0.000 000 000 634 751 367 889 197 801 221 261 633 28 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 634 751 367 889 197 801 221 261 633 28(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
0.000 000 000 634 751 367 889 197 801 221 261 633 28(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: 0.000 000 000 634 751 367 889 197 801 221 261 633 28.

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) 0.000 000 000 634 751 367 889 197 801 221 261 633 28 × 2 = 0 + 0.000 000 001 269 502 735 778 395 602 442 523 266 56;
  • 2) 0.000 000 001 269 502 735 778 395 602 442 523 266 56 × 2 = 0 + 0.000 000 002 539 005 471 556 791 204 885 046 533 12;
  • 3) 0.000 000 002 539 005 471 556 791 204 885 046 533 12 × 2 = 0 + 0.000 000 005 078 010 943 113 582 409 770 093 066 24;
  • 4) 0.000 000 005 078 010 943 113 582 409 770 093 066 24 × 2 = 0 + 0.000 000 010 156 021 886 227 164 819 540 186 132 48;
  • 5) 0.000 000 010 156 021 886 227 164 819 540 186 132 48 × 2 = 0 + 0.000 000 020 312 043 772 454 329 639 080 372 264 96;
  • 6) 0.000 000 020 312 043 772 454 329 639 080 372 264 96 × 2 = 0 + 0.000 000 040 624 087 544 908 659 278 160 744 529 92;
  • 7) 0.000 000 040 624 087 544 908 659 278 160 744 529 92 × 2 = 0 + 0.000 000 081 248 175 089 817 318 556 321 489 059 84;
  • 8) 0.000 000 081 248 175 089 817 318 556 321 489 059 84 × 2 = 0 + 0.000 000 162 496 350 179 634 637 112 642 978 119 68;
  • 9) 0.000 000 162 496 350 179 634 637 112 642 978 119 68 × 2 = 0 + 0.000 000 324 992 700 359 269 274 225 285 956 239 36;
  • 10) 0.000 000 324 992 700 359 269 274 225 285 956 239 36 × 2 = 0 + 0.000 000 649 985 400 718 538 548 450 571 912 478 72;
  • 11) 0.000 000 649 985 400 718 538 548 450 571 912 478 72 × 2 = 0 + 0.000 001 299 970 801 437 077 096 901 143 824 957 44;
  • 12) 0.000 001 299 970 801 437 077 096 901 143 824 957 44 × 2 = 0 + 0.000 002 599 941 602 874 154 193 802 287 649 914 88;
  • 13) 0.000 002 599 941 602 874 154 193 802 287 649 914 88 × 2 = 0 + 0.000 005 199 883 205 748 308 387 604 575 299 829 76;
  • 14) 0.000 005 199 883 205 748 308 387 604 575 299 829 76 × 2 = 0 + 0.000 010 399 766 411 496 616 775 209 150 599 659 52;
  • 15) 0.000 010 399 766 411 496 616 775 209 150 599 659 52 × 2 = 0 + 0.000 020 799 532 822 993 233 550 418 301 199 319 04;
  • 16) 0.000 020 799 532 822 993 233 550 418 301 199 319 04 × 2 = 0 + 0.000 041 599 065 645 986 467 100 836 602 398 638 08;
  • 17) 0.000 041 599 065 645 986 467 100 836 602 398 638 08 × 2 = 0 + 0.000 083 198 131 291 972 934 201 673 204 797 276 16;
  • 18) 0.000 083 198 131 291 972 934 201 673 204 797 276 16 × 2 = 0 + 0.000 166 396 262 583 945 868 403 346 409 594 552 32;
  • 19) 0.000 166 396 262 583 945 868 403 346 409 594 552 32 × 2 = 0 + 0.000 332 792 525 167 891 736 806 692 819 189 104 64;
  • 20) 0.000 332 792 525 167 891 736 806 692 819 189 104 64 × 2 = 0 + 0.000 665 585 050 335 783 473 613 385 638 378 209 28;
  • 21) 0.000 665 585 050 335 783 473 613 385 638 378 209 28 × 2 = 0 + 0.001 331 170 100 671 566 947 226 771 276 756 418 56;
  • 22) 0.001 331 170 100 671 566 947 226 771 276 756 418 56 × 2 = 0 + 0.002 662 340 201 343 133 894 453 542 553 512 837 12;
  • 23) 0.002 662 340 201 343 133 894 453 542 553 512 837 12 × 2 = 0 + 0.005 324 680 402 686 267 788 907 085 107 025 674 24;
  • 24) 0.005 324 680 402 686 267 788 907 085 107 025 674 24 × 2 = 0 + 0.010 649 360 805 372 535 577 814 170 214 051 348 48;
  • 25) 0.010 649 360 805 372 535 577 814 170 214 051 348 48 × 2 = 0 + 0.021 298 721 610 745 071 155 628 340 428 102 696 96;
  • 26) 0.021 298 721 610 745 071 155 628 340 428 102 696 96 × 2 = 0 + 0.042 597 443 221 490 142 311 256 680 856 205 393 92;
