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.

- anonymous

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- SolomonZelman

Do you know what rational and irrational numbers are/

- SolomonZelman

?

- anonymous

no

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## More answers

- anonymous

its an essay question

- 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

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

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

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

I am giving you the definitions neccesary.

- 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

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

- SolomonZelman

where the decomal is not repeating or terminating

- SolomonZelman

Ok, got the definitions of rtional and irrational numbers?

- anonymous

yes

- 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

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

ok

- 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

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

(take your time)

- anonymous

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

- SolomonZelman

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

- SolomonZelman

oh wait, nio

- SolomonZelman

7.756784535583495... is irrational, no?

- anonymous

what?

- anonymous

i thought that irrational was terminating not repeating

- anonymous

i thought that rational was repeating

- 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

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

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

- anonymous

so pi is irrational 3.1415926535.....

- SolomonZelman

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

- SolomonZelman

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

- SolomonZelman

is irrational.

- anonymous

so 7.725

- SolomonZelman

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

- 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

7.7*7.7=59.29
7.9*7.9=62.41
\[\sqrt{60}\]

- SolomonZelman

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

- anonymous

how?

- SolomonZelman

you can use this site:
wolframalpha.com

- SolomonZelman

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

- SolomonZelman

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

- 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

so, \(\sqrt{60}=?\)

- anonymous

\[2\sqrt{15}\]

- anonymous

##### 1 Attachment

- SolomonZelman

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

- SolomonZelman

now round that to nearest hunderedth and you get?

- anonymous

7.75

- SolomonZelman

yes correct

- SolomonZelman

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

- anonymous

thank you

- anonymous

thank you

- SolomonZelman

Yw!

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