Irrational Numbers

[Guest]

I just wanted to clarify this general doubt regarding irrational numbers.

Irrational numbers can be represented on the real number line. Thus, they are a part of the real numbers.

But, if a number like pi (3.14159…) goes on forever, how can we represent it on the number line accurately?

Irrational numbers cannot be expressed as a ratio of two integers (p/q, q ≠ 0), therefore they are not a part of the rational numbers.

But, irrational numbers can be represented as a ratio of two integers right?
For example, pi can be represented as circumference by diameter.
Root 2 can be represented as diagonal by side of a square.

I’m quite confused.

Thank you for your time.

[Guest]

A number "going on forever" has nothing to do with it being on the number line. 1/3, written as a decimal, also goes on forever. Do you have a problem with 1/3 being represented on the number line? $\pi$ is 3.1416926... so lies between 3 and 4. That gives a region on the number line. $\pi$ lies between 3.1 and 3.2. $\pi$ lies between 3.14 and 3.15. Since its decimal form "goes on forever" we can place it more and more accurately to its precise position on the number line.

"But irrational numbers can be represented as integers, right?
For example, pi can be represented as circumference by diameter.
Root 2 can be represented as diagonal by side of a square."

But those are not both integers. Precisely because [tex]]\pi[/tex] is not rational, the radius and circumfernce of a given circle cannot both be integers. Are you clear on what an "integer" is?