  • 27) 0.042 597 443 221 490 142 311 256 680 856 205 393 92 × 2 = 0 + 0.085 194 886 442 980 284 622 513 361 712 410 787 84;
  • 28) 0.085 194 886 442 980 284 622 513 361 712 410 787 84 × 2 = 0 + 0.170 389 772 885 960 569 245 026 723 424 821 575 68;
  • 29) 0.170 389 772 885 960 569 245 026 723 424 821 575 68 × 2 = 0 + 0.340 779 545 771 921 138 490 053 446 849 643 151 36;
  • 30) 0.340 779 545 771 921 138 490 053 446 849 643 151 36 × 2 = 0 + 0.681 559 091 543 842 276 980 106 893 699 286 302 72;
  • 31) 0.681 559 091 543 842 276 980 106 893 699 286 302 72 × 2 = 1 + 0.363 118 183 087 684 553 960 213 787 398 572 605 44;
  • 32) 0.363 118 183 087 684 553 960 213 787 398 572 605 44 × 2 = 0 + 0.726 236 366 175 369 107 920 427 574 797 145 210 88;
  • 33) 0.726 236 366 175 369 107 920 427 574 797 145 210 88 × 2 = 1 + 0.452 472 732 350 738 215 840 855 149 594 290 421 76;
  • 34) 0.452 472 732 350 738 215 840 855 149 594 290 421 76 × 2 = 0 + 0.904 945 464 701 476 431 681 710 299 188 580 843 52;
  • 35) 0.904 945 464 701 476 431 681 710 299 188 580 843 52 × 2 = 1 + 0.809 890 929 402 952 863 363 420 598 377 161 687 04;
  • 36) 0.809 890 929 402 952 863 363 420 598 377 161 687 04 × 2 = 1 + 0.619 781 858 805 905 726 726 841 196 754 323 374 08;
  • 37) 0.619 781 858 805 905 726 726 841 196 754 323 374 08 × 2 = 1 + 0.239 563 717 611 811 453 453 682 393 508 646 748 16;
  • 38) 0.239 563 717 611 811 453 453 682 393 508 646 748 16 × 2 = 0 + 0.479 127 435 223 622 906 907 364 787 017 293 496 32;
  • 39) 0.479 127 435 223 622 906 907 364 787 017 293 496 32 × 2 = 0 + 0.958 254 870 447 245 813 814 729 574 034 586 992 64;
  • 40) 0.958 254 870 447 245 813 814 729 574 034 586 992 64 × 2 = 1 + 0.916 509 740 894 491 627 629 459 148 069 173 985 28;
  • 41) 0.916 509 740 894 491 627 629 459 148 069 173 985 28 × 2 = 1 + 0.833 019 481 788 983 255 258 918 296 138 347 970 56;
  • 42) 0.833 019 481 788 983 255 258 918 296 138 347 970 56 × 2 = 1 + 0.666 038 963 577 966 510 517 836 592 276 695 941 12;
  • 43) 0.666 038 963 577 966 510 517 836 592 276 695 941 12 × 2 = 1 + 0.332 077 927 155 933 021 035 673 184 553 391 882 24;
  • 44) 0.332 077 927 155 933 021 035 673 184 553 391 882 24 × 2 = 0 + 0.664 155 854 311 866 042 071 346 369 106 783 764 48;
  • 45) 0.664 155 854 311 866 042 071 346 369 106 783 764 48 × 2 = 1 + 0.328 311 708 623 732 084 142 692 738 213 567 528 96;
  • 46) 0.328 311 708 623 732 084 142 692 738 213 567 528 96 × 2 = 0 + 0.656 623 417 247 464 168 285 385 476 427 135 057 92;
  • 47) 0.656 623 417 247 464 168 285 385 476 427 135 057 92 × 2 = 1 + 0.313 246 834 494 928 336 570 770 952 854 270 115 84;
  • 48) 0.313 246 834 494 928 336 570 770 952 854 270 115 84 × 2 = 0 + 0.626 493 668 989 856 673 141 541 905 708 540 231 68;
  • 49) 0.626 493 668 989 856 673 141 541 905 708 540 231 68 × 2 = 1 + 0.252 987 337 979 713 346 283 083 811 417 080 463 36;
  • 50) 0.252 987 337 979 713 346 283 083 811 417 080 463 36 × 2 = 0 + 0.505 974 675 959 426 692 566 167 622 834 160 926 72;
  • 51) 0.505 974 675 959 426 692 566 167 622 834 160 926 72 × 2 = 1 + 0.011 949 351 918 853 385 132 335 245 668 321 853 44;
  • 52) 0.011 949 351 918 853 385 132 335 245 668 321 853 44 × 2 = 0 + 0.023 898 703 837 706 770 264 670 491 336 643 706 88;
  • 53) 0.023 898 703 837 706 770 264 670 491 336 643 706 88 × 2 = 0 + 0.047 797 407 675 413 540 529 340 982 673 287 413 76;
  • 54) 0.047 797 407 675 413 540 529 340 982 673 287 413 76 × 2 = 0 + 0.095 594 815 350 827 081 058 681 965 346 574 827 52;
