3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 591 46 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 591 46(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
3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 591 46(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: 3.
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;
  • 3 ÷ 2 = 1 + 1;
  • 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.

3(10) =

11(2)


3. Convert to binary (base 2) the fractional part: 0.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 591 46.

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) 0.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 591 46 × 2 = 0 + 0.283 185 307 179 586 476 925 286 766 559 005 768 394 338 798 750 211 641 949 889 182 92;
  • 2) 0.283 185 307 179 586 476 925 286 766 559 005 768 394 338 798 750 211 641 949 889 182 92 × 2 = 0 + 0.566 370 614 359 172 953 850 573 533 118 011 536 788 677 597 500 423 283 899 778 365 84;
  • 3) 0.566 370 614 359 172 953 850 573 533 118 011 536 788 677 597 500 423 283 899 778 365 84 × 2 = 1 + 0.132 741 228 718 345 907 701 147 066 236 023 073 577 355 195 000 846 567 799 556 731 68;
  • 4) 0.132 741 228 718 345 907 701 147 066 236 023 073 577 355 195 000 846 567 799 556 731 68 × 2 = 0 + 0.265 482 457 436 691 815 402 294 132 472 046 147 154 710 390 001 693 135 599 113 463 36;
  • 5) 0.265 482 457 436 691 815 402 294 132 472 046 147 154 710 390 001 693 135 599 113 463 36 × 2 = 0 + 0.530 964 914 873 383 630 804 588 264 944 092 294 309 420 780 003 386 271 198 226 926 72;
  • 6) 0.530 964 914 873 383 630 804 588 264 944 092 294 309 420 780 003 386 271 198 226 926 72 × 2 = 1 + 0.061 929 829 746 767 261 609 176 529 888 184 588 618 841 560 006 772 542 396 453 853 44;
  • 7) 0.061 929 829 746 767 261 609 176 529 888 184 588 618 841 560 006 772 542 396 453 853 44 × 2 = 0 + 0.123 859 659 493 534 523 218 353 059 776 369 177 237 683 120 013 545 084 792 907 706 88;
  • 8) 0.123 859 659 493 534 523 218 353 059 776 369 177 237 683 120 013 545 084 792 907 706 88 × 2 = 0 + 0.247 719 318 987 069 046 436 706 119 552 738 354 475 366 240 027 090 169 585 815 413 76;
  • 9) 0.247 719 318 987 069 046 436 706 119 552 738 354 475 366 240 027 090 169 585 815 413 76 × 2 = 0 + 0.495 438 637 974 138 092 873 412 239 105 476 708 950 732 480 054 180 339 171 630 827 52;
  • 10) 0.495 438 637 974 138 092 873 412 239 105 476 708 950 732 480 054 180 339 171 630 827 52 × 2 = 0 + 0.990 877 275 948 276 185 746 824 478 210 953 417 901 464 960 108 360 678 343 261 655 04;
  • 11) 0.990 877 275 948 276 185 746 824 478 210 953 417 901 464 960 108 360 678 343 261 655 04 × 2 = 1 + 0.981 754 551 896 552 371 493 648 956 421 906 835 802 929 920 216 721 356 686 523 310 08;
  • 12) 0.981 754 551 896 552 371 493 648 956 421 906 835 802 929 920 216 721 356 686 523 310 08 × 2 = 1 + 0.963 509 103 793 104 742 987 297 912 843 813 671 605 859 840 433 442 713 373 046 620 16;
  • 13) 0.963 509 103 793 104 742 987 297 912 843 813 671 605 859 840 433 442 713 373 046 620 16 × 2 = 1 + 0.927 018 207 586 209 485 974 595 825 687 627 343 211 719 680 866 885 426 746 093 240 32;
  • 14) 0.927 018 207 586 209 485 974 595 825 687 627 343 211 719 680 866 885 426 746 093 240 32 × 2 = 1 + 0.854 036 415 172 418 971 949 191 651 375 254 686 423 439 361 733 770 853 492 186 480 64;
  • 15) 0.854 036 415 172 418 971 949 191 651 375 254 686 423 439 361 733 770 853 492 186 480 64 × 2 = 1 + 0.708 072 830 344 837 943 898 383 302 750 509 372 846 878 723 467 541 706 984 372 961 28;
  • 16) 0.708 072 830 344 837 943 898 383 302 750 509 372 846 878 723 467 541 706 984 372 961 28 × 2 = 1 + 0.416 145 660 689 675 887 796 766 605 501 018 745 693 757 446 935 083 413 968 745 922 56;
  • 17) 0.416 145 660 689 675 887 796 766 605 501 018 745 693 757 446 935 083 413 968 745 922 56 × 2 = 0 + 0.832 291 321 379 351 775 593 533 211 002 037 491 387 514 893 870 166 827 937 491 845 12;
  • 18) 0.832 291 321 379 351 775 593 533 211 002 037 491 387 514 893 870 166 827 937 491 845 12 × 2 = 1 + 0.664 582 642 758 703 551 187 066 422 004 074 982 775 029 787 740 333 655 874 983 690 24;
