q, where q ≠ 0), are not immune to the power of arithmetic! Let's dive into the rules and methods for adding, subtracting, multiplying, and dividing these friendly number fractions.

Addition and Subtraction:

• Same Denominator: If the fractions have the same denominator (like 1/4 and 3/4), simply add or subtract the numerators, keeping the denominator the same. (1/4) + (3/4) = 4/4 = 1.
• Different Denominators: Find the least common multiple (LCM) of the denominators, then convert each fraction to have the LCM as its denominator. Add or subtract the numerators, keeping the common denominator. (1/3) + (2/5) = (5/15) + (6/15) = 11/15.

Multiplication:

Multiply both the numerators and the denominators of each fraction. Simplify the resulting fraction if possible. (2/3) * (4/7) = (2 * 4) / (3 * 7) = 8/21.

Division:

Flip the second fraction (the divisor) to its reciprocal. Then, multiply the first fraction by the reciprocal. Simplify as needed. (3/4) / (5/6) = (3/4) * (6/5) = (3 * 6) / (4 * 5) = 9/10

Rational and Irrational Numbers

Rational Numbers:

• Fractional Friends: These are the numbers you can express as a fraction (p/q, where q ≠ 0), like 1/2, 3/4, or even -5/7.
• Decimals Defined: They can also be represented as terminating or repeating decimals, like 0.75 or 2.34234...
• Arithmetic Champs: Addition, subtraction, multiplication, and division with rational numbers follow specific rules, and the result is always another rational number.
• Examples Everywhere: From ratios in recipes to probabilities in games, rational numbers are at the heart of many real-world applications.

Irrational Numbers:

• Elusive Companions: These are numbers that cannot be expressed as a simple fraction, no matter how hard you try! Examples include pi (3.14159...), the square root of 2 (√2), and the golden ratio (Φ).
• Decimal Mysteries: Their decimal representations never end or repeat in a predictable pattern, keeping them shrouded in a veil of mathematical intrigue.
• A Touch of Infinity: Some irrational numbers, like pi, are believed to extend infinitely without repeating, adding to their mysterious charm.
• Found in Unexpected Places: From the circumference of circles to the proportions of nature's creations, irrational numbers play a surprising role in the world around us