Line segment of length pi (Just a thought I've had)

In summary, the conversation discusses the idea of measuring the length of a line segment with an irrational number, such as pi. It is argued that while a physical line segment cannot be precisely measured, it can be mathematically measured with an interval of length pi. However, at small scales, the uncertainty of atomic dimensions and quantum mechanics would prevent the possibility of measuring exactly pi. The conversation also touches upon the difference between physical lines and geometric lines.
  • #1
eg2333
6
0
If you were to imagine a line segment of length pi, I would guess it would have to be finite. But since pi is an irrational number, it has infinitely many decimals so can't you just keep sort of zooming in on the end of the segment so that it sort of keeps on getting longer indefinitely?

Pi is just an example, but I'm sure any irrational number would bring up the same idea. Any thoughts on this?
 
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  • #2
NOT being able to precisely measure the length of the segment in this case, does not imply that the segment is/gets indefenitely long(er).
 
  • #3
But what do you mean by "precisely measure"? If you are talking about using some kind of measuring device then, of course, it cannot be "precisely measured". No interval can in that sense. If you mean "mathematically", in the same sense that we talk about an interval "of length 1", then it can be "precisely measured". Measure out an interval of length 1 and construct a circle about one end of that interval having the interval as radius. The circle will be of length [itex]]pi[/itex]. An interval of length "pi" is no different from an interval of any other length.
 
  • #4
eg2333 said:
If you were to imagine a line segment of length pi, I would guess it would have to be finite. But since pi is an irrational number, it has infinitely many decimals so can't you just keep sort of zooming in on the end of the segment so that it sort of keeps on getting longer indefinitely?

Well, 3.141592653 is certainly longer than 3.1415, but 3.1415 isn't pi, so no, it is NOT getting longer.
 
  • #5
No, because at sufficiently small scales, atomic dimensions and quantum uncertainty would prevent the possibility of having something mesuring exactly [tex]\pi[/tex]. The same applies for any irrational number.
 
  • #6
JSuarez said:
No, because at sufficiently small scales, atomic dimensions and quantum uncertainty would prevent the possibility of having something mesuring exactly [tex]\pi[/tex]. The same applies for any irrational number.

Huh, are people aware that physical lines and geometric lines are two different things?

Lines and line segments are abstract ideas. They're not what you draw on a piece of paper, nor are they anything that we see. To say that a physical line segment has irrational length is completely meaningless, as it is to say that a physical segment has some precise length.
 

1. What is a line segment of length pi?

A line segment of length pi is a straight line that measures exactly pi units in length. Pi, denoted by the Greek letter π, is an irrational number that represents the ratio of a circle's circumference to its diameter.

2. How is the length of a line segment of pi calculated?

The length of a line segment of pi can be calculated using the formula L=πr, where L is the length of the line segment and r is the radius of the circle. This formula is derived from the definition of pi as the ratio of a circle's circumference to its diameter.

3. Can a line segment of length pi be measured accurately?

No, a line segment of length pi cannot be measured accurately because pi is an irrational number and cannot be expressed as a finite decimal. It has an infinite number of digits after the decimal point, making it impossible to measure precisely.

4. Is a line segment of length pi commonly used in real-life applications?

No, a line segment of length pi is not commonly used in real-life applications. However, pi itself is a crucial mathematical constant that is used in various fields, such as geometry, physics, and engineering.

5. Can a line segment of length pi be visualized?

Yes, a line segment of length pi can be visualized using a ruler and a compass. Draw a circle with a radius of pi units, and then mark two points on the circumference of the circle. The distance between these points will be a line segment of length pi.

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