I have a number machine. I say x gives y if when x goes in, y comes out. For any numbers x and y, by "xy" I mean x followed by y. Here are my two rules.
1x gives x.
For example, 13 gives 3.
If x gives y, then 2x gives yy.
For example, 213 gives 33 since 13 gives 3.
Show that for any number a, there is a number x such that x gives ax.
Your rules are not defined clearly enough by the examples
provided by you:
a. "If x gives y, then 2x gives yy." Does it mean that 3x gives yyy?.
What does xx give?
b. "213 gives 33 since 13 gives 3." True, if a priority is given to apply the conversion rules from right to left;
if not 213=2(13)=1313=
Please clarify the above issues.
Maybe a longer sample chain of transformations is needed.
Edited on December 2, 2014, 8:26 am