1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 161 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 161 7(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
1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 161 7(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: 1.
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;
  • 1 ÷ 2 = 0 + 1;

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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 161 7.

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) 0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 161 7 × 2 = 1 + 0.236 067 977 499 789 696 409 173 668 731 276 235 440 618 359 611 525 724 323 4;
  • 2) 0.236 067 977 499 789 696 409 173 668 731 276 235 440 618 359 611 525 724 323 4 × 2 = 0 + 0.472 135 954 999 579 392 818 347 337 462 552 470 881 236 719 223 051 448 646 8;
  • 3) 0.472 135 954 999 579 392 818 347 337 462 552 470 881 236 719 223 051 448 646 8 × 2 = 0 + 0.944 271 909 999 158 785 636 694 674 925 104 941 762 473 438 446 102 897 293 6;
  • 4) 0.944 271 909 999 158 785 636 694 674 925 104 941 762 473 438 446 102 897 293 6 × 2 = 1 + 0.888 543 819 998 317 571 273 389 349 850 209 883 524 946 876 892 205 794 587 2;
  • 5) 0.888 543 819 998 317 571 273 389 349 850 209 883 524 946 876 892 205 794 587 2 × 2 = 1 + 0.777 087 639 996 635 142 546 778 699 700 419 767 049 893 753 784 411 589 174 4;
  • 6) 0.777 087 639 996 635 142 546 778 699 700 419 767 049 893 753 784 411 589 174 4 × 2 = 1 + 0.554 175 279 993 270 285 093 557 399 400 839 534 099 787 507 568 823 178 348 8;
  • 7) 0.554 175 279 993 270 285 093 557 399 400 839 534 099 787 507 568 823 178 348 8 × 2 = 1 + 0.108 350 559 986 540 570 187 114 798 801 679 068 199 575 015 137 646 356 697 6;
  • 8) 0.108 350 559 986 540 570 187 114 798 801 679 068 199 575 015 137 646 356 697 6 × 2 = 0 + 0.216 701 119 973 081 140 374 229 597 603 358 136 399 150 030 275 292 713 395 2;
  • 9) 0.216 701 119 973 081 140 374 229 597 603 358 136 399 150 030 275 292 713 395 2 × 2 = 0 + 0.433 402 239 946 162 280 748 459 195 206 716 272 798 300 060 550 585 426 790 4;
  • 10) 0.433 402 239 946 162 280 748 459 195 206 716 272 798 300 060 550 585 426 790 4 × 2 = 0 + 0.866 804 479 892 324 561 496 918 390 413 432 545 596 600 121 101 170 853 580 8;
  • 11) 0.866 804 479 892 324 561 496 918 390 413 432 545 596 600 121 101 170 853 580 8 × 2 = 1 + 0.733 608 959 784 649 122 993 836 780 826 865 091 193 200 242 202 341 707 161 6;
  • 12) 0.733 608 959 784 649 122 993 836 780 826 865 091 193 200 242 202 341 707 161 6 × 2 = 1 + 0.467 217 919 569 298 245 987 673 561 653 730 182 386 400 484 404 683 414 323 2;
  • 13) 0.467 217 919 569 298 245 987 673 561 653 730 182 386 400 484 404 683 414 323 2 × 2 = 0 + 0.934 435 839 138 596 491 975 347 123 307 460 364 772 800 968 809 366 828 646 4;
  • 14) 0.934 435 839 138 596 491 975 347 123 307 460 364 772 800 968 809 366 828 646 4 × 2 = 1 + 0.868 871 678 277 192 983 950 694 246 614 920 729 545 601 937 618 733 657 292 8;
  • 15) 0.868 871 678 277 192 983 950 694 246 614 920 729 545 601 937 618 733 657 292 8 × 2 = 1 + 0.737 743 356 554 385 967 901 388 493 229 841 459 091 203 875 237 467 314 585 6;
  • 16) 0.737 743 356 554 385 967 901 388 493 229 841 459 091 203 875 237 467 314 585 6 × 2 = 1 + 0.475 486 713 108 771 935 802 776 986 459 682 918 182 407 750 474 934 629 171 2;
  • 17) 0.475 486 713 108 771 935 802 776 986 459 682 918 182 407 750 474 934 629 171 2 × 2 = 0 + 0.950 973 426 217 543 871 605 553 972 919 365 836 364 815 500 949 869 258 342 4;
  • 18) 0.950 973 426 217 543 871 605 553 972 919 365 836 364 815 500 949 869 258 342 4 × 2 = 1 + 0.901 946 852 435 087 743 211 107 945 838 731 672 729 631 001 899 738 516 684 8;
