1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 135 448 622 62 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 135 448 622 62(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
1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 135 448 622 62(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: 0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 135 448 622 62.

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) 0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 135 448 622 62 × 2 = 1 + 0.236 067 977 499 789 696 409 173 668 731 276 235 440 618 359 611 525 724 270 897 245 24;
  • 2) 0.236 067 977 499 789 696 409 173 668 731 276 235 440 618 359 611 525 724 270 897 245 24 × 2 = 0 + 0.472 135 954 999 579 392 818 347 337 462 552 470 881 236 719 223 051 448 541 794 490 48;
  • 3) 0.472 135 954 999 579 392 818 347 337 462 552 470 881 236 719 223 051 448 541 794 490 48 × 2 = 0 + 0.944 271 909 999 158 785 636 694 674 925 104 941 762 473 438 446 102 897 083 588 980 96;
  • 4) 0.944 271 909 999 158 785 636 694 674 925 104 941 762 473 438 446 102 897 083 588 980 96 × 2 = 1 + 0.888 543 819 998 317 571 273 389 349 850 209 883 524 946 876 892 205 794 167 177 961 92;
  • 5) 0.888 543 819 998 317 571 273 389 349 850 209 883 524 946 876 892 205 794 167 177 961 92 × 2 = 1 + 0.777 087 639 996 635 142 546 778 699 700 419 767 049 893 753 784 411 588 334 355 923 84;
  • 6) 0.777 087 639 996 635 142 546 778 699 700 419 767 049 893 753 784 411 588 334 355 923 84 × 2 = 1 + 0.554 175 279 993 270 285 093 557 399 400 839 534 099 787 507 568 823 176 668 711 847 68;
  • 7) 0.554 175 279 993 270 285 093 557 399 400 839 534 099 787 507 568 823 176 668 711 847 68 × 2 = 1 + 0.108 350 559 986 540 570 187 114 798 801 679 068 199 575 015 137 646 353 337 423 695 36;
  • 8) 0.108 350 559 986 540 570 187 114 798 801 679 068 199 575 015 137 646 353 337 423 695 36 × 2 = 0 + 0.216 701 119 973 081 140 374 229 597 603 358 136 399 150 030 275 292 706 674 847 390 72;
  • 9) 0.216 701 119 973 081 140 374 229 597 603 358 136 399 150 030 275 292 706 674 847 390 72 × 2 = 0 + 0.433 402 239 946 162 280 748 459 195 206 716 272 798 300 060 550 585 413 349 694 781 44;
  • 10) 0.433 402 239 946 162 280 748 459 195 206 716 272 798 300 060 550 585 413 349 694 781 44 × 2 = 0 + 0.866 804 479 892 324 561 496 918 390 413 432 545 596 600 121 101 170 826 699 389 562 88;
  • 11) 0.866 804 479 892 324 561 496 918 390 413 432 545 596 600 121 101 170 826 699 389 562 88 × 2 = 1 + 0.733 608 959 784 649 122 993 836 780 826 865 091 193 200 242 202 341 653 398 779 125 76;
  • 12) 0.733 608 959 784 649 122 993 836 780 826 865 091 193 200 242 202 341 653 398 779 125 76 × 2 = 1 + 0.467 217 919 569 298 245 987 673 561 653 730 182 386 400 484 404 683 306 797 558 251 52;
  • 13) 0.467 217 919 569 298 245 987 673 561 653 730 182 386 400 484 404 683 306 797 558 251 52 × 2 = 0 + 0.934 435 839 138 596 491 975 347 123 307 460 364 772 800 968 809 366 613 595 116 503 04;
  • 14) 0.934 435 839 138 596 491 975 347 123 307 460 364 772 800 968 809 366 613 595 116 503 04 × 2 = 1 + 0.868 871 678 277 192 983 950 694 246 614 920 729 545 601 937 618 733 227 190 233 006 08;
  • 15) 0.868 871 678 277 192 983 950 694 246 614 920 729 545 601 937 618 733 227 190 233 006 08 × 2 = 1 + 0.737 743 356 554 385 967 901 388 493 229 841 459 091 203 875 237 466 454 380 466 012 16;
  • 16) 0.737 743 356 554 385 967 901 388 493 229 841 459 091 203 875 237 466 454 380 466 012 16 × 2 = 1 + 0.475 486 713 108 771 935 802 776 986 459 682 918 182 407 750 474 932 908 760 932 024 32;
  • 17) 0.475 486 713 108 771 935 802 776 986 459 682 918 182 407 750 474 932 908 760 932 024 32 × 2 = 0 + 0.950 973 426 217 543 871 605 553 972 919 365 836 364 815 500 949 865 817 521 864 048 64;
