Editing Round-off error (section)

=== Normalized floating-number system ===

* A floating-point number system is normalized if the leading digit <math>d_{0}</math> is always nonzero unless the number is zero.<ref name="Forrester_2018"/> Since the [[significand]] is <math>d_{0}.d_{1}d_{2}\ldots d_{p-1}</math>, the significand of a nonzero number in a normalized system satisfies <math>1 \leq \text{significand} < \beta ^{p}</math>. Thus, the normalized form of a nonzero [[Institute of Electrical and Electronics Engineers|IEEE]] floating-point number is <math>\pm 1.bb \ldots b \times 2^{E}</math> where <math>b \in {0, 1}</math>. In binary, the leading digit is always <math>1</math> so it is not written out and is called the implicit bit. This gives an extra bit of precision so that the roundoff error caused by representation error is reduced.
* Since floating-point number system <math>F</math> is finite and discrete, it cannot represent all real numbers which means infinite real numbers can only be approximated by some finite numbers through [[rounding|rounding rule]]s. The floating-point approximation of a given real number <math>x</math> by <math>fl(x)</math> can be denoted.
** The total number of normalized floating-point numbers is <math display="block">2(\beta -1)\beta^{p-1} (U-L+1)+1,</math> where
*** <math>2</math> counts choice of sign, being positive or negative
*** <math>(\beta -1)</math> counts choice of the leading digit
*** <math>\beta^{p-1}</math> counts remaining significand digits
*** <math>U-L+1</math> counts choice of exponents
*** <math>1</math> counts the case when the number is <math>0</math>.