0.000 000 000 634 751 367 889 197 801 221 261 633 25 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 25(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 25(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 25.

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 25 × 2 = 0 + 0.000 000 001 269 502 735 778 395 602 442 523 266 5;
  • 2) 0.000 000 001 269 502 735 778 395 602 442 523 266 5 × 2 = 0 + 0.000 000 002 539 005 471 556 791 204 885 046 533;
  • 3) 0.000 000 002 539 005 471 556 791 204 885 046 533 × 2 = 0 + 0.000 000 005 078 010 943 113 582 409 770 093 066;
  • 4) 0.000 000 005 078 010 943 113 582 409 770 093 066 × 2 = 0 + 0.000 000 010 156 021 886 227 164 819 540 186 132;
  • 5) 0.000 000 010 156 021 886 227 164 819 540 186 132 × 2 = 0 + 0.000 000 020 312 043 772 454 329 639 080 372 264;
  • 6) 0.000 000 020 312 043 772 454 329 639 080 372 264 × 2 = 0 + 0.000 000 040 624 087 544 908 659 278 160 744 528;
  • 7) 0.000 000 040 624 087 544 908 659 278 160 744 528 × 2 = 0 + 0.000 000 081 248 175 089 817 318 556 321 489 056;
  • 8) 0.000 000 081 248 175 089 817 318 556 321 489 056 × 2 = 0 + 0.000 000 162 496 350 179 634 637 112 642 978 112;
  • 9) 0.000 000 162 496 350 179 634 637 112 642 978 112 × 2 = 0 + 0.000 000 324 992 700 359 269 274 225 285 956 224;
  • 10) 0.000 000 324 992 700 359 269 274 225 285 956 224 × 2 = 0 + 0.000 000 649 985 400 718 538 548 450 571 912 448;
  • 11) 0.000 000 649 985 400 718 538 548 450 571 912 448 × 2 = 0 + 0.000 001 299 970 801 437 077 096 901 143 824 896;
  • 12) 0.000 001 299 970 801 437 077 096 901 143 824 896 × 2 = 0 + 0.000 002 599 941 602 874 154 193 802 287 649 792;
  • 13) 0.000 002 599 941 602 874 154 193 802 287 649 792 × 2 = 0 + 0.000 005 199 883 205 748 308 387 604 575 299 584;
  • 14) 0.000 005 199 883 205 748 308 387 604 575 299 584 × 2 = 0 + 0.000 010 399 766 411 496 616 775 209 150 599 168;
  • 15) 0.000 010 399 766 411 496 616 775 209 150 599 168 × 2 = 0 + 0.000 020 799 532 822 993 233 550 418 301 198 336;
  • 16) 0.000 020 799 532 822 993 233 550 418 301 198 336 × 2 = 0 + 0.000 041 599 065 645 986 467 100 836 602 396 672;
  • 17) 0.000 041 599 065 645 986 467 100 836 602 396 672 × 2 = 0 + 0.000 083 198 131 291 972 934 201 673 204 793 344;
  • 18) 0.000 083 198 131 291 972 934 201 673 204 793 344 × 2 = 0 + 0.000 166 396 262 583 945 868 403 346 409 586 688;
  • 19) 0.000 166 396 262 583 945 868 403 346 409 586 688 × 2 = 0 + 0.000 332 792 525 167 891 736 806 692 819 173 376;
  • 20) 0.000 332 792 525 167 891 736 806 692 819 173 376 × 2 = 0 + 0.000 665 585 050 335 783 473 613 385 638 346 752;
  • 21) 0.000 665 585 050 335 783 473 613 385 638 346 752 × 2 = 0 + 0.001 331 170 100 671 566 947 226 771 276 693 504;
  • 22) 0.001 331 170 100 671 566 947 226 771 276 693 504 × 2 = 0 + 0.002 662 340 201 343 133 894 453 542 553 387 008;
  • 23) 0.002 662 340 201 343 133 894 453 542 553 387 008 × 2 = 0 + 0.005 324 680 402 686 267 788 907 085 106 774 016;
  • 24) 0.005 324 680 402 686 267 788 907 085 106 774 016 × 2 = 0 + 0.010 649 360 805 372 535 577 814 170 213 548 032;
  • 25) 0.010 649 360 805 372 535 577 814 170 213 548 032 × 2 = 0 + 0.021 298 721 610 745 071 155 628 340 427 096 064;
  • 26) 0.021 298 721 610 745 071 155 628 340 427 096 064 × 2 = 0 + 0.042 597 443 221 490 142 311 256 680 854 192 128;
  • 27) 0.042 597 443 221 490 142 311 256 680 854 192 128 × 2 = 0 + 0.085 194 886 442 980 284 622 513 361 708 384 256;
