1.000 000 021 979 552 668 138 406 918 053 86 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.000 000 021 979 552 668 138 406 918 053 86(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
1.000 000 021 979 552 668 138 406 918 053 86(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: 1.
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;
  • 1 ÷ 2 = 0 + 1;

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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 021 979 552 668 138 406 918 053 86.

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) 0.000 000 021 979 552 668 138 406 918 053 86 × 2 = 0 + 0.000 000 043 959 105 336 276 813 836 107 72;
  • 2) 0.000 000 043 959 105 336 276 813 836 107 72 × 2 = 0 + 0.000 000 087 918 210 672 553 627 672 215 44;
  • 3) 0.000 000 087 918 210 672 553 627 672 215 44 × 2 = 0 + 0.000 000 175 836 421 345 107 255 344 430 88;
  • 4) 0.000 000 175 836 421 345 107 255 344 430 88 × 2 = 0 + 0.000 000 351 672 842 690 214 510 688 861 76;
  • 5) 0.000 000 351 672 842 690 214 510 688 861 76 × 2 = 0 + 0.000 000 703 345 685 380 429 021 377 723 52;
  • 6) 0.000 000 703 345 685 380 429 021 377 723 52 × 2 = 0 + 0.000 001 406 691 370 760 858 042 755 447 04;
  • 7) 0.000 001 406 691 370 760 858 042 755 447 04 × 2 = 0 + 0.000 002 813 382 741 521 716 085 510 894 08;
  • 8) 0.000 002 813 382 741 521 716 085 510 894 08 × 2 = 0 + 0.000 005 626 765 483 043 432 171 021 788 16;
  • 9) 0.000 005 626 765 483 043 432 171 021 788 16 × 2 = 0 + 0.000 011 253 530 966 086 864 342 043 576 32;
  • 10) 0.000 011 253 530 966 086 864 342 043 576 32 × 2 = 0 + 0.000 022 507 061 932 173 728 684 087 152 64;
  • 11) 0.000 022 507 061 932 173 728 684 087 152 64 × 2 = 0 + 0.000 045 014 123 864 347 457 368 174 305 28;
  • 12) 0.000 045 014 123 864 347 457 368 174 305 28 × 2 = 0 + 0.000 090 028 247 728 694 914 736 348 610 56;
  • 13) 0.000 090 028 247 728 694 914 736 348 610 56 × 2 = 0 + 0.000 180 056 495 457 389 829 472 697 221 12;
  • 14) 0.000 180 056 495 457 389 829 472 697 221 12 × 2 = 0 + 0.000 360 112 990 914 779 658 945 394 442 24;
  • 15) 0.000 360 112 990 914 779 658 945 394 442 24 × 2 = 0 + 0.000 720 225 981 829 559 317 890 788 884 48;
  • 16) 0.000 720 225 981 829 559 317 890 788 884 48 × 2 = 0 + 0.001 440 451 963 659 118 635 781 577 768 96;
  • 17) 0.001 440 451 963 659 118 635 781 577 768 96 × 2 = 0 + 0.002 880 903 927 318 237 271 563 155 537 92;
  • 18) 0.002 880 903 927 318 237 271 563 155 537 92 × 2 = 0 + 0.005 761 807 854 636 474 543 126 311 075 84;
  • 19) 0.005 761 807 854 636 474 543 126 311 075 84 × 2 = 0 + 0.011 523 615 709 272 949 086 252 622 151 68;
  • 20) 0.011 523 615 709 272 949 086 252 622 151 68 × 2 = 0 + 0.023 047 231 418 545 898 172 505 244 303 36;
  • 21) 0.023 047 231 418 545 898 172 505 244 303 36 × 2 = 0 + 0.046 094 462 837 091 796 345 010 488 606 72;
  • 22) 0.046 094 462 837 091 796 345 010 488 606 72 × 2 = 0 + 0.092 188 925 674 183 592 690 020 977 213 44;
  • 23) 0.092 188 925 674 183 592 690 020 977 213 44 × 2 = 0 + 0.184 377 851 348 367 185 380 041 954 426 88;
  • 24) 0.184 377 851 348 367 185 380 041 954 426 88 × 2 = 0 + 0.368 755 702 696 734 370 760 083 908 853 76;
  • 25) 0.368 755 702 696 734 370 760 083 908 853 76 × 2 = 0 + 0.737 511 405 393 468 741 520 167 817 707 52;
  • 26) 0.737 511 405 393 468 741 520 167 817 707 52 × 2 = 1 + 0.475 022 810 786 937 483 040 335 635 415 04;
  • 27) 0.475 022 810 786 937 483 040 335 635 415 04 × 2 = 0 + 0.950 045 621 573 874 966 080 671 270 830 08;
  • 28) 0.950 045 621 573 874 966 080 671 270 830 08 × 2 = 1 + 0.900 091 243 147 749 932 161 342 541 660 16;
  • 29) 0.900 091 243 147 749 932 161 342 541 660 16 × 2 = 1 + 0.800 182 486 295 499 864 322 685 083 320 32;
  • 30) 0.800 182 486 295 499 864 322 685 083 320 32 × 2 = 1 + 0.600 364 972 590 999 728 645 370 166 640 64;
  • 31) 0.600 364 972 590 999 728 645 370 166 640 64 × 2 = 1 + 0.200 729 945 181 999 457 290 740 333 281 28;
  • 32) 0.200 729 945 181 999 457 290 740 333 281 28 × 2 = 0 + 0.401 459 890 363 998 914 581 480 666 562 56;
  • 33) 0.401 459 890 363 998 914 581 480 666 562 56 × 2 = 0 + 0.802 919 780 727 997 829 162 961 333 125 12;
  • 34) 0.802 919 780 727 997 829 162 961 333 125 12 × 2 = 1 + 0.605 839 561 455 995 658 325 922 666 250 24;
  • 35) 0.605 839 561 455 995 658 325 922 666 250 24 × 2 = 1 + 0.211 679 122 911 991 316 651 845 332 500 48;
  • 36) 0.211 679 122 911 991 316 651 845 332 500 48 × 2 = 0 + 0.423 358 245 823 982 633 303 690 665 000 96;
  • 37) 0.423 358 245 823 982 633 303 690 665 000 96 × 2 = 0 + 0.846 716 491 647 965 266 607 381 330 001 92;
  • 38) 0.846 716 491 647 965 266 607 381 330 001 92 × 2 = 1 + 0.693 432 983 295 930 533 214 762 660 003 84;
  • 39) 0.693 432 983 295 930 533 214 762 660 003 84 × 2 = 1 + 0.386 865 966 591 861 066 429 525 320 007 68;
  • 40) 0.386 865 966 591 861 066 429 525 320 007 68 × 2 = 0 + 0.773 731 933 183 722 132 859 050 640 015 36;
  • 41) 0.773 731 933 183 722 132 859 050 640 015 36 × 2 = 1 + 0.547 463 866 367 444 265 718 101 280 030 72;
  • 42) 0.547 463 866 367 444 265 718 101 280 030 72 × 2 = 1 + 0.094 927 732 734 888 531 436 202 560 061 44;
  • 43) 0.094 927 732 734 888 531 436 202 560 061 44 × 2 = 0 + 0.189 855 465 469 777 062 872 405 120 122 88;
  • 44) 0.189 855 465 469 777 062 872 405 120 122 88 × 2 = 0 + 0.379 710 930 939 554 125 744 810 240 245 76;
  • 45) 0.379 710 930 939 554 125 744 810 240 245 76 × 2 = 0 + 0.759 421 861 879 108 251 489 620 480 491 52;
  • 46) 0.759 421 861 879 108 251 489 620 480 491 52 × 2 = 1 + 0.518 843 723 758 216 502 979 240 960 983 04;
  • 47) 0.518 843 723 758 216 502 979 240 960 983 04 × 2 = 1 + 0.037 687 447 516 433 005 958 481 921 966 08;
  • 48) 0.037 687 447 516 433 005 958 481 921 966 08 × 2 = 0 + 0.075 374 895 032 866 011 916 963 843 932 16;
  • 49) 0.075 374 895 032 866 011 916 963 843 932 16 × 2 = 0 + 0.150 749 790 065 732 023 833 927 687 864 32;
  • 50) 0.150 749 790 065 732 023 833 927 687 864 32 × 2 = 0 + 0.301 499 580 131 464 047 667 855 375 728 64;
  • 51) 0.301 499 580 131 464 047 667 855 375 728 64 × 2 = 0 + 0.602 999 160 262 928 095 335 710 751 457 28;
  • 52) 0.602 999 160 262 928 095 335 710 751 457 28 × 2 = 1 + 0.205 998 320 525 856 190 671 421 502 914 56;
  • 53) 0.205 998 320 525 856 190 671 421 502 914 56 × 2 = 0 + 0.411 996 641 051 712 381 342 843 005 829 12;

