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''significant digit''
 
''significant digit''
  
A term pertaining to the approximate specification of a [[Real number|real number]]. Let a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s0850901.png" /> be represented in a number system with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s0850902.png" /> by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s0850903.png" />-ary fraction
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s0850904.png" /></td> </tr></table>
+
$$
 +
x  \approx  x  ^ {*}  = ( \alpha _ {n} \dots \alpha _ {1} \alpha _ {0} . \alpha _ {-} 1
 +
\alpha _ {-} 2 \dots \alpha _ {-} n ).
 +
$$
  
In this notation, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s0850905.png" /> is the first non-zero digit, counted from the left. Then all subsequent digits are called significant digits of the approximate number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s0850906.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s0850907.png" /> is said to be correct if the absolute error <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s0850908.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s0850909.png" />, i.e. the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509010.png" />, satisfies the inequality
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509011.png" /></td> </tr></table>
+
$$
 +
\Delta ( x  ^ {*} )  \leq 
 +
\frac{1}{2}
 +
q  ^ {t} .
 +
$$
  
 
In approximate specification of a real number it is usually meaningful to specify only significant digits.
 
In approximate specification of a real number it is usually meaningful to specify only significant digits.
 
 
  
 
====Comments====
 
====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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509013.png" /> significant digits is correct to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509015.png" /> significant digits, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509016.png" />, if the error
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509017.png" /></td> </tr></table>
+
$$
 +
| (( \alpha  ^ {1} \dots \alpha _ {n} ) \times q  ^ {r-} n- m ) - x |
 +
$$
  
is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509018.png" />. Usually <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509019.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509020.png" />. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509021.png" /> is the correct answer to three significant digits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085090/s08509022.png" />.
+
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====
 
====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>
 
<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>

Revision as of 08:13, 6 June 2020


significant digit

A term pertaining to the approximate specification of a 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

[a1] D.M. Young, R.T. Gregory, "A survey of numerical mathematics" , I , Dover, reprint (1988) pp. Chapt. 1
How to Cite This Entry:
Significant figure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Significant_figure&oldid=12701
This article was adapted from an original article by Kh.D. Ikratov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article