Are all irrational numbers rational?

Since pie is irrational, it cannot be exactly represented by a fraction. In summary, the answer is no, there does not exist a fraction for all nonrepeating going on forever decimal values.
  • #1
Skhandelwal
400
3
Since pie is the ratio of the circumference of the circle to its diameter, isn't it possible that there exist a fraction for all nonrepeating going on forever decimal values?
 
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  • #2
Skhandelwal said:
Since pie is the ratio of the circumference of the circle to its diameter, isn't it possible that there exist a fraction for all nonrepeating going on forever decimal values?

Short answer: No.

Tongue-in-cheek answer: Yes, but the fraction would have at least one noninteger.

Longer answer: You're wrongly assuming that a circle with rational circumference and diameter exists.
 
  • #3
? are all humans non human? are all m ortals immortal? are al...
 
  • #4
mathwonk said:
? are all humans non human? are all m ortals immortal? are al...

lol :rofl:
 
  • #5
Skhandelwal said:
Since pie is the ratio of the circumference of the circle to its diameter, isn't it possible that there exist a fraction for all nonrepeating going on forever decimal values?
The definition of "rational number" is that it can be written as a fraction with numerator and denominator integers. The "ratio of the circumference of the circle to its diameter" is not a ratio of integers.
 

1. What is an irrational number?

An irrational number is a real number that cannot be expressed as a ratio of two integers. This means that it cannot be written as a fraction in the form of p/q, where p and q are integers and q is not equal to zero.

2. Are all irrational numbers also real numbers?

Yes, all irrational numbers are also real numbers. Real numbers include both rational and irrational numbers.

3. Can irrational numbers be written in decimal form?

Yes, irrational numbers can be written in decimal form. However, unlike rational numbers, the decimal form of an irrational number is non-terminating and non-repeating.

4. Are there more irrational numbers than rational numbers?

Yes, there are infinitely more irrational numbers than rational numbers. In fact, the set of irrational numbers is uncountable, meaning that there is no way to list all of them in a finite amount of time.

5. Why is it important to know about irrational numbers?

Understanding irrational numbers is important in various fields of study, such as mathematics, physics, and engineering. They are used in calculations and equations to represent precise values that cannot be expressed as rational numbers. Additionally, irrational numbers have unique properties and patterns that have led to important discoveries in mathematics and science.

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