Suppose you have a real number $A$ and approximate it by only $n$ decimal places. Call this number $a$. proof that the upper bound of absolute error of this approximation $|A-a| \le 5 \times 10^{-(n+1)}$.

Can anyone help me to solve this theorem? I thought that i can use induction technique but it doesn't work.


What is the distance between two neighboring numbers with $n$ decimal places? $A$ is no more than half that from the nearest one.

  • $\begingroup$ Thanks, It's intuitive. But can you provide a mathematical proof? $\endgroup$ – Alireza Afzal aghaei Oct 9 at 18:15
  • $\begingroup$ I believe this is a proof. You could add some details, starting with justifying the distance statement from the fact that a number with $n$ decimals is an integer times $10^{-n}$. The minimum spacing of integers of $1$ translates to the spacing of the decimals. Then assume your number is more than $5\cdot 10^{n+1}$ from one end of the interval, show that it is less than that from the other end. $\endgroup$ – Ross Millikan Oct 9 at 18:20

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