  • 19) 0.901 946 852 435 087 743 211 107 945 838 731 672 729 631 001 899 738 516 684 8 × 2 = 1 + 0.803 893 704 870 175 486 422 215 891 677 463 345 459 262 003 799 477 033 369 6;
  • 20) 0.803 893 704 870 175 486 422 215 891 677 463 345 459 262 003 799 477 033 369 6 × 2 = 1 + 0.607 787 409 740 350 972 844 431 783 354 926 690 918 524 007 598 954 066 739 2;
  • 21) 0.607 787 409 740 350 972 844 431 783 354 926 690 918 524 007 598 954 066 739 2 × 2 = 1 + 0.215 574 819 480 701 945 688 863 566 709 853 381 837 048 015 197 908 133 478 4;
  • 22) 0.215 574 819 480 701 945 688 863 566 709 853 381 837 048 015 197 908 133 478 4 × 2 = 0 + 0.431 149 638 961 403 891 377 727 133 419 706 763 674 096 030 395 816 266 956 8;
  • 23) 0.431 149 638 961 403 891 377 727 133 419 706 763 674 096 030 395 816 266 956 8 × 2 = 0 + 0.862 299 277 922 807 782 755 454 266 839 413 527 348 192 060 791 632 533 913 6;
  • 24) 0.862 299 277 922 807 782 755 454 266 839 413 527 348 192 060 791 632 533 913 6 × 2 = 1 + 0.724 598 555 845 615 565 510 908 533 678 827 054 696 384 121 583 265 067 827 2;
  • 25) 0.724 598 555 845 615 565 510 908 533 678 827 054 696 384 121 583 265 067 827 2 × 2 = 1 + 0.449 197 111 691 231 131 021 817 067 357 654 109 392 768 243 166 530 135 654 4;
  • 26) 0.449 197 111 691 231 131 021 817 067 357 654 109 392 768 243 166 530 135 654 4 × 2 = 0 + 0.898 394 223 382 462 262 043 634 134 715 308 218 785 536 486 333 060 271 308 8;
  • 27) 0.898 394 223 382 462 262 043 634 134 715 308 218 785 536 486 333 060 271 308 8 × 2 = 1 + 0.796 788 446 764 924 524 087 268 269 430 616 437 571 072 972 666 120 542 617 6;
  • 28) 0.796 788 446 764 924 524 087 268 269 430 616 437 571 072 972 666 120 542 617 6 × 2 = 1 + 0.593 576 893 529 849 048 174 536 538 861 232 875 142 145 945 332 241 085 235 2;
  • 29) 0.593 576 893 529 849 048 174 536 538 861 232 875 142 145 945 332 241 085 235 2 × 2 = 1 + 0.187 153 787 059 698 096 349 073 077 722 465 750 284 291 890 664 482 170 470 4;
  • 30) 0.187 153 787 059 698 096 349 073 077 722 465 750 284 291 890 664 482 170 470 4 × 2 = 0 + 0.374 307 574 119 396 192 698 146 155 444 931 500 568 583 781 328 964 340 940 8;
  • 31) 0.374 307 574 119 396 192 698 146 155 444 931 500 568 583 781 328 964 340 940 8 × 2 = 0 + 0.748 615 148 238 792 385 396 292 310 889 863 001 137 167 562 657 928 681 881 6;
  • 32) 0.748 615 148 238 792 385 396 292 310 889 863 001 137 167 562 657 928 681 881 6 × 2 = 1 + 0.497 230 296 477 584 770 792 584 621 779 726 002 274 335 125 315 857 363 763 2;
  • 33) 0.497 230 296 477 584 770 792 584 621 779 726 002 274 335 125 315 857 363 763 2 × 2 = 0 + 0.994 460 592 955 169 541 585 169 243 559 452 004 548 670 250 631 714 727 526 4;
  • 34) 0.994 460 592 955 169 541 585 169 243 559 452 004 548 670 250 631 714 727 526 4 × 2 = 1 + 0.988 921 185 910 339 083 170 338 487 118 904 009 097 340 501 263 429 455 052 8;
  • 35) 0.988 921 185 910 339 083 170 338 487 118 904 009 097 340 501 263 429 455 052 8 × 2 = 1 + 0.977 842 371 820 678 166 340 676 974 237 808 018 194 681 002 526 858 910 105 6;
  • 36) 0.977 842 371 820 678 166 340 676 974 237 808 018 194 681 002 526 858 910 105 6 × 2 = 1 + 0.955 684 743 641 356 332 681 353 948 475 616 036 389 362 005 053 717 820 211 2;
  • 37) 0.955 684 743 641 356 332 681 353 948 475 616 036 389 362 005 053 717 820 211 2 × 2 = 1 + 0.911 369 487 282 712 665 362 707 896 951 232 072 778 724 010 107 435 640 422 4;
  • 38) 0.911 369 487 282 712 665 362 707 896 951 232 072 778 724 010 107 435 640 422 4 × 2 = 1 + 0.822 738 974 565 425 330 725 415 793 902 464 145 557 448 020 214 871 280 844 8;
  • 39) 0.822 738 974 565 425 330 725 415 793 902 464 145 557 448 020 214 871 280 844 8 × 2 = 1 + 0.645 477 949 130 850 661 450 831 587 804 928 291 114 896 040 429 742 561 689 6;
