-2.211 829 052 383 358 300 119 548 661 699 653 280 265 896 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -2.211 829 052 383 358 300 119 548 661 699 653 280 265 896(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
-2.211 829 052 383 358 300 119 548 661 699 653 280 265 896(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. Start with the positive version of the number:

|-2.211 829 052 383 358 300 119 548 661 699 653 280 265 896| = 2.211 829 052 383 358 300 119 548 661 699 653 280 265 896


2. First, convert to binary (in base 2) the integer part: 2.
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;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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.

2(10) =

10(2)


4. Convert to binary (base 2) the fractional part: 0.211 829 052 383 358 300 119 548 661 699 653 280 265 896.

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) 0.211 829 052 383 358 300 119 548 661 699 653 280 265 896 × 2 = 0 + 0.423 658 104 766 716 600 239 097 323 399 306 560 531 792;
  • 2) 0.423 658 104 766 716 600 239 097 323 399 306 560 531 792 × 2 = 0 + 0.847 316 209 533 433 200 478 194 646 798 613 121 063 584;
  • 3) 0.847 316 209 533 433 200 478 194 646 798 613 121 063 584 × 2 = 1 + 0.694 632 419 066 866 400 956 389 293 597 226 242 127 168;
  • 4) 0.694 632 419 066 866 400 956 389 293 597 226 242 127 168 × 2 = 1 + 0.389 264 838 133 732 801 912 778 587 194 452 484 254 336;
  • 5) 0.389 264 838 133 732 801 912 778 587 194 452 484 254 336 × 2 = 0 + 0.778 529 676 267 465 603 825 557 174 388 904 968 508 672;
  • 6) 0.778 529 676 267 465 603 825 557 174 388 904 968 508 672 × 2 = 1 + 0.557 059 352 534 931 207 651 114 348 777 809 937 017 344;
  • 7) 0.557 059 352 534 931 207 651 114 348 777 809 937 017 344 × 2 = 1 + 0.114 118 705 069 862 415 302 228 697 555 619 874 034 688;
  • 8) 0.114 118 705 069 862 415 302 228 697 555 619 874 034 688 × 2 = 0 + 0.228 237 410 139 724 830 604 457 395 111 239 748 069 376;
  • 9) 0.228 237 410 139 724 830 604 457 395 111 239 748 069 376 × 2 = 0 + 0.456 474 820 279 449 661 208 914 790 222 479 496 138 752;
  • 10) 0.456 474 820 279 449 661 208 914 790 222 479 496 138 752 × 2 = 0 + 0.912 949 640 558 899 322 417 829 580 444 958 992 277 504;
  • 11) 0.912 949 640 558 899 322 417 829 580 444 958 992 277 504 × 2 = 1 + 0.825 899 281 117 798 644 835 659 160 889 917 984 555 008;
  • 12) 0.825 899 281 117 798 644 835 659 160 889 917 984 555 008 × 2 = 1 + 0.651 798 562 235 597 289 671 318 321 779 835 969 110 016;
  • 13) 0.651 798 562 235 597 289 671 318 321 779 835 969 110 016 × 2 = 1 + 0.303 597 124 471 194 579 342 636 643 559 671 938 220 032;
  • 14) 0.303 597 124 471 194 579 342 636 643 559 671 938 220 032 × 2 = 0 + 0.607 194 248 942 389 158 685 273 287 119 343 876 440 064;
  • 15) 0.607 194 248 942 389 158 685 273 287 119 343 876 440 064 × 2 = 1 + 0.214 388 497 884 778 317 370 546 574 238 687 752 880 128;
  • 16) 0.214 388 497 884 778 317 370 546 574 238 687 752 880 128 × 2 = 0 + 0.428 776 995 769 556 634 741 093 148 477 375 505 760 256;
  • 17) 0.428 776 995 769 556 634 741 093 148 477 375 505 760 256 × 2 = 0 + 0.857 553 991 539 113 269 482 186 296 954 751 011 520 512;
