0.000 000 000 000 000 032 487 359 776 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 032 487 359 776(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
0.000 000 000 000 000 032 487 359 776(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: 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: 0.000 000 000 000 000 032 487 359 776.

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) 0.000 000 000 000 000 032 487 359 776 × 2 = 0 + 0.000 000 000 000 000 064 974 719 552;
  • 2) 0.000 000 000 000 000 064 974 719 552 × 2 = 0 + 0.000 000 000 000 000 129 949 439 104;
  • 3) 0.000 000 000 000 000 129 949 439 104 × 2 = 0 + 0.000 000 000 000 000 259 898 878 208;
  • 4) 0.000 000 000 000 000 259 898 878 208 × 2 = 0 + 0.000 000 000 000 000 519 797 756 416;
  • 5) 0.000 000 000 000 000 519 797 756 416 × 2 = 0 + 0.000 000 000 000 001 039 595 512 832;
  • 6) 0.000 000 000 000 001 039 595 512 832 × 2 = 0 + 0.000 000 000 000 002 079 191 025 664;
  • 7) 0.000 000 000 000 002 079 191 025 664 × 2 = 0 + 0.000 000 000 000 004 158 382 051 328;
  • 8) 0.000 000 000 000 004 158 382 051 328 × 2 = 0 + 0.000 000 000 000 008 316 764 102 656;
  • 9) 0.000 000 000 000 008 316 764 102 656 × 2 = 0 + 0.000 000 000 000 016 633 528 205 312;
  • 10) 0.000 000 000 000 016 633 528 205 312 × 2 = 0 + 0.000 000 000 000 033 267 056 410 624;
  • 11) 0.000 000 000 000 033 267 056 410 624 × 2 = 0 + 0.000 000 000 000 066 534 112 821 248;
  • 12) 0.000 000 000 000 066 534 112 821 248 × 2 = 0 + 0.000 000 000 000 133 068 225 642 496;
  • 13) 0.000 000 000 000 133 068 225 642 496 × 2 = 0 + 0.000 000 000 000 266 136 451 284 992;
  • 14) 0.000 000 000 000 266 136 451 284 992 × 2 = 0 + 0.000 000 000 000 532 272 902 569 984;
  • 15) 0.000 000 000 000 532 272 902 569 984 × 2 = 0 + 0.000 000 000 001 064 545 805 139 968;
  • 16) 0.000 000 000 001 064 545 805 139 968 × 2 = 0 + 0.000 000 000 002 129 091 610 279 936;
  • 17) 0.000 000 000 002 129 091 610 279 936 × 2 = 0 + 0.000 000 000 004 258 183 220 559 872;
  • 18) 0.000 000 000 004 258 183 220 559 872 × 2 = 0 + 0.000 000 000 008 516 366 441 119 744;
  • 19) 0.000 000 000 008 516 366 441 119 744 × 2 = 0 + 0.000 000 000 017 032 732 882 239 488;
  • 20) 0.000 000 000 017 032 732 882 239 488 × 2 = 0 + 0.000 000 000 034 065 465 764 478 976;
  • 21) 0.000 000 000 034 065 465 764 478 976 × 2 = 0 + 0.000 000 000 068 130 931 528 957 952;
  • 22) 0.000 000 000 068 130 931 528 957 952 × 2 = 0 + 0.000 000 000 136 261 863 057 915 904;
  • 23) 0.000 000 000 136 261 863 057 915 904 × 2 = 0 + 0.000 000 000 272 523 726 115 831 808;
  • 24) 0.000 000 000 272 523 726 115 831 808 × 2 = 0 + 0.000 000 000 545 047 452 231 663 616;
  • 25) 0.000 000 000 545 047 452 231 663 616 × 2 = 0 + 0.000 000 001 090 094 904 463 327 232;
  • 26) 0.000 000 001 090 094 904 463 327 232 × 2 = 0 + 0.000 000 002 180 189 808 926 654 464;
  • 27) 0.000 000 002 180 189 808 926 654 464 × 2 = 0 + 0.000 000 004 360 379 617 853 308 928;
  • 28) 0.000 000 004 360 379 617 853 308 928 × 2 = 0 + 0.000 000 008 720 759 235 706 617 856;
  • 29) 0.000 000 008 720 759 235 706 617 856 × 2 = 0 + 0.000 000 017 441 518 471 413 235 712;
  • 30) 0.000 000 017 441 518 471 413 235 712 × 2 = 0 + 0.000 000 034 883 036 942 826 471 424;
  • 31) 0.000 000 034 883 036 942 826 471 424 × 2 = 0 + 0.000 000 069 766 073 885 652 942 848;
  • 32) 0.000 000 069 766 073 885 652 942 848 × 2 = 0 + 0.000 000 139 532 147 771 305 885 696;
  • 33) 0.000 000 139 532 147 771 305 885 696 × 2 = 0 + 0.000 000 279 064 295 542 611 771 392;
