the result of converting numbers between base 10 and 2.

No. It isn't.

YOU are wrong! Wrong! Wrong!

IEEE 64-bit doubles are perfectly capable of representing these values to sufficient accuracy to allow them to be rounded correctly.

See Re: RFC: Large Floating Point Numbers - Rounding Errors (Rounding mode)

No, they are not, this is a borderline case.

Try the following C program and see how the periodic part of the base 2 representation of those rational numbers is truncated and how it converts back to base 10:

#include <stdio.h>

double n[] = { 0.000005, 0.000015, 0.000025, 0.000035, 0.000045, 0.000
+055, 0.000065 };

int main(int argc, char *argv[]) {
    int i;
    for (i = 0; i < sizeof(n)/sizeof(*n); i++) {
        printf("%40.30a => %40.30f\n", n[i], n[i]);
    }
    return 0;
}
[download]

On my amd64 linux box it outputs:

  0x1.4f8b588e368f100000000000000000p-18 =>         0.0000050000000000
+00000409015270
  0x1.f75104d551d6900000000000000000p-17 =>         0.0000150000000000
+00000380012861
  0x1.a36e2eb1c432d00000000000000000p-16 =>         0.0000250000000000
+00001198043401
  0x1.2599ed7c6fbd200000000000000000p-15 =>         0.0000349999999999
+99996933876256
  0x1.797cc39ffd60f00000000000000000p-15 =>         0.0000450000000000
+00002834104479
  0x1.cd5f99c38b04b00000000000000000p-15 =>         0.0000550000000000
+00001958069124
  0x1.10a137f38c54300000000000000000p-14 =>         0.0000649999999999
+99994305770190
[download]

... this is a borderline case.

Because YOU haven't set the rounding mode:

#include <stdio.h>
#include <float.h>

int main( int argc, char **argv ) {
    double n;

    printf( "Setting mode CHOP; controlFP returned: %u\n", 
            _controlfp( _RC_CHOP, _MCW_RC ) 
         );
    for( n = 0.000005; n < 0.0001; n += 0.00001 ) {
        printf( "%.15f %.5f\n", n, n );
    }

    printf( "Setting mode UP; controlFP returned: %u\n", 
             _controlfp( _RC_UP, _MCW_RC ) 
         );
    for( n = 0.000005; n < 0.0001; n += 0.00001 ) {
        printf( "%.15f %.5f\n", n, n );
    }

    printf( "Setting mode DOWN; controlFP returned: %u\n", 
             _controlfp( _RC_DOWN, _MCW_RC ) 
         );
    for( n = 0.000005; n < 0.0001; n += 0.00001 ) {
        printf( "%.15f %.5f\n", n, n );
    }

    printf( "Setting mode NEAR; controlFP returned: %u\n", 
            _controlfp( _RC_NEAR, _MCW_RC ) 
         );
    for( n = 0.000005; n < 0.0001; n += 0.00001 ) {
        printf( "%.15f %.5f\n", n, n );
    }
    return 1;
}
[download]

C:\test>ieeeroundingmode.exe
Setting mode CHOP; controlFP returned: 525087
0.000005000000000 0.00001
0.000015000000000 0.00002
0.000025000000000 0.00003
0.000035000000000 0.00003
0.000045000000000 0.00004
0.000055000000000 0.00005
0.000065000000000 0.00006
0.000075000000000 0.00007
0.000085000000000 0.00008
0.000095000000000 0.00009
Setting mode UP; controlFP returned: 524831
0.000005000000000 0.00001
0.000015000000000 0.00002
0.000025000000000 0.00003
0.000035000000000 0.00004
0.000045000000000 0.00005
0.000055000000000 0.00006
0.000065000000000 0.00007
0.000075000000000 0.00008
0.000085000000000 0.00009
0.000095000000000 0.00010
Setting mode DOWN; controlFP returned: 524575
0.000005000000000 0.00001
0.000015000000000 0.00002
0.000025000000000 0.00003
0.000035000000000 0.00003
0.000045000000000 0.00004
0.000055000000000 0.00005
0.000065000000000 0.00006
0.000075000000000 0.00007
0.000085000000000 0.00008
0.000095000000000 0.00009
Setting mode NEAR; controlFP returned: 524319
0.000005000000000 0.00001
0.000015000000000 0.00002
0.000025000000000 0.00003
0.000035000000000 0.00004
0.000045000000000 0.00005
0.000055000000000 0.00006
0.000065000000000 0.00007
0.000075000000000 0.00008
0.000085000000000 0.00009
0.000095000000000 0.00010
[download]

---

Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.

"Science is about questioning the status quo. Questioning authority".

In the absence of evidence, opinion is indistinguishable from prejudice.