Decimal to 64 Bit IEEE 754 Binary: Convert Number -0.000 000 010 829 423 819 7 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number -0.000 000 010 829 423 819 7(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Start with the positive version of the number:

|-0.000 000 010 829 423 819 7| = 0.000 000 010 829 423 819 7


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 000 010 829 423 819 7.

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) 0.000 000 010 829 423 819 7 × 2 = 0 + 0.000 000 021 658 847 639 4;
  • 2) 0.000 000 021 658 847 639 4 × 2 = 0 + 0.000 000 043 317 695 278 8;
  • 3) 0.000 000 043 317 695 278 8 × 2 = 0 + 0.000 000 086 635 390 557 6;
  • 4) 0.000 000 086 635 390 557 6 × 2 = 0 + 0.000 000 173 270 781 115 2;
  • 5) 0.000 000 173 270 781 115 2 × 2 = 0 + 0.000 000 346 541 562 230 4;
  • 6) 0.000 000 346 541 562 230 4 × 2 = 0 + 0.000 000 693 083 124 460 8;
  • 7) 0.000 000 693 083 124 460 8 × 2 = 0 + 0.000 001 386 166 248 921 6;
  • 8) 0.000 001 386 166 248 921 6 × 2 = 0 + 0.000 002 772 332 497 843 2;
  • 9) 0.000 002 772 332 497 843 2 × 2 = 0 + 0.000 005 544 664 995 686 4;
  • 10) 0.000 005 544 664 995 686 4 × 2 = 0 + 0.000 011 089 329 991 372 8;
  • 11) 0.000 011 089 329 991 372 8 × 2 = 0 + 0.000 022 178 659 982 745 6;
  • 12) 0.000 022 178 659 982 745 6 × 2 = 0 + 0.000 044 357 319 965 491 2;
  • 13) 0.000 044 357 319 965 491 2 × 2 = 0 + 0.000 088 714 639 930 982 4;
  • 14) 0.000 088 714 639 930 982 4 × 2 = 0 + 0.000 177 429 279 861 964 8;
  • 15) 0.000 177 429 279 861 964 8 × 2 = 0 + 0.000 354 858 559 723 929 6;
  • 16) 0.000 354 858 559 723 929 6 × 2 = 0 + 0.000 709 717 119 447 859 2;
  • 17) 0.000 709 717 119 447 859 2 × 2 = 0 + 0.001 419 434 238 895 718 4;
  • 18) 0.001 419 434 238 895 718 4 × 2 = 0 + 0.002 838 868 477 791 436 8;
  • 19) 0.002 838 868 477 791 436 8 × 2 = 0 + 0.005 677 736 955 582 873 6;
  • 20) 0.005 677 736 955 582 873 6 × 2 = 0 + 0.011 355 473 911 165 747 2;
  • 21) 0.011 355 473 911 165 747 2 × 2 = 0 + 0.022 710 947 822 331 494 4;
  • 22) 0.022 710 947 822 331 494 4 × 2 = 0 + 0.045 421 895 644 662 988 8;
  • 23) 0.045 421 895 644 662 988 8 × 2 = 0 + 0.090 843 791 289 325 977 6;
  • 24) 0.090 843 791 289 325 977 6 × 2 = 0 + 0.181 687 582 578 651 955 2;
  • 25) 0.181 687 582 578 651 955 2 × 2 = 0 + 0.363 375 165 157 303 910 4;
  • 26) 0.363 375 165 157 303 910 4 × 2 = 0 + 0.726 750 330 314 607 820 8;
  • 27) 0.726 750 330 314 607 820 8 × 2 = 1 + 0.453 500 660 629 215 641 6;
  • 28) 0.453 500 660 629 215 641 6 × 2 = 0 + 0.907 001 321 258 431 283 2;
  • 29) 0.907 001 321 258 431 283 2 × 2 = 1 + 0.814 002 642 516 862 566 4;
  • 30) 0.814 002 642 516 862 566 4 × 2 = 1 + 0.628 005 285 033 725 132 8;
  • 31) 0.628 005 285 033 725 132 8 × 2 = 1 + 0.256 010 570 067 450 265 6;
  • 32) 0.256 010 570 067 450 265 6 × 2 = 0 + 0.512 021 140 134 900 531 2;
  • 33) 0.512 021 140 134 900 531 2 × 2 = 1 + 0.024 042 280 269 801 062 4;
  • 34) 0.024 042 280 269 801 062 4 × 2 = 0 + 0.048 084 560 539 602 124 8;
  • 35) 0.048 084 560 539 602 124 8 × 2 = 0 + 0.096 169 121 079 204 249 6;
  • 36) 0.096 169 121 079 204 249 6 × 2 = 0 + 0.192 338 242 158 408 499 2;
  • 37) 0.192 338 242 158 408 499 2 × 2 = 0 + 0.384 676 484 316 816 998 4;
