Square Root of an Odd Powered Integer is Always Irrational?

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Discussion Overview

The discussion revolves around whether the square root of an odd powered integer is always irrational. Participants explore this concept through mathematical reasoning, examples, and computational observations, particularly focusing on the behavior of large integers in software like Excel.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question if the square root of an odd powered integer is always irrational, citing examples like ##\sqrt{4^3} = 8##.
  • Others clarify that an integer raised to an odd power that cannot be expressed as a square would yield an irrational number when the square root is taken.
  • One participant mentions that research indicates the square root of any non-square integer is always irrational.
  • Concerns are raised about Excel's handling of large odd powered integers, with claims that it rounds off results, potentially leading to misleading integer values.
  • Participants discuss the limitations of Excel's floating-point precision, noting that it may display results inaccurately due to rounding.
  • There are examples provided of specific calculations in Excel, highlighting discrepancies in expected versus displayed results.
  • Some participants suggest that the underlying implementation of floating-point numbers in Excel may affect the accuracy of large calculations.

Areas of Agreement / Disagreement

Participants express differing views on whether the square root of an odd powered integer is always irrational, with some supporting the idea while others question it based on specific examples. The discussion regarding Excel's computational accuracy also reveals a lack of consensus on how it handles large numbers.

Contextual Notes

Limitations include the dependence on definitions of perfect squares and the specific behavior of computational tools like Excel, which may not accurately represent mathematical results for very large numbers.

  • #31
e2m2a said:
From the research I have done since posting, it says that the square root of any non-square integer is always an irrational number.
Did you notice that if you write any of those irrational square roots as a continued fraction, they always repeat.
 
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  • #32
e2m2a said:
Is it always true that the square root of an odd powered integer will always be irrational?
What about 1?
 
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