  • 55) 0.095 594 815 350 827 081 058 681 965 346 574 827 52 × 2 = 0 + 0.191 189 630 701 654 162 117 363 930 693 149 655 04;
  • 56) 0.191 189 630 701 654 162 117 363 930 693 149 655 04 × 2 = 0 + 0.382 379 261 403 308 324 234 727 861 386 299 310 08;
  • 57) 0.382 379 261 403 308 324 234 727 861 386 299 310 08 × 2 = 0 + 0.764 758 522 806 616 648 469 455 722 772 598 620 16;
  • 58) 0.764 758 522 806 616 648 469 455 722 772 598 620 16 × 2 = 1 + 0.529 517 045 613 233 296 938 911 445 545 197 240 32;
  • 59) 0.529 517 045 613 233 296 938 911 445 545 197 240 32 × 2 = 1 + 0.059 034 091 226 466 593 877 822 891 090 394 480 64;
  • 60) 0.059 034 091 226 466 593 877 822 891 090 394 480 64 × 2 = 0 + 0.118 068 182 452 933 187 755 645 782 180 788 961 28;
  • 61) 0.118 068 182 452 933 187 755 645 782 180 788 961 28 × 2 = 0 + 0.236 136 364 905 866 375 511 291 564 361 577 922 56;
  • 62) 0.236 136 364 905 866 375 511 291 564 361 577 922 56 × 2 = 0 + 0.472 272 729 811 732 751 022 583 128 723 155 845 12;
  • 63) 0.472 272 729 811 732 751 022 583 128 723 155 845 12 × 2 = 0 + 0.944 545 459 623 465 502 045 166 257 446 311 690 24;
  • 64) 0.944 545 459 623 465 502 045 166 257 446 311 690 24 × 2 = 1 + 0.889 090 919 246 931 004 090 332 514 892 623 380 48;
  • 65) 0.889 090 919 246 931 004 090 332 514 892 623 380 48 × 2 = 1 + 0.778 181 838 493 862 008 180 665 029 785 246 760 96;
  • 66) 0.778 181 838 493 862 008 180 665 029 785 246 760 96 × 2 = 1 + 0.556 363 676 987 724 016 361 330 059 570 493 521 92;
  • 67) 0.556 363 676 987 724 016 361 330 059 570 493 521 92 × 2 = 1 + 0.112 727 353 975 448 032 722 660 119 140 987 043 84;
  • 68) 0.112 727 353 975 448 032 722 660 119 140 987 043 84 × 2 = 0 + 0.225 454 707 950 896 065 445 320 238 281 974 087 68;
  • 69) 0.225 454 707 950 896 065 445 320 238 281 974 087 68 × 2 = 0 + 0.450 909 415 901 792 130 890 640 476 563 948 175 36;
  • 70) 0.450 909 415 901 792 130 890 640 476 563 948 175 36 × 2 = 0 + 0.901 818 831 803 584 261 781 280 953 127 896 350 72;
  • 71) 0.901 818 831 803 584 261 781 280 953 127 896 350 72 × 2 = 1 + 0.803 637 663 607 168 523 562 561 906 255 792 701 44;
  • 72) 0.803 637 663 607 168 523 562 561 906 255 792 701 44 × 2 = 1 + 0.607 275 327 214 337 047 125 123 812 511 585 402 88;
  • 73) 0.607 275 327 214 337 047 125 123 812 511 585 402 88 × 2 = 1 + 0.214 550 654 428 674 094 250 247 625 023 170 805 76;
  • 74) 0.214 550 654 428 674 094 250 247 625 023 170 805 76 × 2 = 0 + 0.429 101 308 857 348 188 500 495 250 046 341 611 52;
  • 75) 0.429 101 308 857 348 188 500 495 250 046 341 611 52 × 2 = 0 + 0.858 202 617 714 696 377 000 990 500 092 683 223 04;
  • 76) 0.858 202 617 714 696 377 000 990 500 092 683 223 04 × 2 = 1 + 0.716 405 235 429 392 754 001 981 000 185 366 446 08;
  • 77) 0.716 405 235 429 392 754 001 981 000 185 366 446 08 × 2 = 1 + 0.432 810 470 858 785 508 003 962 000 370 732 892 16;
  • 78) 0.432 810 470 858 785 508 003 962 000 370 732 892 16 × 2 = 0 + 0.865 620 941 717 571 016 007 924 000 741 465 784 32;
  • 79) 0.865 620 941 717 571 016 007 924 000 741 465 784 32 × 2 = 1 + 0.731 241 883 435 142 032 015 848 001 482 931 568 64;
  • 80) 0.731 241 883 435 142 032 015 848 001 482 931 568 64 × 2 = 1 + 0.462 483 766 870 284 064 031 696 002 965 863 137 28;
  • 81) 0.462 483 766 870 284 064 031 696 002 965 863 137 28 × 2 = 0 + 0.924 967 533 740 568 128 063 392 005 931 726 274 56;
  • 82) 0.924 967 533 740 568 128 063 392 005 931 726 274 56 × 2 = 1 + 0.849 935 067 481 136 256 126 784 011 863 452 549 12;
  • 83) 0.849 935 067 481 136 256 126 784 011 863 452 549 12 × 2 = 1 + 0.699 870 134 962 272 512 253 568 023 726 905 098 24;

