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anonymous

  • one year ago

a rational number and an irrational number that are between 7.7 and 7.9. Include the decimal approximation of the irrational number to the nearest hundredth.

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  1. SolomonZelman
    • one year ago
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    Do you know what rational and irrational numbers are/

  2. SolomonZelman
    • one year ago
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    ?

  3. anonymous
    • one year ago
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    no

  4. anonymous
    • one year ago
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    its an essay question

  5. SolomonZelman
    • one year ago
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    rational numbers: \( \color{black}{{\bf ...}~~-4,~~-3,~~-2,~~-1,~~~~0,~~~~1,~~~~2,~~~~3,~~~~4}\) Also it can be a fraction \( \color{black}{3/5,~~~-1/9,~~~~10/7,~~~~~3/9,~~~~1000/81}\) (and so on many other examples...) Also it can be a terminating decimal, or a repeating decimal (because they are technically fractions)

  6. SolomonZelman
    • one year ago
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    if you see: Negative or positive, --- number or fraction, or a decimal, (decimal which terminanting or repeating, BUT NOT some decimal without a pattern) then what you see is a rational number

  7. anonymous
    • one year ago
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    Find a rational number and an irrational number that are between 7.7 and 7.9. Include the decimal approximation of the irrational number to the nearest hundredth.

  8. SolomonZelman
    • one year ago
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    Also \(\sqrt{4}\) is a rational number because, really \(\sqrt{4}=2\) and 2 is rational. And any root that simplifies to a rational is a rational number. (this is all for rational numbers)

  9. SolomonZelman
    • one year ago
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    I am giving you the definitions neccesary.

  10. SolomonZelman
    • one year ago
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    Now and irrational numbers Roots: like \(\sqrt{3}\) , \(\sqrt[6]{16}\), \(\sqrt{98}\) and other roots that don't simplify to a rational number. Also, logarithms if they don't simplify to rational number. Like \(\ln(2),~~\log_{3}(4),~~~\log_{18}(71)\) and others Also, special constant like \( \color{black}{\psi,~e,~\pi~...}\)

  11. SolomonZelman
    • one year ago
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    irrational number is also something like \(4{\bf .}2946188294621536724111...\)

  12. SolomonZelman
    • one year ago
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    where the decomal is not repeating or terminating

  13. SolomonZelman
    • one year ago
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    Ok, got the definitions of rtional and irrational numbers?

  14. anonymous
    • one year ago
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    yes

  15. SolomonZelman
    • one year ago
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    one more, to the definition or RATIONAL number. if your decimal is finite, like 3.4 2.625 -1.88 -7.59 then the number is rational

  16. SolomonZelman
    • one year ago
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    ok, I will give you an example of how to do you problem if I have 4.2 and 4.4 and you will then do it with your numbers.....

  17. anonymous
    • one year ago
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    ok

  18. SolomonZelman
    • one year ago
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    A rational number between 4.2 and 4.4 can for example be any decimal that is greater than 4.2 and less than 4.4 For example 4.3 ------------------------ An irrational number can be a square root that is less than 4.2 and greater than 4.3 For example \(4.2^2=17.64\) \(4.4^2=19.36\) So it can be a square root of any number between 17.64 and 19.36 like \(\sqrt{18}\) Check √18 = 4.242640687 (via calculator) So, now round that to nearest hunderedth √18 ≈ 4.24 (you round it down, because after a 4 is a 2 and 2 is rounded down, not up) so you found your irrational number.

  19. SolomonZelman
    • one year ago
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    correction: A rational number between 4.2 and 4.4 can for example be any \(\color{red}{\rm finite}\) decimal that is greater than 4.2 and less than 4.4

  20. SolomonZelman
    • one year ago
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    (take your time)

  21. anonymous
    • one year ago
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    so a rational number between 7.7 and 7.9 can be 7.756784535583495...

  22. SolomonZelman
    • one year ago
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    ok, yes.... can you tell me how you got this number tho'?

  23. SolomonZelman
    • one year ago
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    oh wait, nio

  24. SolomonZelman
    • one year ago
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    7.756784535583495... is irrational, no?

  25. anonymous
    • one year ago
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    what?

  26. anonymous
    • one year ago
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    i thought that irrational was terminating not repeating

  27. anonymous
    • one year ago
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    i thought that rational was repeating

  28. SolomonZelman
    • one year ago
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    Rational number decimals: Terminating: (like 1.\(\color{red}{34}\)3434343434 .... ) Repeating: (like -2.\(\color{red}{6}\)666666.... ) Also simple decimals that don't go on forever like: \(-2.457\) \(0.5\) \(4.21\) \(-43.873\) \(92.555\)

  29. SolomonZelman
    • one year ago
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    An irrational decimal is a decimal that is NOT terminating and NOT repeating. In other words, it goes forver, and doesn't have any pattern to it what so ever.

  30. SolomonZelman
    • one year ago
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    Can you tell me a rational RATIONAL number that is between \(7.7\) and \(7.8\)? (just give me a simple decimal)

  31. anonymous
    • one year ago
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    so pi is irrational 3.1415926535.....

  32. SolomonZelman
    • one year ago
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    Yes, \(\pi\) (and e, if you heard of it) are definitely irrational

  33. SolomonZelman
    • one year ago
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    also \(\psi\) (infinite series of fibonacci reciprocals)

  34. SolomonZelman
    • one year ago
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    is irrational.

  35. anonymous
    • one year ago
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    so 7.725

  36. SolomonZelman
    • one year ago
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    yes, 7.725 can be a rational number between 7.7 and 7.9 Nice!

  37. SolomonZelman
    • one year ago
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    Now, how about an IRRATIONAL number between 7.7 and 7.9? (remember how I found an IRRATIONAL number between 4.2 and 4.4 ?)

  38. anonymous
    • one year ago
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    7.7*7.7=59.29 7.9*7.9=62.41 \[\sqrt{60}\]

  39. SolomonZelman
    • one year ago
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    yes, now, (using a calculator), calculate \(\sqrt{60}\)

  40. anonymous
    • one year ago
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    how?

  41. SolomonZelman
    • one year ago
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    you can use this site: wolframalpha.com

  42. SolomonZelman
    • one year ago
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    enter there "square root of 60" and it will give you the answer

  43. SolomonZelman
    • one year ago
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    it will simplfy it for you under "Result" but, you need the "Decimal Approximation" that it will give.

  44. SolomonZelman
    • one year ago
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    so, \(\sqrt{60}=?\)

  45. anonymous
    • one year ago
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    \[2\sqrt{15}\]

  46. anonymous
    • one year ago
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  47. SolomonZelman
    • one year ago
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    yes that is the exact result, and the "DECIMAL APPROXIMATION" 9what we need) is \(\sqrt{60} \approx 7.745966692....\)

  48. SolomonZelman
    • one year ago
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    now round that to nearest hunderedth and you get?

  49. anonymous
    • one year ago
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    7.75

  50. SolomonZelman
    • one year ago
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    yes correct

  51. SolomonZelman
    • one year ago
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    you are done i guess. you found rational and irrational numbers as needed;)

  52. anonymous
    • one year ago
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    thank you

  53. anonymous
    • one year ago
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    thank you

  54. SolomonZelman
    • one year ago
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    Yw!

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