0.333 333 333 333 333 314 829 616 255 51 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.333 333 333 333 333 314 829 616 255 51(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
0.333 333 333 333 333 314 829 616 255 51(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: 0.333 333 333 333 333 314 829 616 255 51.

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) 0.333 333 333 333 333 314 829 616 255 51 × 2 = 0 + 0.666 666 666 666 666 629 659 232 511 02;
  • 2) 0.666 666 666 666 666 629 659 232 511 02 × 2 = 1 + 0.333 333 333 333 333 259 318 465 022 04;
  • 3) 0.333 333 333 333 333 259 318 465 022 04 × 2 = 0 + 0.666 666 666 666 666 518 636 930 044 08;
  • 4) 0.666 666 666 666 666 518 636 930 044 08 × 2 = 1 + 0.333 333 333 333 333 037 273 860 088 16;
  • 5) 0.333 333 333 333 333 037 273 860 088 16 × 2 = 0 + 0.666 666 666 666 666 074 547 720 176 32;
  • 6) 0.666 666 666 666 666 074 547 720 176 32 × 2 = 1 + 0.333 333 333 333 332 149 095 440 352 64;
  • 7) 0.333 333 333 333 332 149 095 440 352 64 × 2 = 0 + 0.666 666 666 666 664 298 190 880 705 28;
  • 8) 0.666 666 666 666 664 298 190 880 705 28 × 2 = 1 + 0.333 333 333 333 328 596 381 761 410 56;
  • 9) 0.333 333 333 333 328 596 381 761 410 56 × 2 = 0 + 0.666 666 666 666 657 192 763 522 821 12;
  • 10) 0.666 666 666 666 657 192 763 522 821 12 × 2 = 1 + 0.333 333 333 333 314 385 527 045 642 24;
  • 11) 0.333 333 333 333 314 385 527 045 642 24 × 2 = 0 + 0.666 666 666 666 628 771 054 091 284 48;
  • 12) 0.666 666 666 666 628 771 054 091 284 48 × 2 = 1 + 0.333 333 333 333 257 542 108 182 568 96;
  • 13) 0.333 333 333 333 257 542 108 182 568 96 × 2 = 0 + 0.666 666 666 666 515 084 216 365 137 92;
  • 14) 0.666 666 666 666 515 084 216 365 137 92 × 2 = 1 + 0.333 333 333 333 030 168 432 730 275 84;
  • 15) 0.333 333 333 333 030 168 432 730 275 84 × 2 = 0 + 0.666 666 666 666 060 336 865 460 551 68;
  • 16) 0.666 666 666 666 060 336 865 460 551 68 × 2 = 1 + 0.333 333 333 332 120 673 730 921 103 36;
  • 17) 0.333 333 333 332 120 673 730 921 103 36 × 2 = 0 + 0.666 666 666 664 241 347 461 842 206 72;
  • 18) 0.666 666 666 664 241 347 461 842 206 72 × 2 = 1 + 0.333 333 333 328 482 694 923 684 413 44;
  • 19) 0.333 333 333 328 482 694 923 684 413 44 × 2 = 0 + 0.666 666 666 656 965 389 847 368 826 88;
  • 20) 0.666 666 666 656 965 389 847 368 826 88 × 2 = 1 + 0.333 333 333 313 930 779 694 737 653 76;
  • 21) 0.333 333 333 313 930 779 694 737 653 76 × 2 = 0 + 0.666 666 666 627 861 559 389 475 307 52;
  • 22) 0.666 666 666 627 861 559 389 475 307 52 × 2 = 1 + 0.333 333 333 255 723 118 778 950 615 04;
  • 23) 0.333 333 333 255 723 118 778 950 615 04 × 2 = 0 + 0.666 666 666 511 446 237 557 901 230 08;
  • 24) 0.666 666 666 511 446 237 557 901 230 08 × 2 = 1 + 0.333 333 333 022 892 475 115 802 460 16;
  • 25) 0.333 333 333 022 892 475 115 802 460 16 × 2 = 0 + 0.666 666 666 045 784 950 231 604 920 32;
  • 26) 0.666 666 666 045 784 950 231 604 920 32 × 2 = 1 + 0.333 333 332 091 569 900 463 209 840 64;
  • 27) 0.333 333 332 091 569 900 463 209 840 64 × 2 = 0 + 0.666 666 664 183 139 800 926 419 681 28;
  • 28) 0.666 666 664 183 139 800 926 419 681 28 × 2 = 1 + 0.333 333 328 366 279 601 852 839 362 56;
  • 29) 0.333 333 328 366 279 601 852 839 362 56 × 2 = 0 + 0.666 666 656 732 559 203 705 678 725 12;
  • 30) 0.666 666 656 732 559 203 705 678 725 12 × 2 = 1 + 0.333 333 313 465 118 407 411 357 450 24;
