0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 521 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 521(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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 521(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 521.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 521 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 234 619 042;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 234 619 042 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 469 238 084;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 469 238 084 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 938 476 168;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 938 476 168 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 876 952 336;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 876 952 336 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 753 904 672;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 753 904 672 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 507 809 344;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 507 809 344 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 015 618 688;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 015 618 688 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 031 237 376;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 031 237 376 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 062 474 752;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 062 474 752 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 124 949 504;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 124 949 504 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 944 249 899 008;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 944 249 899 008 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 888 499 798 016;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 888 499 798 016 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 776 999 596 032;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 776 999 596 032 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 553 999 192 064;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 553 999 192 064 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 107 998 384 128;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 107 998 384 128 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 215 996 768 256;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 215 996 768 256 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 431 993 536 512;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 431 993 536 512 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 863 987 073 024;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 863 987 073 024 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 727 974 146 048;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 727 974 146 048 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 891 455 948 292 096;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 891 455 948 292 096 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 782 911 896 584 192;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 782 911 896 584 192 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 565 823 793 168 384;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 565 823 793 168 384 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 131 647 586 336 768;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 131 647 586 336 768 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 263 295 172 673 536;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 263 295 172 673 536 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 526 590 345 347 072;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 526 590 345 347 072 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 053 180 690 694 144;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 053 180 690 694 144 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 106 361 381 388 288;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 106 361 381 388 288 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 212 722 762 776 576;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 212 722 762 776 576 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 425 445 525 553 152;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 425 445 525 553 152 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 816 850 891 051 106 304;
  • 31) 0.825 297 101 522 529 932 158 401 660 816 850 891 051 106 304 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 633 701 782 102 212 608;
  • 32) 0.650 594 203 045 059 864 316 803 321 633 701 782 102 212 608 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 267 403 564 204 425 216;
  • 33) 0.301 188 406 090 119 728 633 606 643 267 403 564 204 425 216 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 534 807 128 408 850 432;
  • 34) 0.602 376 812 180 239 457 267 213 286 534 807 128 408 850 432 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 069 614 256 817 700 864;
  • 35) 0.204 753 624 360 478 914 534 426 573 069 614 256 817 700 864 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 139 228 513 635 401 728;
  • 36) 0.409 507 248 720 957 829 068 853 146 139 228 513 635 401 728 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 278 457 027 270 803 456;
  • 37) 0.819 014 497 441 915 658 137 706 292 278 457 027 270 803 456 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 556 914 054 541 606 912;
  • 38) 0.638 028 994 883 831 316 275 412 584 556 914 054 541 606 912 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 113 828 109 083 213 824;
  • 39) 0.276 057 989 767 662 632 550 825 169 113 828 109 083 213 824 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 227 656 218 166 427 648;
