0.333 333 333 333 333 314 829 616 256 246 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.333 333 333 333 333 314 829 616 256 246(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.333 333 333 333 333 314 829 616 256 246(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.333 333 333 333 333 314 829 616 256 246.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.333 333 333 333 333 314 829 616 256 246 × 2 = 0 + 0.666 666 666 666 666 629 659 232 512 492;
  • 2) 0.666 666 666 666 666 629 659 232 512 492 × 2 = 1 + 0.333 333 333 333 333 259 318 465 024 984;
  • 3) 0.333 333 333 333 333 259 318 465 024 984 × 2 = 0 + 0.666 666 666 666 666 518 636 930 049 968;
  • 4) 0.666 666 666 666 666 518 636 930 049 968 × 2 = 1 + 0.333 333 333 333 333 037 273 860 099 936;
  • 5) 0.333 333 333 333 333 037 273 860 099 936 × 2 = 0 + 0.666 666 666 666 666 074 547 720 199 872;
  • 6) 0.666 666 666 666 666 074 547 720 199 872 × 2 = 1 + 0.333 333 333 333 332 149 095 440 399 744;
  • 7) 0.333 333 333 333 332 149 095 440 399 744 × 2 = 0 + 0.666 666 666 666 664 298 190 880 799 488;
  • 8) 0.666 666 666 666 664 298 190 880 799 488 × 2 = 1 + 0.333 333 333 333 328 596 381 761 598 976;
  • 9) 0.333 333 333 333 328 596 381 761 598 976 × 2 = 0 + 0.666 666 666 666 657 192 763 523 197 952;
  • 10) 0.666 666 666 666 657 192 763 523 197 952 × 2 = 1 + 0.333 333 333 333 314 385 527 046 395 904;
  • 11) 0.333 333 333 333 314 385 527 046 395 904 × 2 = 0 + 0.666 666 666 666 628 771 054 092 791 808;
  • 12) 0.666 666 666 666 628 771 054 092 791 808 × 2 = 1 + 0.333 333 333 333 257 542 108 185 583 616;
  • 13) 0.333 333 333 333 257 542 108 185 583 616 × 2 = 0 + 0.666 666 666 666 515 084 216 371 167 232;
  • 14) 0.666 666 666 666 515 084 216 371 167 232 × 2 = 1 + 0.333 333 333 333 030 168 432 742 334 464;
  • 15) 0.333 333 333 333 030 168 432 742 334 464 × 2 = 0 + 0.666 666 666 666 060 336 865 484 668 928;
  • 16) 0.666 666 666 666 060 336 865 484 668 928 × 2 = 1 + 0.333 333 333 332 120 673 730 969 337 856;
  • 17) 0.333 333 333 332 120 673 730 969 337 856 × 2 = 0 + 0.666 666 666 664 241 347 461 938 675 712;
  • 18) 0.666 666 666 664 241 347 461 938 675 712 × 2 = 1 + 0.333 333 333 328 482 694 923 877 351 424;
  • 19) 0.333 333 333 328 482 694 923 877 351 424 × 2 = 0 + 0.666 666 666 656 965 389 847 754 702 848;
  • 20) 0.666 666 666 656 965 389 847 754 702 848 × 2 = 1 + 0.333 333 333 313 930 779 695 509 405 696;
  • 21) 0.333 333 333 313 930 779 695 509 405 696 × 2 = 0 + 0.666 666 666 627 861 559 391 018 811 392;
  • 22) 0.666 666 666 627 861 559 391 018 811 392 × 2 = 1 + 0.333 333 333 255 723 118 782 037 622 784;
  • 23) 0.333 333 333 255 723 118 782 037 622 784 × 2 = 0 + 0.666 666 666 511 446 237 564 075 245 568;
  • 24) 0.666 666 666 511 446 237 564 075 245 568 × 2 = 1 + 0.333 333 333 022 892 475 128 150 491 136;
  • 25) 0.333 333 333 022 892 475 128 150 491 136 × 2 = 0 + 0.666 666 666 045 784 950 256 300 982 272;
  • 26) 0.666 666 666 045 784 950 256 300 982 272 × 2 = 1 + 0.333 333 332 091 569 900 512 601 964 544;
  • 27) 0.333 333 332 091 569 900 512 601 964 544 × 2 = 0 + 0.666 666 664 183 139 801 025 203 929 088;
  • 28) 0.666 666 664 183 139 801 025 203 929 088 × 2 = 1 + 0.333 333 328 366 279 602 050 407 858 176;
  • 29) 0.333 333 328 366 279 602 050 407 858 176 × 2 = 0 + 0.666 666 656 732 559 204 100 815 716 352;
  • 30) 0.666 666 656 732 559 204 100 815 716 352 × 2 = 1 + 0.333 333 313 465 118 408 201 631 432 704;
  • 31) 0.333 333 313 465 118 408 201 631 432 704 × 2 = 0 + 0.666 666 626 930 236 816 403 262 865 408;
  • 32) 0.666 666 626 930 236 816 403 262 865 408 × 2 = 1 + 0.333 333 253 860 473 632 806 525 730 816;
