Significant figure

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