3.2: Floating Point Numbers
- Page ID
- 53660
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Many fields in scientific computing rely on using decimals and the standard way to store these in a computer is with floating point numbers. Details on floating-point numbers are in Appendix XXXX. Julia has 16-,32- and 64-bit floating point numbers called Float16, Float32 and Float64 and by default on most systems is the Float64.
There are two limitations to any floating-point number. First, the number of digits stored in the number and secondly, the maximum and minimum values. Each built-in type splits the number of bits into storing both and there is a balance between these. A rule of thumb is that
• Float16 stores 4 decimal digits and the max is about 32,000.
• Float32 stores 8 decimal digits and the max is about \(10^{38}\).
• Float64 stores 16 decimal digits and the max is about \(10^{307}\)
We can using the bitstring function in julia to find the binary representation. Notice that
Again, details are in Appendix XXXXX but, in short, a floating- point number is stored in scientific notation with the abscissa, exponent and the sign all combined together.
Unlike integers, most numbers cannot be stored exactly with a floating-point number. For example, 1/3 divides 1 by 3 and results in a floating-point number close to the fraction \(\frac{1}{3}\). In julia this is 0.3333333333333333 and also note that
Notice that there are non-zero bits throughout the number in this case that didn’t occur with 8.625. This is because as a fraction 8.625 has a denominator of 8, which is a power of 2. If a fraction can be written with such a denominator, the number in binary has 0s that pad the right end of the number.
What does this matter? Consider the following:
is not 1, the expected result. This is an example of the limitations of floating-point numbers and 1) either we deal with it or 2) use a different data type (in this case either a BigFloat or Rational would be better).
Note: This occurred because the closest floating point to the fraction 1/9 was just slightly above 1/9 and adding up 9 of those numbers results in the extra amount
Unless you know you have some reason to choose otherwise, choose Float64 for most floating-point numbers. There are still underflow and overflow errors associated with it, but as we will see in Chapter XXXXX, generally round-off error associated with floating-point number is more detrimental to calculations.