  • 34) 0.000 000 279 064 295 542 611 771 392 × 2 = 0 + 0.000 000 558 128 591 085 223 542 784;
  • 35) 0.000 000 558 128 591 085 223 542 784 × 2 = 0 + 0.000 001 116 257 182 170 447 085 568;
  • 36) 0.000 001 116 257 182 170 447 085 568 × 2 = 0 + 0.000 002 232 514 364 340 894 171 136;
  • 37) 0.000 002 232 514 364 340 894 171 136 × 2 = 0 + 0.000 004 465 028 728 681 788 342 272;
  • 38) 0.000 004 465 028 728 681 788 342 272 × 2 = 0 + 0.000 008 930 057 457 363 576 684 544;
  • 39) 0.000 008 930 057 457 363 576 684 544 × 2 = 0 + 0.000 017 860 114 914 727 153 369 088;
  • 40) 0.000 017 860 114 914 727 153 369 088 × 2 = 0 + 0.000 035 720 229 829 454 306 738 176;
  • 41) 0.000 035 720 229 829 454 306 738 176 × 2 = 0 + 0.000 071 440 459 658 908 613 476 352;
  • 42) 0.000 071 440 459 658 908 613 476 352 × 2 = 0 + 0.000 142 880 919 317 817 226 952 704;
  • 43) 0.000 142 880 919 317 817 226 952 704 × 2 = 0 + 0.000 285 761 838 635 634 453 905 408;
  • 44) 0.000 285 761 838 635 634 453 905 408 × 2 = 0 + 0.000 571 523 677 271 268 907 810 816;
  • 45) 0.000 571 523 677 271 268 907 810 816 × 2 = 0 + 0.001 143 047 354 542 537 815 621 632;
  • 46) 0.001 143 047 354 542 537 815 621 632 × 2 = 0 + 0.002 286 094 709 085 075 631 243 264;
  • 47) 0.002 286 094 709 085 075 631 243 264 × 2 = 0 + 0.004 572 189 418 170 151 262 486 528;
  • 48) 0.004 572 189 418 170 151 262 486 528 × 2 = 0 + 0.009 144 378 836 340 302 524 973 056;
  • 49) 0.009 144 378 836 340 302 524 973 056 × 2 = 0 + 0.018 288 757 672 680 605 049 946 112;
  • 50) 0.018 288 757 672 680 605 049 946 112 × 2 = 0 + 0.036 577 515 345 361 210 099 892 224;
  • 51) 0.036 577 515 345 361 210 099 892 224 × 2 = 0 + 0.073 155 030 690 722 420 199 784 448;
  • 52) 0.073 155 030 690 722 420 199 784 448 × 2 = 0 + 0.146 310 061 381 444 840 399 568 896;
  • 53) 0.146 310 061 381 444 840 399 568 896 × 2 = 0 + 0.292 620 122 762 889 680 799 137 792;
  • 54) 0.292 620 122 762 889 680 799 137 792 × 2 = 0 + 0.585 240 245 525 779 361 598 275 584;
  • 55) 0.585 240 245 525 779 361 598 275 584 × 2 = 1 + 0.170 480 491 051 558 723 196 551 168;
  • 56) 0.170 480 491 051 558 723 196 551 168 × 2 = 0 + 0.340 960 982 103 117 446 393 102 336;
  • 57) 0.340 960 982 103 117 446 393 102 336 × 2 = 0 + 0.681 921 964 206 234 892 786 204 672;
  • 58) 0.681 921 964 206 234 892 786 204 672 × 2 = 1 + 0.363 843 928 412 469 785 572 409 344;
  • 59) 0.363 843 928 412 469 785 572 409 344 × 2 = 0 + 0.727 687 856 824 939 571 144 818 688;
  • 60) 0.727 687 856 824 939 571 144 818 688 × 2 = 1 + 0.455 375 713 649 879 142 289 637 376;
  • 61) 0.455 375 713 649 879 142 289 637 376 × 2 = 0 + 0.910 751 427 299 758 284 579 274 752;
  • 62) 0.910 751 427 299 758 284 579 274 752 × 2 = 1 + 0.821 502 854 599 516 569 158 549 504;
  • 63) 0.821 502 854 599 516 569 158 549 504 × 2 = 1 + 0.643 005 709 199 033 138 317 099 008;
  • 64) 0.643 005 709 199 033 138 317 099 008 × 2 = 1 + 0.286 011 418 398 066 276 634 198 016;
  • 65) 0.286 011 418 398 066 276 634 198 016 × 2 = 0 + 0.572 022 836 796 132 553 268 396 032;
  • 66) 0.572 022 836 796 132 553 268 396 032 × 2 = 1 + 0.144 045 673 592 265 106 536 792 064;
  • 67) 0.144 045 673 592 265 106 536 792 064 × 2 = 0 + 0.288 091 347 184 530 213 073 584 128;
  • 68) 0.288 091 347 184 530 213 073 584 128 × 2 = 0 + 0.576 182 694 369 060 426 147 168 256;
  • 69) 0.576 182 694 369 060 426 147 168 256 × 2 = 1 + 0.152 365 388 738 120 852 294 336 512;
  • 70) 0.152 365 388 738 120 852 294 336 512 × 2 = 0 + 0.304 730 777 476 241 704 588 673 024;