  • 40) 0.645 477 949 130 850 661 450 831 587 804 928 291 114 896 040 429 742 561 689 6 × 2 = 1 + 0.290 955 898 261 701 322 901 663 175 609 856 582 229 792 080 859 485 123 379 2;
  • 41) 0.290 955 898 261 701 322 901 663 175 609 856 582 229 792 080 859 485 123 379 2 × 2 = 0 + 0.581 911 796 523 402 645 803 326 351 219 713 164 459 584 161 718 970 246 758 4;
  • 42) 0.581 911 796 523 402 645 803 326 351 219 713 164 459 584 161 718 970 246 758 4 × 2 = 1 + 0.163 823 593 046 805 291 606 652 702 439 426 328 919 168 323 437 940 493 516 8;
  • 43) 0.163 823 593 046 805 291 606 652 702 439 426 328 919 168 323 437 940 493 516 8 × 2 = 0 + 0.327 647 186 093 610 583 213 305 404 878 852 657 838 336 646 875 880 987 033 6;
  • 44) 0.327 647 186 093 610 583 213 305 404 878 852 657 838 336 646 875 880 987 033 6 × 2 = 0 + 0.655 294 372 187 221 166 426 610 809 757 705 315 676 673 293 751 761 974 067 2;
  • 45) 0.655 294 372 187 221 166 426 610 809 757 705 315 676 673 293 751 761 974 067 2 × 2 = 1 + 0.310 588 744 374 442 332 853 221 619 515 410 631 353 346 587 503 523 948 134 4;
  • 46) 0.310 588 744 374 442 332 853 221 619 515 410 631 353 346 587 503 523 948 134 4 × 2 = 0 + 0.621 177 488 748 884 665 706 443 239 030 821 262 706 693 175 007 047 896 268 8;
  • 47) 0.621 177 488 748 884 665 706 443 239 030 821 262 706 693 175 007 047 896 268 8 × 2 = 1 + 0.242 354 977 497 769 331 412 886 478 061 642 525 413 386 350 014 095 792 537 6;
  • 48) 0.242 354 977 497 769 331 412 886 478 061 642 525 413 386 350 014 095 792 537 6 × 2 = 0 + 0.484 709 954 995 538 662 825 772 956 123 285 050 826 772 700 028 191 585 075 2;
  • 49) 0.484 709 954 995 538 662 825 772 956 123 285 050 826 772 700 028 191 585 075 2 × 2 = 0 + 0.969 419 909 991 077 325 651 545 912 246 570 101 653 545 400 056 383 170 150 4;
  • 50) 0.969 419 909 991 077 325 651 545 912 246 570 101 653 545 400 056 383 170 150 4 × 2 = 1 + 0.938 839 819 982 154 651 303 091 824 493 140 203 307 090 800 112 766 340 300 8;
  • 51) 0.938 839 819 982 154 651 303 091 824 493 140 203 307 090 800 112 766 340 300 8 × 2 = 1 + 0.877 679 639 964 309 302 606 183 648 986 280 406 614 181 600 225 532 680 601 6;
  • 52) 0.877 679 639 964 309 302 606 183 648 986 280 406 614 181 600 225 532 680 601 6 × 2 = 1 + 0.755 359 279 928 618 605 212 367 297 972 560 813 228 363 200 451 065 361 203 2;
  • 53) 0.755 359 279 928 618 605 212 367 297 972 560 813 228 363 200 451 065 361 203 2 × 2 = 1 + 0.510 718 559 857 237 210 424 734 595 945 121 626 456 726 400 902 130 722 406 4;

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:


0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 161 7(10) =

0.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(2)

5. Positive number before normalization:

1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 161 7(10) =

1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 161 7(10) =

1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(2) =

1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(2) × 20


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): 0

Mantissa (not normalized):
1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 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) =

1023(10) =

011 1111 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1 =

1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111


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 1111 1111

Mantissa (52 bits) =
1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111


Decimal number 1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 161 7 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111

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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