Answer:
An irrational number can never be represented precisely in decimal form.
Step-by-step explanation:
A number can be precisely represented in decimal form if you can give a rule for the construction of its decimal part,
for example:
2.246973973973973... (an infinite tail of 973's repeated over and over)
7.35 (a tail of zeroes)
If this the case, then the number is a RATIONAL NUMBER, i.e, the QUOTIENT OF TWO INTEGERS.
Let's show this for the first example and then a way to show the general situation will arise naturally.
Suppose N = 2.246973973973...
You can always multiply by a suitable power of 10 until you get a number with only the repeated chain in the tail
for example:
1) [tex]N.10^3=2246.973973973...[/tex]
but also
2) [tex]N.10^6=2246973.973973...[/tex]
Subtracting 1) from 2) we get
[tex]N.10^6-N.10^3=2246973.973973973... - 2246.973973973...[/tex]
Now, the infinite tail disappears
[tex]N.10^6-N.10^3=2246973 - 2246=226747[/tex]
But
[tex]N.10^6-N.10^3=N.(10^6-10^3)=N.(1000000-1000)=999000.N[/tex]
We have then
999000.N=226747
and
[tex]N=\frac{226747}{999000}[/tex]
We do not need to simplify this fraction, because we only wanted to show that N is a quotient of two integers.
We arrive then to the following conclusion:
If an irrational number could be represented precisely in decimal form, then it would have to be the quotient of two integers, which is a contradiction.
So, an irrational number can never be represented precisely in decimal form.
