## 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.

1. SolomonZelman

Do you know what rational and irrational numbers are/

2. SolomonZelman

?

3. anonymous

no

4. anonymous

its an essay question

5. SolomonZelman

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

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

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

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

I am giving you the definitions neccesary.

10. SolomonZelman

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

irrational number is also something like $$4{\bf .}2946188294621536724111...$$

12. SolomonZelman

where the decomal is not repeating or terminating

13. SolomonZelman

Ok, got the definitions of rtional and irrational numbers?

14. anonymous

yes

15. SolomonZelman

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

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

ok

18. SolomonZelman

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

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

21. anonymous

so a rational number between 7.7 and 7.9 can be 7.756784535583495...

22. SolomonZelman

ok, yes.... can you tell me how you got this number tho'?

23. SolomonZelman

oh wait, nio

24. SolomonZelman

7.756784535583495... is irrational, no?

25. anonymous

what?

26. anonymous

i thought that irrational was terminating not repeating

27. anonymous

i thought that rational was repeating

28. SolomonZelman

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

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

Can you tell me a rational RATIONAL number that is between $$7.7$$ and $$7.8$$? (just give me a simple decimal)

31. anonymous

so pi is irrational 3.1415926535.....

32. SolomonZelman

Yes, $$\pi$$ (and e, if you heard of it) are definitely irrational

33. SolomonZelman

also $$\psi$$ (infinite series of fibonacci reciprocals)

34. SolomonZelman

is irrational.

35. anonymous

so 7.725

36. SolomonZelman

yes, 7.725 can be a rational number between 7.7 and 7.9 Nice!

37. SolomonZelman

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

7.7*7.7=59.29 7.9*7.9=62.41 $\sqrt{60}$

39. SolomonZelman

yes, now, (using a calculator), calculate $$\sqrt{60}$$

40. anonymous

how?

41. SolomonZelman

you can use this site: wolframalpha.com

42. SolomonZelman

enter there "square root of 60" and it will give you the answer

43. SolomonZelman

it will simplfy it for you under "Result" but, you need the "Decimal Approximation" that it will give.

44. SolomonZelman
45. SolomonZelman

so, $$\sqrt{60}=?$$

46. anonymous

$2\sqrt{15}$

47. anonymous

48. SolomonZelman

yes that is the exact result, and the "DECIMAL APPROXIMATION" 9what we need) is $$\sqrt{60} \approx 7.745966692....$$

49. SolomonZelman

now round that to nearest hunderedth and you get?

50. anonymous

7.75

51. SolomonZelman

yes correct

52. SolomonZelman

you are done i guess. you found rational and irrational numbers as needed;)

53. anonymous

thank you

54. anonymous

thank you

55. SolomonZelman

Yw!