64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 0.000 000 127 591 192 722 320 4 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 0.000 000 127 591 192 722 320 4(10) converted and written in 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: 0.000 000 127 591 192 722 320 4.

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) 0.000 000 127 591 192 722 320 4 × 2 = 0 + 0.000 000 255 182 385 444 640 8;
  • 2) 0.000 000 255 182 385 444 640 8 × 2 = 0 + 0.000 000 510 364 770 889 281 6;
  • 3) 0.000 000 510 364 770 889 281 6 × 2 = 0 + 0.000 001 020 729 541 778 563 2;
  • 4) 0.000 001 020 729 541 778 563 2 × 2 = 0 + 0.000 002 041 459 083 557 126 4;
  • 5) 0.000 002 041 459 083 557 126 4 × 2 = 0 + 0.000 004 082 918 167 114 252 8;
  • 6) 0.000 004 082 918 167 114 252 8 × 2 = 0 + 0.000 008 165 836 334 228 505 6;
  • 7) 0.000 008 165 836 334 228 505 6 × 2 = 0 + 0.000 016 331 672 668 457 011 2;
  • 8) 0.000 016 331 672 668 457 011 2 × 2 = 0 + 0.000 032 663 345 336 914 022 4;
  • 9) 0.000 032 663 345 336 914 022 4 × 2 = 0 + 0.000 065 326 690 673 828 044 8;
  • 10) 0.000 065 326 690 673 828 044 8 × 2 = 0 + 0.000 130 653 381 347 656 089 6;
  • 11) 0.000 130 653 381 347 656 089 6 × 2 = 0 + 0.000 261 306 762 695 312 179 2;
  • 12) 0.000 261 306 762 695 312 179 2 × 2 = 0 + 0.000 522 613 525 390 624 358 4;
  • 13) 0.000 522 613 525 390 624 358 4 × 2 = 0 + 0.001 045 227 050 781 248 716 8;
  • 14) 0.001 045 227 050 781 248 716 8 × 2 = 0 + 0.002 090 454 101 562 497 433 6;
  • 15) 0.002 090 454 101 562 497 433 6 × 2 = 0 + 0.004 180 908 203 124 994 867 2;
  • 16) 0.004 180 908 203 124 994 867 2 × 2 = 0 + 0.008 361 816 406 249 989 734 4;
  • 17) 0.008 361 816 406 249 989 734 4 × 2 = 0 + 0.016 723 632 812 499 979 468 8;
  • 18) 0.016 723 632 812 499 979 468 8 × 2 = 0 + 0.033 447 265 624 999 958 937 6;
  • 19) 0.033 447 265 624 999 958 937 6 × 2 = 0 + 0.066 894 531 249 999 917 875 2;
  • 20) 0.066 894 531 249 999 917 875 2 × 2 = 0 + 0.133 789 062 499 999 835 750 4;
  • 21) 0.133 789 062 499 999 835 750 4 × 2 = 0 + 0.267 578 124 999 999 671 500 8;
  • 22) 0.267 578 124 999 999 671 500 8 × 2 = 0 + 0.535 156 249 999 999 343 001 6;
  • 23) 0.535 156 249 999 999 343 001 6 × 2 = 1 + 0.070 312 499 999 998 686 003 2;
  • 24) 0.070 312 499 999 998 686 003 2 × 2 = 0 + 0.140 624 999 999 997 372 006 4;
  • 25) 0.140 624 999 999 997 372 006 4 × 2 = 0 + 0.281 249 999 999 994 744 012 8;
  • 26) 0.281 249 999 999 994 744 012 8 × 2 = 0 + 0.562 499 999 999 989 488 025 6;
  • 27) 0.562 499 999 999 989 488 025 6 × 2 = 1 + 0.124 999 999 999 978 976 051 2;
  • 28) 0.124 999 999 999 978 976 051 2 × 2 = 0 + 0.249 999 999 999 957 952 102 4;
  • 29) 0.249 999 999 999 957 952 102 4 × 2 = 0 + 0.499 999 999 999 915 904 204 8;
  • 30) 0.499 999 999 999 915 904 204 8 × 2 = 0 + 0.999 999 999 999 831 808 409 6;
  • 31) 0.999 999 999 999 831 808 409 6 × 2 = 1 + 0.999 999 999 999 663 616 819 2;
  • 32) 0.999 999 999 999 663 616 819 2 × 2 = 1 + 0.999 999 999 999 327 233 638 4;
  • 33) 0.999 999 999 999 327 233 638 4 × 2 = 1 + 0.999 999 999 998 654 467 276 8;
  • 34) 0.999 999 999 998 654 467 276 8 × 2 = 1 + 0.999 999 999 997 308 934 553 6;
  • 35) 0.999 999 999 997 308 934 553 6 × 2 = 1 + 0.999 999 999 994 617 869 107 2;
  • 36) 0.999 999 999 994 617 869 107 2 × 2 = 1 + 0.999 999 999 989 235 738 214 4;
  • 37) 0.999 999 999 989 235 738 214 4 × 2 = 1 + 0.999 999 999 978 471 476 428 8;
  • 38) 0.999 999 999 978 471 476 428 8 × 2 = 1 + 0.999 999 999 956 942 952 857 6;
  • 39) 0.999 999 999 956 942 952 857 6 × 2 = 1 + 0.999 999 999 913 885 905 715 2;
  • 40) 0.999 999 999 913 885 905 715 2 × 2 = 1 + 0.999 999 999 827 771 811 430 4;
  • 41) 0.999 999 999 827 771 811 430 4 × 2 = 1 + 0.999 999 999 655 543 622 860 8;
  • 42) 0.999 999 999 655 543 622 860 8 × 2 = 1 + 0.999 999 999 311 087 245 721 6;
  • 43) 0.999 999 999 311 087 245 721 6 × 2 = 1 + 0.999 999 998 622 174 491 443 2;
  • 44) 0.999 999 998 622 174 491 443 2 × 2 = 1 + 0.999 999 997 244 348 982 886 4;
  • 45) 0.999 999 997 244 348 982 886 4 × 2 = 1 + 0.999 999 994 488 697 965 772 8;
  • 46) 0.999 999 994 488 697 965 772 8 × 2 = 1 + 0.999 999 988 977 395 931 545 6;
  • 47) 0.999 999 988 977 395 931 545 6 × 2 = 1 + 0.999 999 977 954 791 863 091 2;
