Question

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.

Answer 1

Do you know what rational and irrational numbers are/

Answer 2

?

Answer 3

no

Answer 4

its an essay question

Answer 5

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)

Answer 6

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

Answer 7

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.

Answer 8

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)

Answer 9

I am giving you the definitions neccesary.

Answer 10

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 \( \color{black}{\psi,~e,~\pi~...}\)

Answer 11

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

Answer 12

where the decomal is not repeating or terminating

Answer 13

Ok, got the definitions of rtional and irrational numbers?

Answer 14

yes

Answer 15

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

Answer 16

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.....

Answer 17

ok

Answer 18

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.

Answer 19

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

Answer 20

(take your time)

Answer 21

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

Answer 22

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

Answer 23

oh wait, nio

Answer 24

7.756784535583495... is irrational, no?

Answer 25

what?

Answer 26

i thought that irrational was terminating not repeating

Answer 27

i thought that rational was repeating

Answer 28

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:
\(-2.457\)
\(0.5\)
\(4.21\)
\(-43.873\)
\(92.555\)

Answer 29

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.

Answer 30

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

Answer 31

so pi is irrational 3.1415926535.....

Answer 32

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

Answer 33

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

Answer 34

is irrational.

Answer 35

so 7.725

Answer 36

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

Nice!

Answer 37

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 ?)

Answer 38

7.7*7.7=59.29
7.9*7.9=62.41

\[\sqrt{60}\]

Answer 39

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

Answer 40

how?

Answer 41

you can use this site:
wolframalpha.com

Answer 42

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

Answer 43

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

Answer 44

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

Answer 45

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

Answer 46

\[2\sqrt{15}\]

Answer 47

yes that is the exact result, and the "DECIMAL APPROXIMATION" 9what we need)
is

\(\sqrt{60} \approx 7.745966692....\)

Answer 48

now round that to nearest hunderedth and you get?

Answer 49

7.75

Answer 50

yes correct

Answer 51

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

Answer 52

thank you

Answer 53

thank you

Answer 54

Yw!