  • 19) 0.664 582 642 758 703 551 187 066 422 004 074 982 775 029 787 740 333 655 874 983 690 24 × 2 = 1 + 0.329 165 285 517 407 102 374 132 844 008 149 965 550 059 575 480 667 311 749 967 380 48;
  • 20) 0.329 165 285 517 407 102 374 132 844 008 149 965 550 059 575 480 667 311 749 967 380 48 × 2 = 0 + 0.658 330 571 034 814 204 748 265 688 016 299 931 100 119 150 961 334 623 499 934 760 96;
  • 21) 0.658 330 571 034 814 204 748 265 688 016 299 931 100 119 150 961 334 623 499 934 760 96 × 2 = 1 + 0.316 661 142 069 628 409 496 531 376 032 599 862 200 238 301 922 669 246 999 869 521 92;
  • 22) 0.316 661 142 069 628 409 496 531 376 032 599 862 200 238 301 922 669 246 999 869 521 92 × 2 = 0 + 0.633 322 284 139 256 818 993 062 752 065 199 724 400 476 603 845 338 493 999 739 043 84;
  • 23) 0.633 322 284 139 256 818 993 062 752 065 199 724 400 476 603 845 338 493 999 739 043 84 × 2 = 1 + 0.266 644 568 278 513 637 986 125 504 130 399 448 800 953 207 690 676 987 999 478 087 68;
  • 24) 0.266 644 568 278 513 637 986 125 504 130 399 448 800 953 207 690 676 987 999 478 087 68 × 2 = 0 + 0.533 289 136 557 027 275 972 251 008 260 798 897 601 906 415 381 353 975 998 956 175 36;
  • 25) 0.533 289 136 557 027 275 972 251 008 260 798 897 601 906 415 381 353 975 998 956 175 36 × 2 = 1 + 0.066 578 273 114 054 551 944 502 016 521 597 795 203 812 830 762 707 951 997 912 350 72;
  • 26) 0.066 578 273 114 054 551 944 502 016 521 597 795 203 812 830 762 707 951 997 912 350 72 × 2 = 0 + 0.133 156 546 228 109 103 889 004 033 043 195 590 407 625 661 525 415 903 995 824 701 44;
  • 27) 0.133 156 546 228 109 103 889 004 033 043 195 590 407 625 661 525 415 903 995 824 701 44 × 2 = 0 + 0.266 313 092 456 218 207 778 008 066 086 391 180 815 251 323 050 831 807 991 649 402 88;
  • 28) 0.266 313 092 456 218 207 778 008 066 086 391 180 815 251 323 050 831 807 991 649 402 88 × 2 = 0 + 0.532 626 184 912 436 415 556 016 132 172 782 361 630 502 646 101 663 615 983 298 805 76;
  • 29) 0.532 626 184 912 436 415 556 016 132 172 782 361 630 502 646 101 663 615 983 298 805 76 × 2 = 1 + 0.065 252 369 824 872 831 112 032 264 345 564 723 261 005 292 203 327 231 966 597 611 52;
  • 30) 0.065 252 369 824 872 831 112 032 264 345 564 723 261 005 292 203 327 231 966 597 611 52 × 2 = 0 + 0.130 504 739 649 745 662 224 064 528 691 129 446 522 010 584 406 654 463 933 195 223 04;
  • 31) 0.130 504 739 649 745 662 224 064 528 691 129 446 522 010 584 406 654 463 933 195 223 04 × 2 = 0 + 0.261 009 479 299 491 324 448 129 057 382 258 893 044 021 168 813 308 927 866 390 446 08;
  • 32) 0.261 009 479 299 491 324 448 129 057 382 258 893 044 021 168 813 308 927 866 390 446 08 × 2 = 0 + 0.522 018 958 598 982 648 896 258 114 764 517 786 088 042 337 626 617 855 732 780 892 16;
  • 33) 0.522 018 958 598 982 648 896 258 114 764 517 786 088 042 337 626 617 855 732 780 892 16 × 2 = 1 + 0.044 037 917 197 965 297 792 516 229 529 035 572 176 084 675 253 235 711 465 561 784 32;
  • 34) 0.044 037 917 197 965 297 792 516 229 529 035 572 176 084 675 253 235 711 465 561 784 32 × 2 = 0 + 0.088 075 834 395 930 595 585 032 459 058 071 144 352 169 350 506 471 422 931 123 568 64;
  • 35) 0.088 075 834 395 930 595 585 032 459 058 071 144 352 169 350 506 471 422 931 123 568 64 × 2 = 0 + 0.176 151 668 791 861 191 170 064 918 116 142 288 704 338 701 012 942 845 862 247 137 28;
  • 36) 0.176 151 668 791 861 191 170 064 918 116 142 288 704 338 701 012 942 845 862 247 137 28 × 2 = 0 + 0.352 303 337 583 722 382 340 129 836 232 284 577 408 677 402 025 885 691 724 494 274 56;
  • 37) 0.352 303 337 583 722 382 340 129 836 232 284 577 408 677 402 025 885 691 724 494 274 56 × 2 = 0 + 0.704 606 675 167 444 764 680 259 672 464 569 154 817 354 804 051 771 383 448 988 549 12;
  • 38) 0.704 606 675 167 444 764 680 259 672 464 569 154 817 354 804 051 771 383 448 988 549 12 × 2 = 1 + 0.409 213 350 334 889 529 360 519 344 929 138 309 634 709 608 103 542 766 897 977 098 24;
  • 39) 0.409 213 350 334 889 529 360 519 344 929 138 309 634 709 608 103 542 766 897 977 098 24 × 2 = 0 + 0.818 426 700 669 779 058 721 038 689 858 276 619 269 419 216 207 085 533 795 954 196 48;
  • 40) 0.818 426 700 669 779 058 721 038 689 858 276 619 269 419 216 207 085 533 795 954 196 48 × 2 = 1 + 0.636 853 401 339 558 117 442 077 379 716 553 238 538 838 432 414 171 067 591 908 392 96;
