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Re: Bug 1+4/10

• To: mathgroup at smc.vnet.net
• Subject: [mg120045] Re: Bug 1+4/10
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Thu, 7 Jul 2011 07:28:46 -0400 (EDT)
• References: <201107060940.FAA29406@smc.vnet.net>
• Reply-to: drmajorbob at yahoo.com

1.4 is stored in a finite number of bits, despite the fact that its  
expansion in binary is repeating:

{bits, digitsToLeft} = RealDigits[input = 1.4, 2];

{{1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1,
    1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0,
   0, 1, 1, 0, 0, 1, 1, 0}, 1}

The number stored (as opposed to the number written down) is  
(conceptually):

stored = FromDigits[bits, 2]/2^(-digitsToLeft + Length@bits)

3152519739159347/2251799813685248

The stored number, as you see, is not precisely 1 + 4/10.

Set to various precisions and displayed, "stored" looks like this:

Table[{k, SetPrecision[stored, k]}, {k, 30}]

{{1, 1.}, {2, 1.4}, {3, 1.40}, {4, 1.400}, {5, 1.4000}, {6,
   1.40000}, {7, 1.400000}, {8, 1.4000000}, {9, 1.40000000}, {10,
   1.400000000}, {11, 1.4000000000}, {12, 1.40000000000}, {13,
   1.400000000000}, {14, 1.4000000000000}, {15, 1.40000000000000}, {16,
    1.400000000000000}, {17, 1.3999999999999999}, {18,
   1.39999999999999991}, {19, 1.399999999999999911}, {20,
   1.3999999999999999112}, {21, 1.39999999999999991118}, {22,
   1.399999999999999911182}, {23, 1.3999999999999999111822}, {24,
   1.39999999999999991118216}, {25, 1.399999999999999911182158}, {26,
   1.3999999999999999111821580}, {27,
   1.39999999999999991118215803}, {28,
   1.399999999999999911182158030}, {29,
   1.3999999999999999111821580300}, {30,
   1.39999999999999991118215802999}}

As precision increases, the displayed number comes closer and closer to  
the "stored" number, which is NOT necessarily closer to 1 + 4/10.  
SetPrecision works with what it has, in other words... not with what  
you're thinking. It treats the first argument as a calculated result, not  
a raw textual input.

(The "stored" number above is a Rational, and is NOT used in Mathematica  
calculations; computer arithmetic is used on a machine Real, with all its  
attendant inaccuracies... and SPEED.)

A setPrecision function can be written to work with raw input, but you'd  
have to feed it "1.4" as a String (or some such), since entering 1.4 alone  
immediately results in a machine precision Real. That function would be no  
use with calculated results, so we'd still need SetPrecision, as is, for  
general purposes.

You already know how to input and use 1 + 4/10, which is stored as a  
Rational:

1 + 1/4 // FullForm

Rational[5,4]

Here you can see the repeating binary digits mentioned earlier:

RealDigits[1 + 4/10, 2]

{{1, {0, 1, 1, 0}}, 1}

Bottom line? Rationals are exact. Reals are not.

Get used to it!

Bobby

On Wed, 06 Jul 2011 04:40:16 -0500, slawek <slawek at host.pl> wrote:

> Let check
>
> In[1]:= 1.4 == 1 + 4/10
> Out[1]= True
>
> In[2]:= a = SetPrecision[1.4, 30]
> Out[2]= 1.39999999999999991118215802999
>
> In[3]:= b = SetPrecision[1 + 4/10, 30]
> Out[3]= 1.40000000000000000000000000000
>
> No comment is needed.
>
> slawek
>
>


-- 
DrMajorBob at yahoo.com

• References:
  • Bug 1+4/10
    • From: "slawek" <slawek@host.pl>