  • 33) 0.333 333 253 860 473 632 806 525 730 816 × 2 = 0 + 0.666 666 507 720 947 265 613 051 461 632;
  • 34) 0.666 666 507 720 947 265 613 051 461 632 × 2 = 1 + 0.333 333 015 441 894 531 226 102 923 264;
  • 35) 0.333 333 015 441 894 531 226 102 923 264 × 2 = 0 + 0.666 666 030 883 789 062 452 205 846 528;
  • 36) 0.666 666 030 883 789 062 452 205 846 528 × 2 = 1 + 0.333 332 061 767 578 124 904 411 693 056;
  • 37) 0.333 332 061 767 578 124 904 411 693 056 × 2 = 0 + 0.666 664 123 535 156 249 808 823 386 112;
  • 38) 0.666 664 123 535 156 249 808 823 386 112 × 2 = 1 + 0.333 328 247 070 312 499 617 646 772 224;
  • 39) 0.333 328 247 070 312 499 617 646 772 224 × 2 = 0 + 0.666 656 494 140 624 999 235 293 544 448;
  • 40) 0.666 656 494 140 624 999 235 293 544 448 × 2 = 1 + 0.333 312 988 281 249 998 470 587 088 896;
  • 41) 0.333 312 988 281 249 998 470 587 088 896 × 2 = 0 + 0.666 625 976 562 499 996 941 174 177 792;
  • 42) 0.666 625 976 562 499 996 941 174 177 792 × 2 = 1 + 0.333 251 953 124 999 993 882 348 355 584;
  • 43) 0.333 251 953 124 999 993 882 348 355 584 × 2 = 0 + 0.666 503 906 249 999 987 764 696 711 168;
  • 44) 0.666 503 906 249 999 987 764 696 711 168 × 2 = 1 + 0.333 007 812 499 999 975 529 393 422 336;
  • 45) 0.333 007 812 499 999 975 529 393 422 336 × 2 = 0 + 0.666 015 624 999 999 951 058 786 844 672;
  • 46) 0.666 015 624 999 999 951 058 786 844 672 × 2 = 1 + 0.332 031 249 999 999 902 117 573 689 344;
  • 47) 0.332 031 249 999 999 902 117 573 689 344 × 2 = 0 + 0.664 062 499 999 999 804 235 147 378 688;
  • 48) 0.664 062 499 999 999 804 235 147 378 688 × 2 = 1 + 0.328 124 999 999 999 608 470 294 757 376;
  • 49) 0.328 124 999 999 999 608 470 294 757 376 × 2 = 0 + 0.656 249 999 999 999 216 940 589 514 752;
  • 50) 0.656 249 999 999 999 216 940 589 514 752 × 2 = 1 + 0.312 499 999 999 998 433 881 179 029 504;
  • 51) 0.312 499 999 999 998 433 881 179 029 504 × 2 = 0 + 0.624 999 999 999 996 867 762 358 059 008;
  • 52) 0.624 999 999 999 996 867 762 358 059 008 × 2 = 1 + 0.249 999 999 999 993 735 524 716 118 016;
  • 53) 0.249 999 999 999 993 735 524 716 118 016 × 2 = 0 + 0.499 999 999 999 987 471 049 432 236 032;
  • 54) 0.499 999 999 999 987 471 049 432 236 032 × 2 = 0 + 0.999 999 999 999 974 942 098 864 472 064;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.333 333 333 333 333 314 829 616 256 246(10) =

0.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 00(2)

5. Positive number before normalization:

0.333 333 333 333 333 314 829 616 256 246(10) =

0.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 00(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 2 positions to the right, so that only one non zero digit remains to the left of it:


0.333 333 333 333 333 314 829 616 256 246(10) =

0.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 00(2) =

0.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 00(2) × 20 =

1.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100(2) × 2-2


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -2

Mantissa (not normalized):
1.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-2 + 2(11-1) - 1 =

(-2 + 1 023)(10) =

1 021(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 021 ÷ 2 = 510 + 1;
  • 510 ÷ 2 = 255 + 0;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1021(10) =

011 1111 1101(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100 =

0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1111 1101

Mantissa (52 bits) =
0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100


Decimal number 0.333 333 333 333 333 314 829 616 256 246 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1101 - 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100