  • 28) 0.085 194 886 442 980 284 622 513 361 708 384 256 × 2 = 0 + 0.170 389 772 885 960 569 245 026 723 416 768 512;
  • 29) 0.170 389 772 885 960 569 245 026 723 416 768 512 × 2 = 0 + 0.340 779 545 771 921 138 490 053 446 833 537 024;
  • 30) 0.340 779 545 771 921 138 490 053 446 833 537 024 × 2 = 0 + 0.681 559 091 543 842 276 980 106 893 667 074 048;
  • 31) 0.681 559 091 543 842 276 980 106 893 667 074 048 × 2 = 1 + 0.363 118 183 087 684 553 960 213 787 334 148 096;
  • 32) 0.363 118 183 087 684 553 960 213 787 334 148 096 × 2 = 0 + 0.726 236 366 175 369 107 920 427 574 668 296 192;
  • 33) 0.726 236 366 175 369 107 920 427 574 668 296 192 × 2 = 1 + 0.452 472 732 350 738 215 840 855 149 336 592 384;
  • 34) 0.452 472 732 350 738 215 840 855 149 336 592 384 × 2 = 0 + 0.904 945 464 701 476 431 681 710 298 673 184 768;
  • 35) 0.904 945 464 701 476 431 681 710 298 673 184 768 × 2 = 1 + 0.809 890 929 402 952 863 363 420 597 346 369 536;
  • 36) 0.809 890 929 402 952 863 363 420 597 346 369 536 × 2 = 1 + 0.619 781 858 805 905 726 726 841 194 692 739 072;
  • 37) 0.619 781 858 805 905 726 726 841 194 692 739 072 × 2 = 1 + 0.239 563 717 611 811 453 453 682 389 385 478 144;
  • 38) 0.239 563 717 611 811 453 453 682 389 385 478 144 × 2 = 0 + 0.479 127 435 223 622 906 907 364 778 770 956 288;
  • 39) 0.479 127 435 223 622 906 907 364 778 770 956 288 × 2 = 0 + 0.958 254 870 447 245 813 814 729 557 541 912 576;
  • 40) 0.958 254 870 447 245 813 814 729 557 541 912 576 × 2 = 1 + 0.916 509 740 894 491 627 629 459 115 083 825 152;
  • 41) 0.916 509 740 894 491 627 629 459 115 083 825 152 × 2 = 1 + 0.833 019 481 788 983 255 258 918 230 167 650 304;
  • 42) 0.833 019 481 788 983 255 258 918 230 167 650 304 × 2 = 1 + 0.666 038 963 577 966 510 517 836 460 335 300 608;
  • 43) 0.666 038 963 577 966 510 517 836 460 335 300 608 × 2 = 1 + 0.332 077 927 155 933 021 035 672 920 670 601 216;
  • 44) 0.332 077 927 155 933 021 035 672 920 670 601 216 × 2 = 0 + 0.664 155 854 311 866 042 071 345 841 341 202 432;
  • 45) 0.664 155 854 311 866 042 071 345 841 341 202 432 × 2 = 1 + 0.328 311 708 623 732 084 142 691 682 682 404 864;
  • 46) 0.328 311 708 623 732 084 142 691 682 682 404 864 × 2 = 0 + 0.656 623 417 247 464 168 285 383 365 364 809 728;
  • 47) 0.656 623 417 247 464 168 285 383 365 364 809 728 × 2 = 1 + 0.313 246 834 494 928 336 570 766 730 729 619 456;
  • 48) 0.313 246 834 494 928 336 570 766 730 729 619 456 × 2 = 0 + 0.626 493 668 989 856 673 141 533 461 459 238 912;
  • 49) 0.626 493 668 989 856 673 141 533 461 459 238 912 × 2 = 1 + 0.252 987 337 979 713 346 283 066 922 918 477 824;
  • 50) 0.252 987 337 979 713 346 283 066 922 918 477 824 × 2 = 0 + 0.505 974 675 959 426 692 566 133 845 836 955 648;
  • 51) 0.505 974 675 959 426 692 566 133 845 836 955 648 × 2 = 1 + 0.011 949 351 918 853 385 132 267 691 673 911 296;
  • 52) 0.011 949 351 918 853 385 132 267 691 673 911 296 × 2 = 0 + 0.023 898 703 837 706 770 264 535 383 347 822 592;
  • 53) 0.023 898 703 837 706 770 264 535 383 347 822 592 × 2 = 0 + 0.047 797 407 675 413 540 529 070 766 695 645 184;
  • 54) 0.047 797 407 675 413 540 529 070 766 695 645 184 × 2 = 0 + 0.095 594 815 350 827 081 058 141 533 391 290 368;
  • 55) 0.095 594 815 350 827 081 058 141 533 391 290 368 × 2 = 0 + 0.191 189 630 701 654 162 116 283 066 782 580 736;
  • 56) 0.191 189 630 701 654 162 116 283 066 782 580 736 × 2 = 0 + 0.382 379 261 403 308 324 232 566 133 565 161 472;
  • 57) 0.382 379 261 403 308 324 232 566 133 565 161 472 × 2 = 0 + 0.764 758 522 806 616 648 465 132 267 130 322 944;
