How to Understand the Real Number Line

Step 1: Distinction Between Rational and Irrational Numbers

• Rational Numbers (Q): These can be expressed as the quotient or fraction \(\frac{p}{q}\)​ of two integers, where \(p\) (numerator) and \(q\) (denominator) are integers and \(q≠0\). Examples include \(\frac{1}{2}, 5, -3\), etc.
• Irrational Numbers: Numbers that can’t be written as simple fractions or ratios. This set includes numbers like \(π\) and \(\sqrt{2}\)​, which don’t terminate or repeat in their decimal form.

Step 2: Bridging The Gap – The Union

Step 3: Visualizing the Infinite: The Real Number Line

Step 4: Diving Deeper: Intervals and Bounds

• Open Intervals: Represented as \((a, b)\), it includes all numbers between \(a\) and \(b\) but not the endpoints themselves.
• Closed Intervals: \([a, b]\) includes all the numbers between and including \(a\) and \(b\).
• Infinite Intervals: These can be open or closed on one side and stretch infinitely on the other, like \([a, ∞)\) or \((-∞, b]\).

Step 5: Density Property

Step 6: Beyond the Real – A Glimpse

Step 7: Conclusion

Examples:

Locate \(−2.5\)​ on the number line.

Solution:

Consider the number \(−2.5\). This is a rational number, and on the real number line, it would be located to the left of \(0\) and halfway between \(-2\) and \(-3\).