  • 40) 0.552 115 979 535 325 265 101 650 338 227 656 218 166 427 648 × 2 = 1 + 0.104 231 959 070 650 530 203 300 676 455 312 436 332 855 296;
  • 41) 0.104 231 959 070 650 530 203 300 676 455 312 436 332 855 296 × 2 = 0 + 0.208 463 918 141 301 060 406 601 352 910 624 872 665 710 592;
  • 42) 0.208 463 918 141 301 060 406 601 352 910 624 872 665 710 592 × 2 = 0 + 0.416 927 836 282 602 120 813 202 705 821 249 745 331 421 184;
  • 43) 0.416 927 836 282 602 120 813 202 705 821 249 745 331 421 184 × 2 = 0 + 0.833 855 672 565 204 241 626 405 411 642 499 490 662 842 368;
  • 44) 0.833 855 672 565 204 241 626 405 411 642 499 490 662 842 368 × 2 = 1 + 0.667 711 345 130 408 483 252 810 823 284 998 981 325 684 736;
  • 45) 0.667 711 345 130 408 483 252 810 823 284 998 981 325 684 736 × 2 = 1 + 0.335 422 690 260 816 966 505 621 646 569 997 962 651 369 472;
  • 46) 0.335 422 690 260 816 966 505 621 646 569 997 962 651 369 472 × 2 = 0 + 0.670 845 380 521 633 933 011 243 293 139 995 925 302 738 944;
  • 47) 0.670 845 380 521 633 933 011 243 293 139 995 925 302 738 944 × 2 = 1 + 0.341 690 761 043 267 866 022 486 586 279 991 850 605 477 888;
  • 48) 0.341 690 761 043 267 866 022 486 586 279 991 850 605 477 888 × 2 = 0 + 0.683 381 522 086 535 732 044 973 172 559 983 701 210 955 776;
  • 49) 0.683 381 522 086 535 732 044 973 172 559 983 701 210 955 776 × 2 = 1 + 0.366 763 044 173 071 464 089 946 345 119 967 402 421 911 552;
  • 50) 0.366 763 044 173 071 464 089 946 345 119 967 402 421 911 552 × 2 = 0 + 0.733 526 088 346 142 928 179 892 690 239 934 804 843 823 104;
  • 51) 0.733 526 088 346 142 928 179 892 690 239 934 804 843 823 104 × 2 = 1 + 0.467 052 176 692 285 856 359 785 380 479 869 609 687 646 208;
  • 52) 0.467 052 176 692 285 856 359 785 380 479 869 609 687 646 208 × 2 = 0 + 0.934 104 353 384 571 712 719 570 760 959 739 219 375 292 416;
  • 53) 0.934 104 353 384 571 712 719 570 760 959 739 219 375 292 416 × 2 = 1 + 0.868 208 706 769 143 425 439 141 521 919 478 438 750 584 832;
  • 54) 0.868 208 706 769 143 425 439 141 521 919 478 438 750 584 832 × 2 = 1 + 0.736 417 413 538 286 850 878 283 043 838 956 877 501 169 664;
  • 55) 0.736 417 413 538 286 850 878 283 043 838 956 877 501 169 664 × 2 = 1 + 0.472 834 827 076 573 701 756 566 087 677 913 755 002 339 328;
  • 56) 0.472 834 827 076 573 701 756 566 087 677 913 755 002 339 328 × 2 = 0 + 0.945 669 654 153 147 403 513 132 175 355 827 510 004 678 656;
  • 57) 0.945 669 654 153 147 403 513 132 175 355 827 510 004 678 656 × 2 = 1 + 0.891 339 308 306 294 807 026 264 350 711 655 020 009 357 312;
  • 58) 0.891 339 308 306 294 807 026 264 350 711 655 020 009 357 312 × 2 = 1 + 0.782 678 616 612 589 614 052 528 701 423 310 040 018 714 624;
  • 59) 0.782 678 616 612 589 614 052 528 701 423 310 040 018 714 624 × 2 = 1 + 0.565 357 233 225 179 228 105 057 402 846 620 080 037 429 248;
  • 60) 0.565 357 233 225 179 228 105 057 402 846 620 080 037 429 248 × 2 = 1 + 0.130 714 466 450 358 456 210 114 805 693 240 160 074 858 496;
  • 61) 0.130 714 466 450 358 456 210 114 805 693 240 160 074 858 496 × 2 = 0 + 0.261 428 932 900 716 912 420 229 611 386 480 320 149 716 992;
  • 62) 0.261 428 932 900 716 912 420 229 611 386 480 320 149 716 992 × 2 = 0 + 0.522 857 865 801 433 824 840 459 222 772 960 640 299 433 984;
  • 63) 0.522 857 865 801 433 824 840 459 222 772 960 640 299 433 984 × 2 = 1 + 0.045 715 731 602 867 649 680 918 445 545 921 280 598 867 968;
  • 64) 0.045 715 731 602 867 649 680 918 445 545 921 280 598 867 968 × 2 = 0 + 0.091 431 463 205 735 299 361 836 891 091 842 561 197 735 936;
  • 65) 0.091 431 463 205 735 299 361 836 891 091 842 561 197 735 936 × 2 = 0 + 0.182 862 926 411 470 598 723 673 782 183 685 122 395 471 872;
  • 66) 0.182 862 926 411 470 598 723 673 782 183 685 122 395 471 872 × 2 = 0 + 0.365 725 852 822 941 197 447 347 564 367 370 244 790 943 744;
  • 67) 0.365 725 852 822 941 197 447 347 564 367 370 244 790 943 744 × 2 = 0 + 0.731 451 705 645 882 394 894 695 128 734 740 489 581 887 488;
  • 68) 0.731 451 705 645 882 394 894 695 128 734 740 489 581 887 488 × 2 = 1 + 0.462 903 411 291 764 789 789 390 257 469 480 979 163 774 976;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 521(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

5. Positive number before normalization:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 521(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 16 positions to the right, so that only one non zero digit remains to the left of it:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 521(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 20 =

1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 2-16


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): -16

Mantissa (not normalized):
1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-16 + 2(11-1) - 1 =

(-16 + 1 023)(10) =

1 007(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;
  • 1 007 ÷ 2 = 503 + 1;
  • 503 ÷ 2 = 251 + 1;
  • 251 ÷ 2 = 125 + 1;
  • 125 ÷ 2 = 62 + 1;
  • 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) =

1007(10) =

011 1110 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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001 =

0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 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 1110 1111

Mantissa (52 bits) =
0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


Decimal number 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 521 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1110 1111 - 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001

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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