  • 71) 0.304 730 777 476 241 704 588 673 024 × 2 = 0 + 0.609 461 554 952 483 409 177 346 048;
  • 72) 0.609 461 554 952 483 409 177 346 048 × 2 = 1 + 0.218 923 109 904 966 818 354 692 096;
  • 73) 0.218 923 109 904 966 818 354 692 096 × 2 = 0 + 0.437 846 219 809 933 636 709 384 192;
  • 74) 0.437 846 219 809 933 636 709 384 192 × 2 = 0 + 0.875 692 439 619 867 273 418 768 384;
  • 75) 0.875 692 439 619 867 273 418 768 384 × 2 = 1 + 0.751 384 879 239 734 546 837 536 768;
  • 76) 0.751 384 879 239 734 546 837 536 768 × 2 = 1 + 0.502 769 758 479 469 093 675 073 536;
  • 77) 0.502 769 758 479 469 093 675 073 536 × 2 = 1 + 0.005 539 516 958 938 187 350 147 072;
  • 78) 0.005 539 516 958 938 187 350 147 072 × 2 = 0 + 0.011 079 033 917 876 374 700 294 144;
  • 79) 0.011 079 033 917 876 374 700 294 144 × 2 = 0 + 0.022 158 067 835 752 749 400 588 288;
  • 80) 0.022 158 067 835 752 749 400 588 288 × 2 = 0 + 0.044 316 135 671 505 498 801 176 576;
  • 81) 0.044 316 135 671 505 498 801 176 576 × 2 = 0 + 0.088 632 271 343 010 997 602 353 152;
  • 82) 0.088 632 271 343 010 997 602 353 152 × 2 = 0 + 0.177 264 542 686 021 995 204 706 304;
  • 83) 0.177 264 542 686 021 995 204 706 304 × 2 = 0 + 0.354 529 085 372 043 990 409 412 608;
  • 84) 0.354 529 085 372 043 990 409 412 608 × 2 = 0 + 0.709 058 170 744 087 980 818 825 216;
  • 85) 0.709 058 170 744 087 980 818 825 216 × 2 = 1 + 0.418 116 341 488 175 961 637 650 432;
  • 86) 0.418 116 341 488 175 961 637 650 432 × 2 = 0 + 0.836 232 682 976 351 923 275 300 864;
  • 87) 0.836 232 682 976 351 923 275 300 864 × 2 = 1 + 0.672 465 365 952 703 846 550 601 728;
  • 88) 0.672 465 365 952 703 846 550 601 728 × 2 = 1 + 0.344 930 731 905 407 693 101 203 456;
  • 89) 0.344 930 731 905 407 693 101 203 456 × 2 = 0 + 0.689 861 463 810 815 386 202 406 912;
  • 90) 0.689 861 463 810 815 386 202 406 912 × 2 = 1 + 0.379 722 927 621 630 772 404 813 824;
  • 91) 0.379 722 927 621 630 772 404 813 824 × 2 = 0 + 0.759 445 855 243 261 544 809 627 648;
  • 92) 0.759 445 855 243 261 544 809 627 648 × 2 = 1 + 0.518 891 710 486 523 089 619 255 296;
  • 93) 0.518 891 710 486 523 089 619 255 296 × 2 = 1 + 0.037 783 420 973 046 179 238 510 592;
  • 94) 0.037 783 420 973 046 179 238 510 592 × 2 = 0 + 0.075 566 841 946 092 358 477 021 184;
  • 95) 0.075 566 841 946 092 358 477 021 184 × 2 = 0 + 0.151 133 683 892 184 716 954 042 368;
  • 96) 0.151 133 683 892 184 716 954 042 368 × 2 = 0 + 0.302 267 367 784 369 433 908 084 736;
  • 97) 0.302 267 367 784 369 433 908 084 736 × 2 = 0 + 0.604 534 735 568 738 867 816 169 472;
  • 98) 0.604 534 735 568 738 867 816 169 472 × 2 = 1 + 0.209 069 471 137 477 735 632 338 944;
  • 99) 0.209 069 471 137 477 735 632 338 944 × 2 = 0 + 0.418 138 942 274 955 471 264 677 888;
  • 100) 0.418 138 942 274 955 471 264 677 888 × 2 = 0 + 0.836 277 884 549 910 942 529 355 776;
  • 101) 0.836 277 884 549 910 942 529 355 776 × 2 = 1 + 0.672 555 769 099 821 885 058 711 552;
  • 102) 0.672 555 769 099 821 885 058 711 552 × 2 = 1 + 0.345 111 538 199 643 770 117 423 104;
  • 103) 0.345 111 538 199 643 770 117 423 104 × 2 = 0 + 0.690 223 076 399 287 540 234 846 208;
  • 104) 0.690 223 076 399 287 540 234 846 208 × 2 = 1 + 0.380 446 152 798 575 080 469 692 416;
  • 105) 0.380 446 152 798 575 080 469 692 416 × 2 = 0 + 0.760 892 305 597 150 160 939 384 832;
  • 106) 0.760 892 305 597 150 160 939 384 832 × 2 = 1 + 0.521 784 611 194 300 321 878 769 664;
  • 107) 0.521 784 611 194 300 321 878 769 664 × 2 = 1 + 0.043 569 222 388 600 643 757 539 328;