  • 18) 0.857 553 991 539 113 269 482 186 296 954 751 011 520 512 × 2 = 1 + 0.715 107 983 078 226 538 964 372 593 909 502 023 041 024;
  • 19) 0.715 107 983 078 226 538 964 372 593 909 502 023 041 024 × 2 = 1 + 0.430 215 966 156 453 077 928 745 187 819 004 046 082 048;
  • 20) 0.430 215 966 156 453 077 928 745 187 819 004 046 082 048 × 2 = 0 + 0.860 431 932 312 906 155 857 490 375 638 008 092 164 096;
  • 21) 0.860 431 932 312 906 155 857 490 375 638 008 092 164 096 × 2 = 1 + 0.720 863 864 625 812 311 714 980 751 276 016 184 328 192;
  • 22) 0.720 863 864 625 812 311 714 980 751 276 016 184 328 192 × 2 = 1 + 0.441 727 729 251 624 623 429 961 502 552 032 368 656 384;
  • 23) 0.441 727 729 251 624 623 429 961 502 552 032 368 656 384 × 2 = 0 + 0.883 455 458 503 249 246 859 923 005 104 064 737 312 768;
  • 24) 0.883 455 458 503 249 246 859 923 005 104 064 737 312 768 × 2 = 1 + 0.766 910 917 006 498 493 719 846 010 208 129 474 625 536;
  • 25) 0.766 910 917 006 498 493 719 846 010 208 129 474 625 536 × 2 = 1 + 0.533 821 834 012 996 987 439 692 020 416 258 949 251 072;
  • 26) 0.533 821 834 012 996 987 439 692 020 416 258 949 251 072 × 2 = 1 + 0.067 643 668 025 993 974 879 384 040 832 517 898 502 144;
  • 27) 0.067 643 668 025 993 974 879 384 040 832 517 898 502 144 × 2 = 0 + 0.135 287 336 051 987 949 758 768 081 665 035 797 004 288;
  • 28) 0.135 287 336 051 987 949 758 768 081 665 035 797 004 288 × 2 = 0 + 0.270 574 672 103 975 899 517 536 163 330 071 594 008 576;
  • 29) 0.270 574 672 103 975 899 517 536 163 330 071 594 008 576 × 2 = 0 + 0.541 149 344 207 951 799 035 072 326 660 143 188 017 152;
  • 30) 0.541 149 344 207 951 799 035 072 326 660 143 188 017 152 × 2 = 1 + 0.082 298 688 415 903 598 070 144 653 320 286 376 034 304;
  • 31) 0.082 298 688 415 903 598 070 144 653 320 286 376 034 304 × 2 = 0 + 0.164 597 376 831 807 196 140 289 306 640 572 752 068 608;
  • 32) 0.164 597 376 831 807 196 140 289 306 640 572 752 068 608 × 2 = 0 + 0.329 194 753 663 614 392 280 578 613 281 145 504 137 216;
  • 33) 0.329 194 753 663 614 392 280 578 613 281 145 504 137 216 × 2 = 0 + 0.658 389 507 327 228 784 561 157 226 562 291 008 274 432;
  • 34) 0.658 389 507 327 228 784 561 157 226 562 291 008 274 432 × 2 = 1 + 0.316 779 014 654 457 569 122 314 453 124 582 016 548 864;
  • 35) 0.316 779 014 654 457 569 122 314 453 124 582 016 548 864 × 2 = 0 + 0.633 558 029 308 915 138 244 628 906 249 164 033 097 728;
  • 36) 0.633 558 029 308 915 138 244 628 906 249 164 033 097 728 × 2 = 1 + 0.267 116 058 617 830 276 489 257 812 498 328 066 195 456;
  • 37) 0.267 116 058 617 830 276 489 257 812 498 328 066 195 456 × 2 = 0 + 0.534 232 117 235 660 552 978 515 624 996 656 132 390 912;
  • 38) 0.534 232 117 235 660 552 978 515 624 996 656 132 390 912 × 2 = 1 + 0.068 464 234 471 321 105 957 031 249 993 312 264 781 824;
  • 39) 0.068 464 234 471 321 105 957 031 249 993 312 264 781 824 × 2 = 0 + 0.136 928 468 942 642 211 914 062 499 986 624 529 563 648;
  • 40) 0.136 928 468 942 642 211 914 062 499 986 624 529 563 648 × 2 = 0 + 0.273 856 937 885 284 423 828 124 999 973 249 059 127 296;
  • 41) 0.273 856 937 885 284 423 828 124 999 973 249 059 127 296 × 2 = 0 + 0.547 713 875 770 568 847 656 249 999 946 498 118 254 592;
