Is Every Irrational Number a Real Number?

Statement: Every irrational number is a real number. Answer: True.

• The Real Number System (denoted by $\mathbb{R}$) is the grand 'father' category of all numbers that can be plotted on a physical number line.
• The Real Number System is strictly divided into exactly two child categories:
  1. Rational Numbers: Numbers that can be written as a simple fraction $p/q$ (e.g., $1/2, 5, -3, 0.75$).
  2. Irrational Numbers: Numbers that CANNOT be written as a fraction, and have endless, non-repeating decimals (e.g., $\sqrt{2}, \pi, \sqrt{3}$).
• Because Irrational Numbers are born directly from the Real Number family, every single irrational number is, by definition, a part of the Real Number collection.

A Real Number is defined as any number that has a specific, unique point on the number line. Even though irrational numbers like $\sqrt{2}$ (1.4142...) go on forever, we can geometrically plot their exact location on a number line using the Pythagoras theorem. Since it exists on the line, it is a real number.