  • 38) 0.384 676 484 316 816 998 4 × 2 = 0 + 0.769 352 968 633 633 996 8;
  • 39) 0.769 352 968 633 633 996 8 × 2 = 1 + 0.538 705 937 267 267 993 6;
  • 40) 0.538 705 937 267 267 993 6 × 2 = 1 + 0.077 411 874 534 535 987 2;
  • 41) 0.077 411 874 534 535 987 2 × 2 = 0 + 0.154 823 749 069 071 974 4;
  • 42) 0.154 823 749 069 071 974 4 × 2 = 0 + 0.309 647 498 138 143 948 8;
  • 43) 0.309 647 498 138 143 948 8 × 2 = 0 + 0.619 294 996 276 287 897 6;
  • 44) 0.619 294 996 276 287 897 6 × 2 = 1 + 0.238 589 992 552 575 795 2;
  • 45) 0.238 589 992 552 575 795 2 × 2 = 0 + 0.477 179 985 105 151 590 4;
  • 46) 0.477 179 985 105 151 590 4 × 2 = 0 + 0.954 359 970 210 303 180 8;
  • 47) 0.954 359 970 210 303 180 8 × 2 = 1 + 0.908 719 940 420 606 361 6;
  • 48) 0.908 719 940 420 606 361 6 × 2 = 1 + 0.817 439 880 841 212 723 2;
  • 49) 0.817 439 880 841 212 723 2 × 2 = 1 + 0.634 879 761 682 425 446 4;
  • 50) 0.634 879 761 682 425 446 4 × 2 = 1 + 0.269 759 523 364 850 892 8;
  • 51) 0.269 759 523 364 850 892 8 × 2 = 0 + 0.539 519 046 729 701 785 6;
  • 52) 0.539 519 046 729 701 785 6 × 2 = 1 + 0.079 038 093 459 403 571 2;
  • 53) 0.079 038 093 459 403 571 2 × 2 = 0 + 0.158 076 186 918 807 142 4;
  • 54) 0.158 076 186 918 807 142 4 × 2 = 0 + 0.316 152 373 837 614 284 8;
  • 55) 0.316 152 373 837 614 284 8 × 2 = 0 + 0.632 304 747 675 228 569 6;
  • 56) 0.632 304 747 675 228 569 6 × 2 = 1 + 0.264 609 495 350 457 139 2;
  • 57) 0.264 609 495 350 457 139 2 × 2 = 0 + 0.529 218 990 700 914 278 4;
  • 58) 0.529 218 990 700 914 278 4 × 2 = 1 + 0.058 437 981 401 828 556 8;
  • 59) 0.058 437 981 401 828 556 8 × 2 = 0 + 0.116 875 962 803 657 113 6;
  • 60) 0.116 875 962 803 657 113 6 × 2 = 0 + 0.233 751 925 607 314 227 2;
  • 61) 0.233 751 925 607 314 227 2 × 2 = 0 + 0.467 503 851 214 628 454 4;
  • 62) 0.467 503 851 214 628 454 4 × 2 = 0 + 0.935 007 702 429 256 908 8;
  • 63) 0.935 007 702 429 256 908 8 × 2 = 1 + 0.870 015 404 858 513 817 6;
  • 64) 0.870 015 404 858 513 817 6 × 2 = 1 + 0.740 030 809 717 027 635 2;
  • 65) 0.740 030 809 717 027 635 2 × 2 = 1 + 0.480 061 619 434 055 270 4;
  • 66) 0.480 061 619 434 055 270 4 × 2 = 0 + 0.960 123 238 868 110 540 8;
  • 67) 0.960 123 238 868 110 540 8 × 2 = 1 + 0.920 246 477 736 221 081 6;
  • 68) 0.920 246 477 736 221 081 6 × 2 = 1 + 0.840 492 955 472 442 163 2;
  • 69) 0.840 492 955 472 442 163 2 × 2 = 1 + 0.680 985 910 944 884 326 4;
  • 70) 0.680 985 910 944 884 326 4 × 2 = 1 + 0.361 971 821 889 768 652 8;
  • 71) 0.361 971 821 889 768 652 8 × 2 = 0 + 0.723 943 643 779 537 305 6;
  • 72) 0.723 943 643 779 537 305 6 × 2 = 1 + 0.447 887 287 559 074 611 2;
  • 73) 0.447 887 287 559 074 611 2 × 2 = 0 + 0.895 774 575 118 149 222 4;
  • 74) 0.895 774 575 118 149 222 4 × 2 = 1 + 0.791 549 150 236 298 444 8;
  • 75) 0.791 549 150 236 298 444 8 × 2 = 1 + 0.583 098 300 472 596 889 6;
  • 76) 0.583 098 300 472 596 889 6 × 2 = 1 + 0.166 196 600 945 193 779 2;
  • 77) 0.166 196 600 945 193 779 2 × 2 = 0 + 0.332 393 201 890 387 558 4;
  • 78) 0.332 393 201 890 387 558 4 × 2 = 0 + 0.664 786 403 780 775 116 8;
  • 79) 0.664 786 403 780 775 116 8 × 2 = 1 + 0.329 572 807 561 550 233 6;