  • 42) 0.547 713 875 770 568 847 656 249 999 946 498 118 254 592 × 2 = 1 + 0.095 427 751 541 137 695 312 499 999 892 996 236 509 184;
  • 43) 0.095 427 751 541 137 695 312 499 999 892 996 236 509 184 × 2 = 0 + 0.190 855 503 082 275 390 624 999 999 785 992 473 018 368;
  • 44) 0.190 855 503 082 275 390 624 999 999 785 992 473 018 368 × 2 = 0 + 0.381 711 006 164 550 781 249 999 999 571 984 946 036 736;
  • 45) 0.381 711 006 164 550 781 249 999 999 571 984 946 036 736 × 2 = 0 + 0.763 422 012 329 101 562 499 999 999 143 969 892 073 472;
  • 46) 0.763 422 012 329 101 562 499 999 999 143 969 892 073 472 × 2 = 1 + 0.526 844 024 658 203 124 999 999 998 287 939 784 146 944;
  • 47) 0.526 844 024 658 203 124 999 999 998 287 939 784 146 944 × 2 = 1 + 0.053 688 049 316 406 249 999 999 996 575 879 568 293 888;
  • 48) 0.053 688 049 316 406 249 999 999 996 575 879 568 293 888 × 2 = 0 + 0.107 376 098 632 812 499 999 999 993 151 759 136 587 776;
  • 49) 0.107 376 098 632 812 499 999 999 993 151 759 136 587 776 × 2 = 0 + 0.214 752 197 265 624 999 999 999 986 303 518 273 175 552;
  • 50) 0.214 752 197 265 624 999 999 999 986 303 518 273 175 552 × 2 = 0 + 0.429 504 394 531 249 999 999 999 972 607 036 546 351 104;
  • 51) 0.429 504 394 531 249 999 999 999 972 607 036 546 351 104 × 2 = 0 + 0.859 008 789 062 499 999 999 999 945 214 073 092 702 208;
  • 52) 0.859 008 789 062 499 999 999 999 945 214 073 092 702 208 × 2 = 1 + 0.718 017 578 124 999 999 999 999 890 428 146 185 404 416;
  • 53) 0.718 017 578 124 999 999 999 999 890 428 146 185 404 416 × 2 = 1 + 0.436 035 156 249 999 999 999 999 780 856 292 370 808 832;

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:


0.211 829 052 383 358 300 119 548 661 699 653 280 265 896(10) =

0.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2)

6. Positive number before normalization:

2.211 829 052 383 358 300 119 548 661 699 653 280 265 896(10) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the left, so that only one non zero digit remains to the left of it:


2.211 829 052 383 358 300 119 548 661 699 653 280 265 896(10) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2) × 20 =

1.0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11(2) × 21


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): 1

Mantissa (not normalized):
1.0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

1 + 2(11-1) - 1 =

(1 + 1 023)(10) =

1 024(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;
  • 1 024 ÷ 2 = 512 + 0;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

1024(10) =

100 0000 0000(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, 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. 0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11 =

0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000


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) =
100 0000 0000

Mantissa (52 bits) =
0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000


Decimal number -2.211 829 052 383 358 300 119 548 661 699 653 280 265 896 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 0000 - 0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000

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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