anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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SolomonZelman
  • SolomonZelman
Do you know what rational and irrational numbers are/
SolomonZelman
  • SolomonZelman
?
anonymous
  • anonymous
no

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anonymous
  • anonymous
its an essay question
SolomonZelman
  • 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)
SolomonZelman
  • 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
anonymous
  • 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.
SolomonZelman
  • 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)
SolomonZelman
  • SolomonZelman
I am giving you the definitions neccesary.
SolomonZelman
  • 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 \( \color{black}{\psi,~e,~\pi~...}\)
SolomonZelman
  • SolomonZelman
irrational number is also something like \(4{\bf .}2946188294621536724111...\)
SolomonZelman
  • SolomonZelman
where the decomal is not repeating or terminating
SolomonZelman
  • SolomonZelman
Ok, got the definitions of rtional and irrational numbers?
anonymous
  • anonymous
yes
SolomonZelman
  • 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
SolomonZelman
  • 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.....
anonymous
  • anonymous
ok
SolomonZelman
  • 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.
SolomonZelman
  • 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
SolomonZelman
  • SolomonZelman
(take your time)
anonymous
  • anonymous
so a rational number between 7.7 and 7.9 can be 7.756784535583495...
SolomonZelman
  • SolomonZelman
ok, yes.... can you tell me how you got this number tho'?
SolomonZelman
  • SolomonZelman
oh wait, nio
SolomonZelman
  • SolomonZelman
7.756784535583495... is irrational, no?
anonymous
  • anonymous
what?
anonymous
  • anonymous
i thought that irrational was terminating not repeating
anonymous
  • anonymous
i thought that rational was repeating
SolomonZelman
  • 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: \(-2.457\) \(0.5\) \(4.21\) \(-43.873\) \(92.555\)
SolomonZelman
  • 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.
SolomonZelman
  • SolomonZelman
Can you tell me a rational RATIONAL number that is between \(7.7\) and \(7.8\)? (just give me a simple decimal)
anonymous
  • anonymous
so pi is irrational 3.1415926535.....
SolomonZelman
  • SolomonZelman
Yes, \(\pi\) (and e, if you heard of it) are definitely irrational
SolomonZelman
  • SolomonZelman
also \(\psi\) (infinite series of fibonacci reciprocals)
SolomonZelman
  • SolomonZelman
is irrational.
anonymous
  • anonymous
so 7.725
SolomonZelman
  • SolomonZelman
yes, 7.725 can be a rational number between 7.7 and 7.9 Nice!
SolomonZelman
  • 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 ?)
anonymous
  • anonymous
7.7*7.7=59.29 7.9*7.9=62.41 \[\sqrt{60}\]
SolomonZelman
  • SolomonZelman
yes, now, (using a calculator), calculate \(\sqrt{60}\)
anonymous
  • anonymous
how?
SolomonZelman
  • SolomonZelman
you can use this site: wolframalpha.com
SolomonZelman
  • SolomonZelman
enter there "square root of 60" and it will give you the answer
SolomonZelman
  • SolomonZelman
it will simplfy it for you under "Result" but, you need the "Decimal Approximation" that it will give.
SolomonZelman
  • SolomonZelman
Or google can do it for you, as well: https://www.google.com/search?site=&source=hp&q=square+root+of+60&oq=square+root+of+60&gs_l=hp.3..19j0l10.1111.4707.0.5000.22.15.0.2.2.0.923.2182.1j4j0j2j6-1.8.0....0...1c.1.64.hp..13.9.1262.0.wj4VxQsctJM
SolomonZelman
  • SolomonZelman
so, \(\sqrt{60}=?\)
anonymous
  • anonymous
\[2\sqrt{15}\]
anonymous
  • anonymous
1 Attachment
SolomonZelman
  • SolomonZelman
yes that is the exact result, and the "DECIMAL APPROXIMATION" 9what we need) is \(\sqrt{60} \approx 7.745966692....\)
SolomonZelman
  • SolomonZelman
now round that to nearest hunderedth and you get?
anonymous
  • anonymous
7.75
SolomonZelman
  • SolomonZelman
yes correct
SolomonZelman
  • SolomonZelman
you are done i guess. you found rational and irrational numbers as needed;)
anonymous
  • anonymous
thank you
anonymous
  • anonymous
thank you
SolomonZelman
  • SolomonZelman
Yw!

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