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What is the default precision of ` when not specifying a digit after it?

Posted 8 months ago
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If the symbol ` has no digits after it, what is the precision of the number understood to be?

Here is a the sample code:

DKReplace = {70.6`, 74.2`, 80.1`, 75.3`, 55.6`, 55.5`, 62.6`, 64.9`, 
80.1`, 102.5`};

Thank you,


2 Replies

Numbers entered with a backtick ` but without a specified precision are treated as machine precision numbers, which is to say, they are treated just like an undecorated number with a decimal point. A backtick without a number specifying precision is useful when applied to an integer without a decimal point. With a number having a decimal point, the backtick does nothing.

You can determine the precision of a number using the Precision function:

In[1]:= Precision[70.6]
Out[1]= MachinePrecision
In[2]:= Precision[70.6`]
Out[2]= MachinePrecision

The symbol MachinePrecision is treated as a constant and is set to the value of $MachinePrecision, which begs the question, "What is $MachinePrecision?" From the documentation:

The typical arrangement is that all machine‐precision numbers in the Wolfram Language are represented as "double‐precision floating‐point numbers" in the underlying computer system. On most current computers, such numbers contain a total of 64 binary bits, typically yielding 16 decimal digits of mantissa.

Thank you very much for your excellent explanation!


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