1.000 000 021 979 552 668 138 406 918 054 708 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.000 000 021 979 552 668 138 406 918 054 708(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
1.000 000 021 979 552 668 138 406 918 054 708(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: 0.000 000 021 979 552 668 138 406 918 054 708.

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) 0.000 000 021 979 552 668 138 406 918 054 708 × 2 = 0 + 0.000 000 043 959 105 336 276 813 836 109 416;
  • 2) 0.000 000 043 959 105 336 276 813 836 109 416 × 2 = 0 + 0.000 000 087 918 210 672 553 627 672 218 832;
  • 3) 0.000 000 087 918 210 672 553 627 672 218 832 × 2 = 0 + 0.000 000 175 836 421 345 107 255 344 437 664;
  • 4) 0.000 000 175 836 421 345 107 255 344 437 664 × 2 = 0 + 0.000 000 351 672 842 690 214 510 688 875 328;
  • 5) 0.000 000 351 672 842 690 214 510 688 875 328 × 2 = 0 + 0.000 000 703 345 685 380 429 021 377 750 656;
  • 6) 0.000 000 703 345 685 380 429 021 377 750 656 × 2 = 0 + 0.000 001 406 691 370 760 858 042 755 501 312;
  • 7) 0.000 001 406 691 370 760 858 042 755 501 312 × 2 = 0 + 0.000 002 813 382 741 521 716 085 511 002 624;
  • 8) 0.000 002 813 382 741 521 716 085 511 002 624 × 2 = 0 + 0.000 005 626 765 483 043 432 171 022 005 248;
  • 9) 0.000 005 626 765 483 043 432 171 022 005 248 × 2 = 0 + 0.000 011 253 530 966 086 864 342 044 010 496;
  • 10) 0.000 011 253 530 966 086 864 342 044 010 496 × 2 = 0 + 0.000 022 507 061 932 173 728 684 088 020 992;
  • 11) 0.000 022 507 061 932 173 728 684 088 020 992 × 2 = 0 + 0.000 045 014 123 864 347 457 368 176 041 984;
  • 12) 0.000 045 014 123 864 347 457 368 176 041 984 × 2 = 0 + 0.000 090 028 247 728 694 914 736 352 083 968;
  • 13) 0.000 090 028 247 728 694 914 736 352 083 968 × 2 = 0 + 0.000 180 056 495 457 389 829 472 704 167 936;
  • 14) 0.000 180 056 495 457 389 829 472 704 167 936 × 2 = 0 + 0.000 360 112 990 914 779 658 945 408 335 872;
  • 15) 0.000 360 112 990 914 779 658 945 408 335 872 × 2 = 0 + 0.000 720 225 981 829 559 317 890 816 671 744;
  • 16) 0.000 720 225 981 829 559 317 890 816 671 744 × 2 = 0 + 0.001 440 451 963 659 118 635 781 633 343 488;
  • 17) 0.001 440 451 963 659 118 635 781 633 343 488 × 2 = 0 + 0.002 880 903 927 318 237 271 563 266 686 976;
  • 18) 0.002 880 903 927 318 237 271 563 266 686 976 × 2 = 0 + 0.005 761 807 854 636 474 543 126 533 373 952;
  • 19) 0.005 761 807 854 636 474 543 126 533 373 952 × 2 = 0 + 0.011 523 615 709 272 949 086 253 066 747 904;
  • 20) 0.011 523 615 709 272 949 086 253 066 747 904 × 2 = 0 + 0.023 047 231 418 545 898 172 506 133 495 808;
  • 21) 0.023 047 231 418 545 898 172 506 133 495 808 × 2 = 0 + 0.046 094 462 837 091 796 345 012 266 991 616;
  • 22) 0.046 094 462 837 091 796 345 012 266 991 616 × 2 = 0 + 0.092 188 925 674 183 592 690 024 533 983 232;
  • 23) 0.092 188 925 674 183 592 690 024 533 983 232 × 2 = 0 + 0.184 377 851 348 367 185 380 049 067 966 464;
  • 24) 0.184 377 851 348 367 185 380 049 067 966 464 × 2 = 0 + 0.368 755 702 696 734 370 760 098 135 932 928;
  • 25) 0.368 755 702 696 734 370 760 098 135 932 928 × 2 = 0 + 0.737 511 405 393 468 741 520 196 271 865 856;
  • 26) 0.737 511 405 393 468 741 520 196 271 865 856 × 2 = 1 + 0.475 022 810 786 937 483 040 392 543 731 712;
  • 27) 0.475 022 810 786 937 483 040 392 543 731 712 × 2 = 0 + 0.950 045 621 573 874 966 080 785 087 463 424;
  • 28) 0.950 045 621 573 874 966 080 785 087 463 424 × 2 = 1 + 0.900 091 243 147 749 932 161 570 174 926 848;