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:


0.000 000 021 979 552 668 138 406 918 053 86(10) =

0.0000 0000 0000 0000 0000 0000 0101 1110 0110 0110 1100 0110 0001 0(2)

5. Positive number before normalization:

1.000 000 021 979 552 668 138 406 918 053 86(10) =

1.0000 0000 0000 0000 0000 0000 0101 1110 0110 0110 1100 0110 0001 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.000 000 021 979 552 668 138 406 918 053 86(10) =

1.0000 0000 0000 0000 0000 0000 0101 1110 0110 0110 1100 0110 0001 0(2) =

1.0000 0000 0000 0000 0000 0000 0101 1110 0110 0110 1100 0110 0001 0(2) × 20


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): 0

Mantissa (not normalized):
1.0000 0000 0000 0000 0000 0000 0101 1110 0110 0110 1100 0110 0001 0


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 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) =

1023(10) =

011 1111 1111(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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0000 0000 0000 0000 0000 0000 0101 1110 0110 0110 1100 0110 0001 0 =

0000 0000 0000 0000 0000 0000 0101 1110 0110 0110 1100 0110 0001


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 1111

Mantissa (52 bits) =
0000 0000 0000 0000 0000 0000 0101 1110 0110 0110 1100 0110 0001


Decimal number 1.000 000 021 979 552 668 138 406 918 053 86 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 0000 0000 0000 0000 0000 0000 0101 1110 0110 0110 1100 0110 0001

» Convert decimal 1.000 000 021 979 552 668 138 406 918 053 71 to 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