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).


5. 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:


0.000 000 010 829 423 819 7(10) =

0.0000 0000 0000 0000 0000 0000 0010 1110 1000 0011 0001 0011 1101 0001 0100 0011 1011 1101 0111 001(2)

6. Positive number before normalization:

0.000 000 010 829 423 819 7(10) =

0.0000 0000 0000 0000 0000 0000 0010 1110 1000 0011 0001 0011 1101 0001 0100 0011 1011 1101 0111 001(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 27 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 010 829 423 819 7(10) =

0.0000 0000 0000 0000 0000 0000 0010 1110 1000 0011 0001 0011 1101 0001 0100 0011 1011 1101 0111 001(2) =

0.0000 0000 0000 0000 0000 0000 0010 1110 1000 0011 0001 0011 1101 0001 0100 0011 1011 1101 0111 001(2) × 20 =

1.0111 0100 0001 1000 1001 1110 1000 1010 0001 1101 1110 1011 1001(2) × 2-27


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): -27

Mantissa (not normalized):
1.0111 0100 0001 1000 1001 1110 1000 1010 0001 1101 1110 1011 1001


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-27 + 2(11-1) - 1 =

(-27 + 1 023)(10) =

996(10)


10. 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;
  • 996 ÷ 2 = 498 + 0;
  • 498 ÷ 2 = 249 + 0;
  • 249 ÷ 2 = 124 + 1;
  • 124 ÷ 2 = 62 + 0;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =

996(10) =

011 1110 0100(2)


12. 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. 0111 0100 0001 1000 1001 1110 1000 1010 0001 1101 1110 1011 1001 =

0111 0100 0001 1000 1001 1110 1000 1010 0001 1101 1110 1011 1001


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

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

Exponent (11 bits) =
011 1110 0100

Mantissa (52 bits) =
0111 0100 0001 1000 1001 1110 1000 1010 0001 1101 1110 1011 1001


The base ten decimal number -0.000 000 010 829 423 819 7 converted and written in 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1110 0100 - 0111 0100 0001 1000 1001 1110 1000 1010 0001 1101 1110 1011 1001

» Decimal number in base ten -0.000 000 010 829 423 829 3 converted and written as 64 bit double precision IEEE 754 binary floating point representation standard

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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