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''significant digit''

A term pertaining to the approximate specification of a [[Real number|real number]]. Let a real number  $  x $
be represented in a number system with base  $  q $
by a  $  q $-
ary fraction

$$ 
x  \approx  x  ^ {*}  =  ( \alpha _ {n} \dots \alpha _ {1} \alpha _ {0} .  \alpha _ {-1}
\alpha _ {-2} \dots \alpha _ {-n} ).
$$

In this notation, suppose that  $  \alpha _ {s} $
is the first non-zero digit, counted from the left. Then all subsequent digits are called significant digits of the approximate number  $  x  ^ {*} $.

A significant digit  $  \alpha _ {t} $
is said to be correct if the absolute error  $  \Delta ( x  ^ {*} ) $
of  $  x  ^ {*} $, 
i.e. the difference  $  | x - x  ^ {*} | $, 
satisfies the inequality

$$ 
\Delta ( x  ^ {*} )  \leq   
\frac{1}{2}
 q  ^ {t} .
$$

In approximate specification of a real number it is usually meaningful to specify only significant digits.

====Comments====
One speaks of doing a calculation to three significant digits if each calculated result is rounded so that there are no more than three digits after and including the first non-zero digit. One says that an approximate number resulting from a calculation (in mathematics) or from a measurement (in sciences and technology)  $  x  ^ {*} = ( \alpha _ {1} \dots \alpha _ {r} ) \times q  ^ {-m} $
with  $  r $
significant digits is correct to  $  n $
significant digits,  $  n \leq  r $, 
if the error

$$ 
| (( \alpha  ^ {1} \dots \alpha _ {n} ) \times q  ^ {r- n- m}) - x |
$$

is less than  $  q  ^ {r-} n- m /2 $. 
Usually  $  q = 10 $
or  $  2 $. 
For example,  $  0.0308 $
is the correct answer to three significant digits of  $  \sqrt {x  ^ {2} + 1 } - 1 = 0 $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.M. Young,   R.T. Gregory,   "A survey of numerical mathematics" , '''I''' , Dover, reprint  (1988)  pp. Chapt. 1</TD></TR></table>