  • 41) 0.636 853 401 339 558 117 442 077 379 716 553 238 538 838 432 414 171 067 591 908 392 96 × 2 = 1 + 0.273 706 802 679 116 234 884 154 759 433 106 477 077 676 864 828 342 135 183 816 785 92;
  • 42) 0.273 706 802 679 116 234 884 154 759 433 106 477 077 676 864 828 342 135 183 816 785 92 × 2 = 0 + 0.547 413 605 358 232 469 768 309 518 866 212 954 155 353 729 656 684 270 367 633 571 84;
  • 43) 0.547 413 605 358 232 469 768 309 518 866 212 954 155 353 729 656 684 270 367 633 571 84 × 2 = 1 + 0.094 827 210 716 464 939 536 619 037 732 425 908 310 707 459 313 368 540 735 267 143 68;
  • 44) 0.094 827 210 716 464 939 536 619 037 732 425 908 310 707 459 313 368 540 735 267 143 68 × 2 = 0 + 0.189 654 421 432 929 879 073 238 075 464 851 816 621 414 918 626 737 081 470 534 287 36;
  • 45) 0.189 654 421 432 929 879 073 238 075 464 851 816 621 414 918 626 737 081 470 534 287 36 × 2 = 0 + 0.379 308 842 865 859 758 146 476 150 929 703 633 242 829 837 253 474 162 941 068 574 72;
  • 46) 0.379 308 842 865 859 758 146 476 150 929 703 633 242 829 837 253 474 162 941 068 574 72 × 2 = 0 + 0.758 617 685 731 719 516 292 952 301 859 407 266 485 659 674 506 948 325 882 137 149 44;
  • 47) 0.758 617 685 731 719 516 292 952 301 859 407 266 485 659 674 506 948 325 882 137 149 44 × 2 = 1 + 0.517 235 371 463 439 032 585 904 603 718 814 532 971 319 349 013 896 651 764 274 298 88;
  • 48) 0.517 235 371 463 439 032 585 904 603 718 814 532 971 319 349 013 896 651 764 274 298 88 × 2 = 1 + 0.034 470 742 926 878 065 171 809 207 437 629 065 942 638 698 027 793 303 528 548 597 76;
  • 49) 0.034 470 742 926 878 065 171 809 207 437 629 065 942 638 698 027 793 303 528 548 597 76 × 2 = 0 + 0.068 941 485 853 756 130 343 618 414 875 258 131 885 277 396 055 586 607 057 097 195 52;
  • 50) 0.068 941 485 853 756 130 343 618 414 875 258 131 885 277 396 055 586 607 057 097 195 52 × 2 = 0 + 0.137 882 971 707 512 260 687 236 829 750 516 263 770 554 792 111 173 214 114 194 391 04;
  • 51) 0.137 882 971 707 512 260 687 236 829 750 516 263 770 554 792 111 173 214 114 194 391 04 × 2 = 0 + 0.275 765 943 415 024 521 374 473 659 501 032 527 541 109 584 222 346 428 228 388 782 08;
  • 52) 0.275 765 943 415 024 521 374 473 659 501 032 527 541 109 584 222 346 428 228 388 782 08 × 2 = 0 + 0.551 531 886 830 049 042 748 947 319 002 065 055 082 219 168 444 692 856 456 777 564 16;
  • 53) 0.551 531 886 830 049 042 748 947 319 002 065 055 082 219 168 444 692 856 456 777 564 16 × 2 = 1 + 0.103 063 773 660 098 085 497 894 638 004 130 110 164 438 336 889 385 712 913 555 128 32;

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:


0.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 591 46(10) =

0.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2)

5. Positive number before normalization:

3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 591 46(10) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the left, so that only one non zero digit remains to the left of it:


3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 591 46(10) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2) × 20 =

1.1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 01(2) × 21


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): 1

Mantissa (not normalized):
1.1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 01


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

1 + 2(11-1) - 1 =

(1 + 1 023)(10) =

1 024(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;
  • 1 024 ÷ 2 = 512 + 0;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

1024(10) =

100 0000 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, 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 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 01 =

1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000


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) =
100 0000 0000

Mantissa (52 bits) =
1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000


Decimal number 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 591 46 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000

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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