The discussion centers on the representation of π as an irrational number and its implications in mathematics. Participants assert that π, as the ratio of a circle's circumference to its diameter, cannot be expressed as a ratio of two rational numbers, similar to √2. The conversation highlights the limitations of the base 10 numeration system in fully capturing the value of π, emphasizing that while π is a definite value, it cannot be precisely described within our current numerical framework. The necessity of irrational numbers in mathematics is underscored, as they fill the gaps in the number line and ensure the completeness of mathematical theorems.
PREREQUISITESMathematicians, educators, students in advanced mathematics, and anyone interested in the foundational concepts of irrational numbers and their role in mathematical theory.
π is special because of its geometrical importance as the universal ratio of the circumference to the diameter of a circle.Premanand said:If pi is a part of the area of a perfect circle, which I assume we can construct. Why does it have uncertainty?.
Thank you. But it still puzzles me why we have such numbers? If the side has a irrational number in its measurements, like the square root of 2 you mentioned or the value of pi in its area, what does it really say us? Is it the limitation of our number system or is it something special about the geometry that I cannot comprehend ? because one side of the triangle is 1 units which we know for sure and the other is also 1 units but the third one according to us is 1.41421356237... and it never ends...Adel Makram said:π is special because of its geometrical importance as the universal ratio of the circumference to the diameter of a circle.
It is also an irrational number such as √2 which can not be put as a ratio of rational numbers or a number consists of decimals of certain pattern. If you draw a right triangle with units sides, then√2 becomes the largest side. √2 is a number and it is well defined in the paper where you draw the triangle but still can not be put as a ratio of 2 rational numbers.
Thanks Jonny. But you have mentioned '(the number line would have too many "gaps")'. By this we assume the gap between two numbers. Say in the number line of integers 1 2 3 4 ... and the gap between them occupied by rational like 1 (1.0-1.9) 2 (2.0 - 2.9) 3 (3.0 - 3.9) ... and the gap between the rational which has infinite possibilities is occupied by irrational ex (1.0 - 1.999999999999999999999...) which never ends. Is it not more like approximation of accuracy? Say if I calculate the area of circle and use pi r square in my equation, depending on which value I use for pi (3.14 or 3.14159) my accuracy will increase. Is it not the same when I say my side of triangle is 1 unit ? When I measure accurately enough my side of the triangle can be 1.001 or even 1.000000000001. So that inaccuracy is related with all numbers we quote be it integer or rational or irrational. Why does it come specifically to hypotenuse of a triangle or area of circle then ? Why should I say my radius is a finite rational number but my area is infinite irrational number?JonnyG said:I'm sure others can give better answers but:
1) I am pretty sure that regardless of what base you choose, there will be irrational numbers
2) mathematics would be boring without the irrational numbers. If all we had were the rationals, we would lose a lot of our theorems (the number line would have too many "gaps")
JonnyG said:I'm sure others can give better answers but:
1) I am pretty sure that regardless of what base you choose, there will be irrational numbers
2) mathematics would be boring without the irrational numbers. If all we had were the rationals, we would lose a lot of our theorems (the number line would have too many "gaps")
Thank you. I am sorry that I confused measurements in engineering and mathematics. I have this habit. But I think I get your answer. 'The fact that we cannot write every decimal position in the "base 10 numeration system" only says that our numeration system is insufficient' - it means that pi is a definite value (ratio of area to circumference) , but couldn't be definitely described by our number system. Same is the case with root of 2 (which is somewhere between 1.41 and 1.42) .. In some other system pi would have a definite description , but as Jonny said would be incomplete in other terms... And that Our numeration system is not good enough to describe the area of circle in absolute terms and just mentioning it as some value of pi completes it... Does it mean all irrational numbers are limitations... Or rather limitation adjustersHallsofIvy said:You seem to be very confused about numbers. There is no uncertainty in \pi- we know exactly what it is. The fact that we cannot write every decimal position in the "base 10 numeration system" only says that our numeration system is insufficient, not that we do not know what \pi is. Further 1.99999999... (never ending) also has a very specific value- it is 2. When you are talking about mathematics there is no "measuring" involved so we don't need to worry about "accuracy". If you are talking about specific physics or engineering applications of mathematics, that is a different question- but it is no longer mathematics.
Yes, Jonny, the definition of "rational" and "irrational" numbers is independent of the numeration system
What sequences?HallsofIvy said:That is a technical term referring to convergence of sequences. Just as JonnnyG said, there exist Cauchy sequences of rational numbers that do not converge to any rationa numbers. So the set of rational numbers is not "complete". The "number line", by which we normally mean all real number, both rational and irrational, is complete.
Sequences of numbers, of course. A sequence of numbers, a_1, a_2, ..., a_n, ... is called a "Cauchy sequence" as long as |a_n- a_m| goes to 0 as m and n go to infinity. It can be shown that any convergent sequence is a "Cauchy sequence" and that any Cauchy sequence will converge to a real number but a Cauchy sequence of rational numbers will not necessarily converge to a rational number.Josh S Thompson said:What sequences?