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:


0.000 000 000 000 000 032 487 359 776(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0101 0111 0100 1001 0011 1000 0000 1011 0101 1000 0100 1101 011(2)

5. Positive number before normalization:

0.000 000 000 000 000 032 487 359 776(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0101 0111 0100 1001 0011 1000 0000 1011 0101 1000 0100 1101 011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 55 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 032 487 359 776(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0101 0111 0100 1001 0011 1000 0000 1011 0101 1000 0100 1101 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 0101 0111 0100 1001 0011 1000 0000 1011 0101 1000 0100 1101 011(2) × 20 =

1.0010 1011 1010 0100 1001 1100 0000 0101 1010 1100 0010 0110 1011(2) × 2-55


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): -55

Mantissa (not normalized):
1.0010 1011 1010 0100 1001 1100 0000 0101 1010 1100 0010 0110 1011


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-55 + 2(11-1) - 1 =

(-55 + 1 023)(10) =

968(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;
  • 968 ÷ 2 = 484 + 0;
  • 484 ÷ 2 = 242 + 0;
  • 242 ÷ 2 = 121 + 0;
  • 121 ÷ 2 = 60 + 1;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

968(10) =

011 1100 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. 0010 1011 1010 0100 1001 1100 0000 0101 1010 1100 0010 0110 1011 =

0010 1011 1010 0100 1001 1100 0000 0101 1010 1100 0010 0110 1011


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 1100 1000

Mantissa (52 bits) =
0010 1011 1010 0100 1001 1100 0000 0101 1010 1100 0010 0110 1011


Decimal number 0.000 000 000 000 000 032 487 359 776 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 1000 - 0010 1011 1010 0100 1001 1100 0000 0101 1010 1100 0010 0110 1011

» Convert decimal 0.000 000 000 000 000 032 487 359 76 to 64 bit double precision IEEE 754 binary floating point representation standard

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» Month 10, 2026 [October]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: October - by our visitors

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