  • 48) 0.999 999 977 954 791 863 091 2 × 2 = 1 + 0.999 999 955 909 583 726 182 4;
  • 49) 0.999 999 955 909 583 726 182 4 × 2 = 1 + 0.999 999 911 819 167 452 364 8;
  • 50) 0.999 999 911 819 167 452 364 8 × 2 = 1 + 0.999 999 823 638 334 904 729 6;
  • 51) 0.999 999 823 638 334 904 729 6 × 2 = 1 + 0.999 999 647 276 669 809 459 2;
  • 52) 0.999 999 647 276 669 809 459 2 × 2 = 1 + 0.999 999 294 553 339 618 918 4;
  • 53) 0.999 999 294 553 339 618 918 4 × 2 = 1 + 0.999 998 589 106 679 237 836 8;
  • 54) 0.999 998 589 106 679 237 836 8 × 2 = 1 + 0.999 997 178 213 358 475 673 6;
  • 55) 0.999 997 178 213 358 475 673 6 × 2 = 1 + 0.999 994 356 426 716 951 347 2;
  • 56) 0.999 994 356 426 716 951 347 2 × 2 = 1 + 0.999 988 712 853 433 902 694 4;
  • 57) 0.999 988 712 853 433 902 694 4 × 2 = 1 + 0.999 977 425 706 867 805 388 8;
  • 58) 0.999 977 425 706 867 805 388 8 × 2 = 1 + 0.999 954 851 413 735 610 777 6;
  • 59) 0.999 954 851 413 735 610 777 6 × 2 = 1 + 0.999 909 702 827 471 221 555 2;
  • 60) 0.999 909 702 827 471 221 555 2 × 2 = 1 + 0.999 819 405 654 942 443 110 4;
  • 61) 0.999 819 405 654 942 443 110 4 × 2 = 1 + 0.999 638 811 309 884 886 220 8;
  • 62) 0.999 638 811 309 884 886 220 8 × 2 = 1 + 0.999 277 622 619 769 772 441 6;
  • 63) 0.999 277 622 619 769 772 441 6 × 2 = 1 + 0.998 555 245 239 539 544 883 2;
  • 64) 0.998 555 245 239 539 544 883 2 × 2 = 1 + 0.997 110 490 479 079 089 766 4;
  • 65) 0.997 110 490 479 079 089 766 4 × 2 = 1 + 0.994 220 980 958 158 179 532 8;
  • 66) 0.994 220 980 958 158 179 532 8 × 2 = 1 + 0.988 441 961 916 316 359 065 6;
  • 67) 0.988 441 961 916 316 359 065 6 × 2 = 1 + 0.976 883 923 832 632 718 131 2;
  • 68) 0.976 883 923 832 632 718 131 2 × 2 = 1 + 0.953 767 847 665 265 436 262 4;
  • 69) 0.953 767 847 665 265 436 262 4 × 2 = 1 + 0.907 535 695 330 530 872 524 8;
  • 70) 0.907 535 695 330 530 872 524 8 × 2 = 1 + 0.815 071 390 661 061 745 049 6;
  • 71) 0.815 071 390 661 061 745 049 6 × 2 = 1 + 0.630 142 781 322 123 490 099 2;
  • 72) 0.630 142 781 322 123 490 099 2 × 2 = 1 + 0.260 285 562 644 246 980 198 4;
  • 73) 0.260 285 562 644 246 980 198 4 × 2 = 0 + 0.520 571 125 288 493 960 396 8;
  • 74) 0.520 571 125 288 493 960 396 8 × 2 = 1 + 0.041 142 250 576 987 920 793 6;
  • 75) 0.041 142 250 576 987 920 793 6 × 2 = 0 + 0.082 284 501 153 975 841 587 2;

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...)


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:


0.000 000 127 591 192 722 320 4(10) =

0.0000 0000 0000 0000 0000 0010 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 010(2)


5. Positive number before normalization:

0.000 000 127 591 192 722 320 4(10) =

0.0000 0000 0000 0000 0000 0010 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 23 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 127 591 192 722 320 4(10) =

0.0000 0000 0000 0000 0000 0010 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 010(2) =

0.0000 0000 0000 0000 0000 0010 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 010(2) × 20 =

1.0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010(2) × 2-23


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): -23

Mantissa (not normalized):
1.0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-23 + 2(11-1) - 1 =

(-23 + 1 023)(10) =

1 000(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;
  • 1 000 ÷ 2 = 500 + 0;
  • 500 ÷ 2 = 250 + 0;
  • 250 ÷ 2 = 125 + 0;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 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) =

1000(10) =

011 1110 1000(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. 0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010 =

0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010


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 1110 1000

Mantissa (52 bits) =
0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010


The base ten decimal number 0.000 000 127 591 192 722 320 4 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 011 1110 1000 - 0001 0001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1010

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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