  • 29) 0.900 091 243 147 749 932 161 570 174 926 848 × 2 = 1 + 0.800 182 486 295 499 864 323 140 349 853 696;
  • 30) 0.800 182 486 295 499 864 323 140 349 853 696 × 2 = 1 + 0.600 364 972 590 999 728 646 280 699 707 392;
  • 31) 0.600 364 972 590 999 728 646 280 699 707 392 × 2 = 1 + 0.200 729 945 181 999 457 292 561 399 414 784;
  • 32) 0.200 729 945 181 999 457 292 561 399 414 784 × 2 = 0 + 0.401 459 890 363 998 914 585 122 798 829 568;
  • 33) 0.401 459 890 363 998 914 585 122 798 829 568 × 2 = 0 + 0.802 919 780 727 997 829 170 245 597 659 136;
  • 34) 0.802 919 780 727 997 829 170 245 597 659 136 × 2 = 1 + 0.605 839 561 455 995 658 340 491 195 318 272;
  • 35) 0.605 839 561 455 995 658 340 491 195 318 272 × 2 = 1 + 0.211 679 122 911 991 316 680 982 390 636 544;
  • 36) 0.211 679 122 911 991 316 680 982 390 636 544 × 2 = 0 + 0.423 358 245 823 982 633 361 964 781 273 088;
  • 37) 0.423 358 245 823 982 633 361 964 781 273 088 × 2 = 0 + 0.846 716 491 647 965 266 723 929 562 546 176;
  • 38) 0.846 716 491 647 965 266 723 929 562 546 176 × 2 = 1 + 0.693 432 983 295 930 533 447 859 125 092 352;
  • 39) 0.693 432 983 295 930 533 447 859 125 092 352 × 2 = 1 + 0.386 865 966 591 861 066 895 718 250 184 704;
  • 40) 0.386 865 966 591 861 066 895 718 250 184 704 × 2 = 0 + 0.773 731 933 183 722 133 791 436 500 369 408;
  • 41) 0.773 731 933 183 722 133 791 436 500 369 408 × 2 = 1 + 0.547 463 866 367 444 267 582 873 000 738 816;
  • 42) 0.547 463 866 367 444 267 582 873 000 738 816 × 2 = 1 + 0.094 927 732 734 888 535 165 746 001 477 632;
  • 43) 0.094 927 732 734 888 535 165 746 001 477 632 × 2 = 0 + 0.189 855 465 469 777 070 331 492 002 955 264;
  • 44) 0.189 855 465 469 777 070 331 492 002 955 264 × 2 = 0 + 0.379 710 930 939 554 140 662 984 005 910 528;
  • 45) 0.379 710 930 939 554 140 662 984 005 910 528 × 2 = 0 + 0.759 421 861 879 108 281 325 968 011 821 056;
  • 46) 0.759 421 861 879 108 281 325 968 011 821 056 × 2 = 1 + 0.518 843 723 758 216 562 651 936 023 642 112;
  • 47) 0.518 843 723 758 216 562 651 936 023 642 112 × 2 = 1 + 0.037 687 447 516 433 125 303 872 047 284 224;
  • 48) 0.037 687 447 516 433 125 303 872 047 284 224 × 2 = 0 + 0.075 374 895 032 866 250 607 744 094 568 448;
  • 49) 0.075 374 895 032 866 250 607 744 094 568 448 × 2 = 0 + 0.150 749 790 065 732 501 215 488 189 136 896;
  • 50) 0.150 749 790 065 732 501 215 488 189 136 896 × 2 = 0 + 0.301 499 580 131 465 002 430 976 378 273 792;
  • 51) 0.301 499 580 131 465 002 430 976 378 273 792 × 2 = 0 + 0.602 999 160 262 930 004 861 952 756 547 584;
  • 52) 0.602 999 160 262 930 004 861 952 756 547 584 × 2 = 1 + 0.205 998 320 525 860 009 723 905 513 095 168;
  • 53) 0.205 998 320 525 860 009 723 905 513 095 168 × 2 = 0 + 0.411 996 641 051 720 019 447 811 026 190 336;

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:


0.000 000 021 979 552 668 138 406 918 054 708(10) =

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

5. Positive number before normalization:

1.000 000 021 979 552 668 138 406 918 054 708(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:


1.000 000 021 979 552 668 138 406 918 054 708(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 1.000 000 021 979 552 668 138 406 918 054 708 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 1.000 000 021 979 552 668 138 406 918 054 691 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