What Will Happen Without Decimal Expansion? After a discussion on the complexity of decimal expansion (such as $0.\bar{9}=1$), some of my students (middle school) decided to throw away the decimal expansion of some numbers! Namely, the numbers which their decimal expansion is not terminating.
I understand that their decision leads to some problems: We know that decimal expansions of many fractions are not terminating, and no irrational number can be represented as a terminating decimal expansion. 
But what are the real disasters which occur if we limit the decimal expansions to the numbers with terminating ones? I look for answers which can be comprehended by students in middle school.
Thanks.
 A: Well, depends what the meaning of "disaster" is. 
In the rationals, $ \mathbb Q$, every non-zero quantity is invertible.  Any fraction can be expressed as a finite or infinite repeating decimal.  Thus $1.5$ has as inverse $0.(6)$.  In a more theoretical POV, any linear transformation with coefficients in $ \mathbb Q$ acting on elements of $ \mathbb Q$ will yield results in $ \mathbb Q$, and if invertible, the inverse will have the same property.
However, if you give up infinite repeating decimals not all numbers will have inverses, in a relatively goofy manner.  E.g., $0.5$ will have an inverse, $1.5$ won't, and $2.5$ will again have one, then no $x.5$ will have inverses until $12.5$ etc, etc, etc.  Actually numbers of the type $2^k5^m$, where $k, m$ integer will have finite decimal inverses, none others will.  However in base $12$ only numbers of the type $2^k3^m$, where $k, m$ integer will have finite decimal inverses while in base p (p - prime) only numbers of the type $p^k$, where $k$ integer will have finite decimal inverses.  
So not only you lose invertibility of all nonzero quantities, what quantities are invertible becomes dependent on the basis in which you express your rational quantities.  More theoretical, you lose the general invertibility of linear transformations formulated in the universe (set, class) of rational quantities.

OTOH, care with infinite expansions will help your students avoid confusing irrational numbers with their infinite decimal expansions, a topic of acerbic controversy between professional mathematicians.
A: If you restrict to finite decimal expansions, then every number lives in the ring $\mathbb{Z}[1/10]$. I'm not sure it's a disaster, but it's definitely not the real numbers...
A: Hint.-I would do it this way:
You can start with a lesson about countable and uncountable sets.
Rinse thoroughly that $\mathbb Q $ is countable and $\mathbb R$ is uncountable.
Then tell them the numbers with finite decimal expansion form a "very small"   subset of $\mathbb Q$.
Play with that "very small" (it is infinite and equals $\mathbb Z[\frac {1}{10}]$).
End with the most of the real numbers
left out of consideration when you see just finite decimal expansion.
