is 10.8472 a rational number

Question

is 10.4872 a rational number

Answer 1

Yes it is!

First, let's define a "rational" number - it is a number that can be written in a form of a fraction "p/q", where p and q are any integers (whole numbers). Some examples are:

0.5 = 1/2

0.333 = 1/3

and so on.

Thus, if we can write decimal number is a form of a fraction, it is a rational number. Let's try that for 10.8472:

10.8472 = 10 + 8472/10000, i.e. it is 10 wholes and 8472 tenthousandths (8472 is numerator and 10 000 is denominator in this fraction).

Now, we can convert 10 to fraction, too, by multiplying 10 with 10000 (fraction denominator). We can write number 10 as (10 * 10000)/10000.

So our fraction 10+ 8472/10000 becomes:

(10 * 10000)/10000 + 8472/10000=

100000/10000 + 8472/10000=

 108472/10000=

 10.8472

Thus, 10.8472 can be written as a fraction of two numbers (108472/10000), and is therefore a rational number.

Please note that this fraction can be furhter simplified if we devide both numerator and denominator with the same number (in this case 2):

108472/2 = 54236

10000/2 = 5000

so it is 54236/5000

Repeat

54236/2 = 27118

5000/2 = 2500

now it becomes 27118/2500

Repeat

27118/2 = 13559

2500/2 = 1250

Finally we reach point where there is no common divisor, and fraction cannot be simplified beyon 13559/1250. If you divide these two numbers you will reach the same number we started with: 10.8472.

In conclusion, any terminating decimal (decimal with an end) is a rational number.