Rational Numbers

What Are Rational Numbers?

A rational number is any number that can be expressed in the form of a fraction $( \frac{m}{n} )$, where $( m )$ and ( n ) are integers and $( n \neq 0 )$. Examples include:

• $( \frac{1}{2} )$
• $( -\frac{3}{4} )$
• 5 (which can be written as $( \frac{5}{1} ))$

In rational numbers, both positive and negative fractions exist. If $( n )$ is negative, the rational number is negative.

Textual Diagram: Rational Number Representation

Positive Rational Numbers: 1/2, 3/4, 5/6
Negative Rational Numbers: -1/2, -3/4, -5/6
Integers as Rational Numbers: 5 = 5/1, -3 = -3/1

Representation on the Number Line

Rational numbers can be represented on a number line. For instance:

• $( \frac{1}{2} )$ lies halfway between 0 and 1.
• $( -\frac{3}{4} )$ lies three-quarters of the way from 0 to -1.

Example:

0  ------ 1/2 -------1

-1 ------- -3/4 -----0

Equivalent Fractions and Cross-Multiplying

Two rational numbers $( \frac{m}{n} ) and ( \frac{r}{s} )$ are considered equivalent if:

$m \times s = n \times r$

This is known as cross-multiplying.

Example:

• $( \frac{2}{3} = \frac{4}{6} ) because ( 2 \times 6 = 3 \times 4 = 12 )$.

Textual Diagram:

Cross-multiplying Example: 

  2 x 6 = 3 x 4
  12 = 12 (True, so they are equivalent)

Addition and Subtraction of Rational Numbers

Addition with Common Denominators:

If two rational numbers share a common denominator, you simply add the numerators.

$\frac{a}{n} + \frac{b}{n} = \frac{a + b}{n}$

Example:

$\frac{3}{5} + \frac{2}{5} = \frac{3 + 2}{5} = \frac{5}{5} = 1$

Addition with Different Denominators:

When denominators differ, you need to find a common denominator:

$\frac{m}{n} + \frac{r}{s} = \frac{ms + nr}{ns}$

Example:

$\frac{1}{2} + \frac{1}{3} = \frac{1 \times 3 + 1 \times 2}{2 \times 3} = \frac{3 + 2}{6} = \frac{5}{6}$

Subtraction:

The process is similar to addition, except you subtract the numerators:

$\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$

Textual Diagram for Addition and Subtraction:

3/4   -   1/4
 |----|----|----|
 0    1/4  2/4  3/4 1

Result: 2/4 = 1/2

Multiplication and Division of Rational Numbers

Multiplication:

To multiply two rational numbers, multiply the numerators and denominators:

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

Example:

$\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$

Division:

To divide two rational numbers, multiply by the reciprocal:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Example:

$\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$

Textual Diagram for Multiplication:

2/3 x 3/4

(2 x 3)/(3 x 4) = 6/12 = 1/2

Simplifying Rational Numbers

Rational numbers should be presented in their simplest form by dividing the numerator and the denominator by their greatest common divisor (GCD).

Example:

$\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$

Textual Diagram: Simplification

Original: 8/12
Step 1: Find GCD = 4
Step 2: Divide 8/12 by 4
Result: 2/3

Conclusion

Rational numbers play a critical role in mathematics, providing a way to represent quantities, perform calculations, and understand concepts such as probability and data distributions in AI/ML/Data Science. Mastering these concepts will help you build a strong foundation for more advanced topics.

Keep practicing, and you’ll soon be comfortable with rational numbers!

This blog post covers the essentials of rational numbers with examples, diagrams, and explanations that make it comprehensive and easy to follow. You can directly add this to your GitHub Pages blog by placing it in the Markdown file format.