  • 18) 0.950 973 426 217 543 871 605 553 972 919 365 836 364 815 500 949 865 817 521 864 048 64 × 2 = 1 + 0.901 946 852 435 087 743 211 107 945 838 731 672 729 631 001 899 731 635 043 728 097 28;
  • 19) 0.901 946 852 435 087 743 211 107 945 838 731 672 729 631 001 899 731 635 043 728 097 28 × 2 = 1 + 0.803 893 704 870 175 486 422 215 891 677 463 345 459 262 003 799 463 270 087 456 194 56;
  • 20) 0.803 893 704 870 175 486 422 215 891 677 463 345 459 262 003 799 463 270 087 456 194 56 × 2 = 1 + 0.607 787 409 740 350 972 844 431 783 354 926 690 918 524 007 598 926 540 174 912 389 12;
  • 21) 0.607 787 409 740 350 972 844 431 783 354 926 690 918 524 007 598 926 540 174 912 389 12 × 2 = 1 + 0.215 574 819 480 701 945 688 863 566 709 853 381 837 048 015 197 853 080 349 824 778 24;
  • 22) 0.215 574 819 480 701 945 688 863 566 709 853 381 837 048 015 197 853 080 349 824 778 24 × 2 = 0 + 0.431 149 638 961 403 891 377 727 133 419 706 763 674 096 030 395 706 160 699 649 556 48;
  • 23) 0.431 149 638 961 403 891 377 727 133 419 706 763 674 096 030 395 706 160 699 649 556 48 × 2 = 0 + 0.862 299 277 922 807 782 755 454 266 839 413 527 348 192 060 791 412 321 399 299 112 96;
  • 24) 0.862 299 277 922 807 782 755 454 266 839 413 527 348 192 060 791 412 321 399 299 112 96 × 2 = 1 + 0.724 598 555 845 615 565 510 908 533 678 827 054 696 384 121 582 824 642 798 598 225 92;
  • 25) 0.724 598 555 845 615 565 510 908 533 678 827 054 696 384 121 582 824 642 798 598 225 92 × 2 = 1 + 0.449 197 111 691 231 131 021 817 067 357 654 109 392 768 243 165 649 285 597 196 451 84;
  • 26) 0.449 197 111 691 231 131 021 817 067 357 654 109 392 768 243 165 649 285 597 196 451 84 × 2 = 0 + 0.898 394 223 382 462 262 043 634 134 715 308 218 785 536 486 331 298 571 194 392 903 68;
  • 27) 0.898 394 223 382 462 262 043 634 134 715 308 218 785 536 486 331 298 571 194 392 903 68 × 2 = 1 + 0.796 788 446 764 924 524 087 268 269 430 616 437 571 072 972 662 597 142 388 785 807 36;
  • 28) 0.796 788 446 764 924 524 087 268 269 430 616 437 571 072 972 662 597 142 388 785 807 36 × 2 = 1 + 0.593 576 893 529 849 048 174 536 538 861 232 875 142 145 945 325 194 284 777 571 614 72;
  • 29) 0.593 576 893 529 849 048 174 536 538 861 232 875 142 145 945 325 194 284 777 571 614 72 × 2 = 1 + 0.187 153 787 059 698 096 349 073 077 722 465 750 284 291 890 650 388 569 555 143 229 44;
  • 30) 0.187 153 787 059 698 096 349 073 077 722 465 750 284 291 890 650 388 569 555 143 229 44 × 2 = 0 + 0.374 307 574 119 396 192 698 146 155 444 931 500 568 583 781 300 777 139 110 286 458 88;
  • 31) 0.374 307 574 119 396 192 698 146 155 444 931 500 568 583 781 300 777 139 110 286 458 88 × 2 = 0 + 0.748 615 148 238 792 385 396 292 310 889 863 001 137 167 562 601 554 278 220 572 917 76;
  • 32) 0.748 615 148 238 792 385 396 292 310 889 863 001 137 167 562 601 554 278 220 572 917 76 × 2 = 1 + 0.497 230 296 477 584 770 792 584 621 779 726 002 274 335 125 203 108 556 441 145 835 52;
  • 33) 0.497 230 296 477 584 770 792 584 621 779 726 002 274 335 125 203 108 556 441 145 835 52 × 2 = 0 + 0.994 460 592 955 169 541 585 169 243 559 452 004 548 670 250 406 217 112 882 291 671 04;
  • 34) 0.994 460 592 955 169 541 585 169 243 559 452 004 548 670 250 406 217 112 882 291 671 04 × 2 = 1 + 0.988 921 185 910 339 083 170 338 487 118 904 009 097 340 500 812 434 225 764 583 342 08;
  • 35) 0.988 921 185 910 339 083 170 338 487 118 904 009 097 340 500 812 434 225 764 583 342 08 × 2 = 1 + 0.977 842 371 820 678 166 340 676 974 237 808 018 194 681 001 624 868 451 529 166 684 16;
  • 36) 0.977 842 371 820 678 166 340 676 974 237 808 018 194 681 001 624 868 451 529 166 684 16 × 2 = 1 + 0.955 684 743 641 356 332 681 353 948 475 616 036 389 362 003 249 736 903 058 333 368 32;
  • 37) 0.955 684 743 641 356 332 681 353 948 475 616 036 389 362 003 249 736 903 058 333 368 32 × 2 = 1 + 0.911 369 487 282 712 665 362 707 896 951 232 072 778 724 006 499 473 806 116 666 736 64;