  • 58) 0.764 758 522 806 616 648 465 132 267 130 322 944 × 2 = 1 + 0.529 517 045 613 233 296 930 264 534 260 645 888;
  • 59) 0.529 517 045 613 233 296 930 264 534 260 645 888 × 2 = 1 + 0.059 034 091 226 466 593 860 529 068 521 291 776;
  • 60) 0.059 034 091 226 466 593 860 529 068 521 291 776 × 2 = 0 + 0.118 068 182 452 933 187 721 058 137 042 583 552;
  • 61) 0.118 068 182 452 933 187 721 058 137 042 583 552 × 2 = 0 + 0.236 136 364 905 866 375 442 116 274 085 167 104;
  • 62) 0.236 136 364 905 866 375 442 116 274 085 167 104 × 2 = 0 + 0.472 272 729 811 732 750 884 232 548 170 334 208;
  • 63) 0.472 272 729 811 732 750 884 232 548 170 334 208 × 2 = 0 + 0.944 545 459 623 465 501 768 465 096 340 668 416;
  • 64) 0.944 545 459 623 465 501 768 465 096 340 668 416 × 2 = 1 + 0.889 090 919 246 931 003 536 930 192 681 336 832;
  • 65) 0.889 090 919 246 931 003 536 930 192 681 336 832 × 2 = 1 + 0.778 181 838 493 862 007 073 860 385 362 673 664;
  • 66) 0.778 181 838 493 862 007 073 860 385 362 673 664 × 2 = 1 + 0.556 363 676 987 724 014 147 720 770 725 347 328;
  • 67) 0.556 363 676 987 724 014 147 720 770 725 347 328 × 2 = 1 + 0.112 727 353 975 448 028 295 441 541 450 694 656;
  • 68) 0.112 727 353 975 448 028 295 441 541 450 694 656 × 2 = 0 + 0.225 454 707 950 896 056 590 883 082 901 389 312;
  • 69) 0.225 454 707 950 896 056 590 883 082 901 389 312 × 2 = 0 + 0.450 909 415 901 792 113 181 766 165 802 778 624;
  • 70) 0.450 909 415 901 792 113 181 766 165 802 778 624 × 2 = 0 + 0.901 818 831 803 584 226 363 532 331 605 557 248;
  • 71) 0.901 818 831 803 584 226 363 532 331 605 557 248 × 2 = 1 + 0.803 637 663 607 168 452 727 064 663 211 114 496;
  • 72) 0.803 637 663 607 168 452 727 064 663 211 114 496 × 2 = 1 + 0.607 275 327 214 336 905 454 129 326 422 228 992;
  • 73) 0.607 275 327 214 336 905 454 129 326 422 228 992 × 2 = 1 + 0.214 550 654 428 673 810 908 258 652 844 457 984;
  • 74) 0.214 550 654 428 673 810 908 258 652 844 457 984 × 2 = 0 + 0.429 101 308 857 347 621 816 517 305 688 915 968;
  • 75) 0.429 101 308 857 347 621 816 517 305 688 915 968 × 2 = 0 + 0.858 202 617 714 695 243 633 034 611 377 831 936;
  • 76) 0.858 202 617 714 695 243 633 034 611 377 831 936 × 2 = 1 + 0.716 405 235 429 390 487 266 069 222 755 663 872;
  • 77) 0.716 405 235 429 390 487 266 069 222 755 663 872 × 2 = 1 + 0.432 810 470 858 780 974 532 138 445 511 327 744;
  • 78) 0.432 810 470 858 780 974 532 138 445 511 327 744 × 2 = 0 + 0.865 620 941 717 561 949 064 276 891 022 655 488;
  • 79) 0.865 620 941 717 561 949 064 276 891 022 655 488 × 2 = 1 + 0.731 241 883 435 123 898 128 553 782 045 310 976;
  • 80) 0.731 241 883 435 123 898 128 553 782 045 310 976 × 2 = 1 + 0.462 483 766 870 247 796 257 107 564 090 621 952;
  • 81) 0.462 483 766 870 247 796 257 107 564 090 621 952 × 2 = 0 + 0.924 967 533 740 495 592 514 215 128 181 243 904;
  • 82) 0.924 967 533 740 495 592 514 215 128 181 243 904 × 2 = 1 + 0.849 935 067 480 991 185 028 430 256 362 487 808;
  • 83) 0.849 935 067 480 991 185 028 430 256 362 487 808 × 2 = 1 + 0.699 870 134 961 982 370 056 860 512 724 975 616;

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 25(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 25(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 25(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 25 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 94 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