  • 31) 0.333 333 313 465 118 407 411 357 450 24 × 2 = 0 + 0.666 666 626 930 236 814 822 714 900 48;
  • 32) 0.666 666 626 930 236 814 822 714 900 48 × 2 = 1 + 0.333 333 253 860 473 629 645 429 800 96;
  • 33) 0.333 333 253 860 473 629 645 429 800 96 × 2 = 0 + 0.666 666 507 720 947 259 290 859 601 92;
  • 34) 0.666 666 507 720 947 259 290 859 601 92 × 2 = 1 + 0.333 333 015 441 894 518 581 719 203 84;
  • 35) 0.333 333 015 441 894 518 581 719 203 84 × 2 = 0 + 0.666 666 030 883 789 037 163 438 407 68;
  • 36) 0.666 666 030 883 789 037 163 438 407 68 × 2 = 1 + 0.333 332 061 767 578 074 326 876 815 36;
  • 37) 0.333 332 061 767 578 074 326 876 815 36 × 2 = 0 + 0.666 664 123 535 156 148 653 753 630 72;
  • 38) 0.666 664 123 535 156 148 653 753 630 72 × 2 = 1 + 0.333 328 247 070 312 297 307 507 261 44;
  • 39) 0.333 328 247 070 312 297 307 507 261 44 × 2 = 0 + 0.666 656 494 140 624 594 615 014 522 88;
  • 40) 0.666 656 494 140 624 594 615 014 522 88 × 2 = 1 + 0.333 312 988 281 249 189 230 029 045 76;
  • 41) 0.333 312 988 281 249 189 230 029 045 76 × 2 = 0 + 0.666 625 976 562 498 378 460 058 091 52;
  • 42) 0.666 625 976 562 498 378 460 058 091 52 × 2 = 1 + 0.333 251 953 124 996 756 920 116 183 04;
  • 43) 0.333 251 953 124 996 756 920 116 183 04 × 2 = 0 + 0.666 503 906 249 993 513 840 232 366 08;
  • 44) 0.666 503 906 249 993 513 840 232 366 08 × 2 = 1 + 0.333 007 812 499 987 027 680 464 732 16;
  • 45) 0.333 007 812 499 987 027 680 464 732 16 × 2 = 0 + 0.666 015 624 999 974 055 360 929 464 32;
  • 46) 0.666 015 624 999 974 055 360 929 464 32 × 2 = 1 + 0.332 031 249 999 948 110 721 858 928 64;
  • 47) 0.332 031 249 999 948 110 721 858 928 64 × 2 = 0 + 0.664 062 499 999 896 221 443 717 857 28;
  • 48) 0.664 062 499 999 896 221 443 717 857 28 × 2 = 1 + 0.328 124 999 999 792 442 887 435 714 56;
  • 49) 0.328 124 999 999 792 442 887 435 714 56 × 2 = 0 + 0.656 249 999 999 584 885 774 871 429 12;
  • 50) 0.656 249 999 999 584 885 774 871 429 12 × 2 = 1 + 0.312 499 999 999 169 771 549 742 858 24;
  • 51) 0.312 499 999 999 169 771 549 742 858 24 × 2 = 0 + 0.624 999 999 998 339 543 099 485 716 48;
  • 52) 0.624 999 999 998 339 543 099 485 716 48 × 2 = 1 + 0.249 999 999 996 679 086 198 971 432 96;
  • 53) 0.249 999 999 996 679 086 198 971 432 96 × 2 = 0 + 0.499 999 999 993 358 172 397 942 865 92;
  • 54) 0.499 999 999 993 358 172 397 942 865 92 × 2 = 0 + 0.999 999 999 986 716 344 795 885 731 84;

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:


0.333 333 333 333 333 314 829 616 255 51(10) =

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

5. Positive number before normalization:

0.333 333 333 333 333 314 829 616 255 51(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:


0.333 333 333 333 333 314 829 616 255 51(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 0.333 333 333 333 333 314 829 616 255 51 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

Share

» Convert decimal 0.333 333 333 333 333 314 829 616 255 89 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 08, 2026 [August]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: August - by our visitors

» Improve your social skills: The Attention Economy - The Battlefield For Your Focus. Gemini Notebook YouTube Video of 7.50 minutes

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