  • 38) 0.911 369 487 282 712 665 362 707 896 951 232 072 778 724 006 499 473 806 116 666 736 64 × 2 = 1 + 0.822 738 974 565 425 330 725 415 793 902 464 145 557 448 012 998 947 612 233 333 473 28;
  • 39) 0.822 738 974 565 425 330 725 415 793 902 464 145 557 448 012 998 947 612 233 333 473 28 × 2 = 1 + 0.645 477 949 130 850 661 450 831 587 804 928 291 114 896 025 997 895 224 466 666 946 56;
  • 40) 0.645 477 949 130 850 661 450 831 587 804 928 291 114 896 025 997 895 224 466 666 946 56 × 2 = 1 + 0.290 955 898 261 701 322 901 663 175 609 856 582 229 792 051 995 790 448 933 333 893 12;
  • 41) 0.290 955 898 261 701 322 901 663 175 609 856 582 229 792 051 995 790 448 933 333 893 12 × 2 = 0 + 0.581 911 796 523 402 645 803 326 351 219 713 164 459 584 103 991 580 897 866 667 786 24;
  • 42) 0.581 911 796 523 402 645 803 326 351 219 713 164 459 584 103 991 580 897 866 667 786 24 × 2 = 1 + 0.163 823 593 046 805 291 606 652 702 439 426 328 919 168 207 983 161 795 733 335 572 48;
  • 43) 0.163 823 593 046 805 291 606 652 702 439 426 328 919 168 207 983 161 795 733 335 572 48 × 2 = 0 + 0.327 647 186 093 610 583 213 305 404 878 852 657 838 336 415 966 323 591 466 671 144 96;
  • 44) 0.327 647 186 093 610 583 213 305 404 878 852 657 838 336 415 966 323 591 466 671 144 96 × 2 = 0 + 0.655 294 372 187 221 166 426 610 809 757 705 315 676 672 831 932 647 182 933 342 289 92;
  • 45) 0.655 294 372 187 221 166 426 610 809 757 705 315 676 672 831 932 647 182 933 342 289 92 × 2 = 1 + 0.310 588 744 374 442 332 853 221 619 515 410 631 353 345 663 865 294 365 866 684 579 84;
  • 46) 0.310 588 744 374 442 332 853 221 619 515 410 631 353 345 663 865 294 365 866 684 579 84 × 2 = 0 + 0.621 177 488 748 884 665 706 443 239 030 821 262 706 691 327 730 588 731 733 369 159 68;
  • 47) 0.621 177 488 748 884 665 706 443 239 030 821 262 706 691 327 730 588 731 733 369 159 68 × 2 = 1 + 0.242 354 977 497 769 331 412 886 478 061 642 525 413 382 655 461 177 463 466 738 319 36;
  • 48) 0.242 354 977 497 769 331 412 886 478 061 642 525 413 382 655 461 177 463 466 738 319 36 × 2 = 0 + 0.484 709 954 995 538 662 825 772 956 123 285 050 826 765 310 922 354 926 933 476 638 72;
  • 49) 0.484 709 954 995 538 662 825 772 956 123 285 050 826 765 310 922 354 926 933 476 638 72 × 2 = 0 + 0.969 419 909 991 077 325 651 545 912 246 570 101 653 530 621 844 709 853 866 953 277 44;
  • 50) 0.969 419 909 991 077 325 651 545 912 246 570 101 653 530 621 844 709 853 866 953 277 44 × 2 = 1 + 0.938 839 819 982 154 651 303 091 824 493 140 203 307 061 243 689 419 707 733 906 554 88;
  • 51) 0.938 839 819 982 154 651 303 091 824 493 140 203 307 061 243 689 419 707 733 906 554 88 × 2 = 1 + 0.877 679 639 964 309 302 606 183 648 986 280 406 614 122 487 378 839 415 467 813 109 76;
  • 52) 0.877 679 639 964 309 302 606 183 648 986 280 406 614 122 487 378 839 415 467 813 109 76 × 2 = 1 + 0.755 359 279 928 618 605 212 367 297 972 560 813 228 244 974 757 678 830 935 626 219 52;
  • 53) 0.755 359 279 928 618 605 212 367 297 972 560 813 228 244 974 757 678 830 935 626 219 52 × 2 = 1 + 0.510 718 559 857 237 210 424 734 595 945 121 626 456 489 949 515 357 661 871 252 439 04;

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:


0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 135 448 622 62(10) =

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

5. Positive number before normalization:

1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 135 448 622 62(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:


1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 135 448 622 62(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 1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 762 862 135 448 622 62 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