SOLUTION: is seven an irrational number?

Algebra ->  Real-numbers -> SOLUTION: is seven an irrational number?      Log On


   

Question 102231: is seven an irrational number?
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
No because 7 can be written as 14/2 or 21/3 etc.
Because it can be written as a ratio it is called rational.
---------------
Cheers,
Stan H.

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Is seven an irrational number?

No for it is RATIOnal, because it is the RATIO of integers 7 to 1.

Any number is RATIOnal if it can be written "an integer over an integer"

These are rational numbers:

1%2F2, 3%2F5, 5%2F3, 8764061%2F2208567, 0%2F1, 7%2F1

7.85674 is a rational number because we can begin by placing it over 1,
like this 7.85671%2F1.00000 and then we can moving the decimal on top and 
bottom five places and we have 785671%2F100000 and that is "an integer
over an integer". 

An example of an irrational number is 

2.67667666766667666667666666766666667666666667···

where after each 7 is one more 6 than was after the last 7 and this goes
on forever.  It does not repeat the same block of digits forever, as
each new block contains one more 6 than the last.

However a decimal such as

2.66776677667766776677667766776677···

is a RATONAL number because it repeats the block of digits 6677 forever.

Any termination decimal or non-terminating decimal WHICH REPEATS THE SAME
block of digits forever can be shown to represent a rational number.

Incidentally, the never ending decimal above,

2.66776677667766776677667766776677···

is RATIONAL because it is equivalent to the fraction:

2425%2F909 which is "an integer over an integer".

Don't believe it?  Watch this long division:

            2.667766
     909)2425.000000
         1818       
          607 0    
          545 4    
           61 60   
           54 54   
            7 060   
            6 363  
              6970  
              6363 
               6070
               5454
                6160    
                5454
                 706

Look at the division and you can see that you
will keep getting the same block of digits 
6677 forever.

Rational numbers are numbers that can be written
as COMMON FRACTIONS, either proper of improper.

Irrational number are those which can't be so
written.

sqrt%282%29 is an irrational number. It starts off:

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737212644121497099935831432226659275055927557999505011527820605714701095599716059702745345968620147285... 

It never stops and it never repeats the same set of digits over and over
forever.  Check this website and see it carried to a million decimal
places.  

http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt2.1mil

pi is another irrational number.  It starts off:

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609...

It never stops and it never repeats the same set of digits over and over
forever.  Check out this website:   

http://pi.ytmnd.com/

and hear a bunch of the digits of pi sung.

Edwin