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:


0.000 000 000 634 751 367 889 197 801 221 261 633 28(10) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2)

5. Positive number before normalization:

0.000 000 000 634 751 367 889 197 801 221 261 633 28(10) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 31 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 634 751 367 889 197 801 221 261 633 28(10) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2) =

0.0000 0000 0000 0000 0000 0000 0000 0010 1011 1001 1110 1010 1010 0000 0110 0001 1110 0011 1001 1011 011(2) × 20 =

1.0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011(2) × 2-31


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): -31

Mantissa (not normalized):
1.0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-31 + 2(11-1) - 1 =

(-31 + 1 023)(10) =

992(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;
  • 992 ÷ 2 = 496 + 0;
  • 496 ÷ 2 = 248 + 0;
  • 248 ÷ 2 = 124 + 0;
  • 124 ÷ 2 = 62 + 0;
  • 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) =

992(10) =

011 1110 0000(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. 0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011 =

0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 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 1110 0000

Mantissa (52 bits) =
0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011


Decimal number 0.000 000 000 634 751 367 889 197 801 221 261 633 28 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1110 0000 - 0101 1100 1111 0101 0101 0000 0011 0000 1111 0001 1100 1101 1011

» Convert decimal 0.000 000 000 634 751 367 889 197 801 221 261 633 53 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» 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