0.000 000 000 000 000 012 345 687 894 572 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 572 8(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
0.000 000 000 000 000 012 345 687 894 572 8(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: 0.000 000 000 000 000 012 345 687 894 572 8.

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) 0.000 000 000 000 000 012 345 687 894 572 8 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 145 6;
  • 2) 0.000 000 000 000 000 024 691 375 789 145 6 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 291 2;
  • 3) 0.000 000 000 000 000 049 382 751 578 291 2 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 582 4;
  • 4) 0.000 000 000 000 000 098 765 503 156 582 4 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 164 8;
  • 5) 0.000 000 000 000 000 197 531 006 313 164 8 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 329 6;
  • 6) 0.000 000 000 000 000 395 062 012 626 329 6 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 659 2;
  • 7) 0.000 000 000 000 000 790 124 025 252 659 2 × 2 = 0 + 0.000 000 000 000 001 580 248 050 505 318 4;
  • 8) 0.000 000 000 000 001 580 248 050 505 318 4 × 2 = 0 + 0.000 000 000 000 003 160 496 101 010 636 8;
  • 9) 0.000 000 000 000 003 160 496 101 010 636 8 × 2 = 0 + 0.000 000 000 000 006 320 992 202 021 273 6;
  • 10) 0.000 000 000 000 006 320 992 202 021 273 6 × 2 = 0 + 0.000 000 000 000 012 641 984 404 042 547 2;
  • 11) 0.000 000 000 000 012 641 984 404 042 547 2 × 2 = 0 + 0.000 000 000 000 025 283 968 808 085 094 4;
  • 12) 0.000 000 000 000 025 283 968 808 085 094 4 × 2 = 0 + 0.000 000 000 000 050 567 937 616 170 188 8;
  • 13) 0.000 000 000 000 050 567 937 616 170 188 8 × 2 = 0 + 0.000 000 000 000 101 135 875 232 340 377 6;
  • 14) 0.000 000 000 000 101 135 875 232 340 377 6 × 2 = 0 + 0.000 000 000 000 202 271 750 464 680 755 2;
  • 15) 0.000 000 000 000 202 271 750 464 680 755 2 × 2 = 0 + 0.000 000 000 000 404 543 500 929 361 510 4;
  • 16) 0.000 000 000 000 404 543 500 929 361 510 4 × 2 = 0 + 0.000 000 000 000 809 087 001 858 723 020 8;
  • 17) 0.000 000 000 000 809 087 001 858 723 020 8 × 2 = 0 + 0.000 000 000 001 618 174 003 717 446 041 6;
  • 18) 0.000 000 000 001 618 174 003 717 446 041 6 × 2 = 0 + 0.000 000 000 003 236 348 007 434 892 083 2;
  • 19) 0.000 000 000 003 236 348 007 434 892 083 2 × 2 = 0 + 0.000 000 000 006 472 696 014 869 784 166 4;
  • 20) 0.000 000 000 006 472 696 014 869 784 166 4 × 2 = 0 + 0.000 000 000 012 945 392 029 739 568 332 8;
  • 21) 0.000 000 000 012 945 392 029 739 568 332 8 × 2 = 0 + 0.000 000 000 025 890 784 059 479 136 665 6;
  • 22) 0.000 000 000 025 890 784 059 479 136 665 6 × 2 = 0 + 0.000 000 000 051 781 568 118 958 273 331 2;
  • 23) 0.000 000 000 051 781 568 118 958 273 331 2 × 2 = 0 + 0.000 000 000 103 563 136 237 916 546 662 4;
  • 24) 0.000 000 000 103 563 136 237 916 546 662 4 × 2 = 0 + 0.000 000 000 207 126 272 475 833 093 324 8;
  • 25) 0.000 000 000 207 126 272 475 833 093 324 8 × 2 = 0 + 0.000 000 000 414 252 544 951 666 186 649 6;
  • 26) 0.000 000 000 414 252 544 951 666 186 649 6 × 2 = 0 + 0.000 000 000 828 505 089 903 332 373 299 2;
  • 27) 0.000 000 000 828 505 089 903 332 373 299 2 × 2 = 0 + 0.000 000 001 657 010 179 806 664 746 598 4;
  • 28) 0.000 000 001 657 010 179 806 664 746 598 4 × 2 = 0 + 0.000 000 003 314 020 359 613 329 493 196 8;
  • 29) 0.000 000 003 314 020 359 613 329 493 196 8 × 2 = 0 + 0.000 000 006 628 040 719 226 658 986 393 6;
  • 30) 0.000 000 006 628 040 719 226 658 986 393 6 × 2 = 0 + 0.000 000 013 256 081 438 453 317 972 787 2;
  • 31) 0.000 000 013 256 081 438 453 317 972 787 2 × 2 = 0 + 0.000 000 026 512 162 876 906 635 945 574 4;
  • 32) 0.000 000 026 512 162 876 906 635 945 574 4 × 2 = 0 + 0.000 000 053 024 325 753 813 271 891 148 8;
  • 33) 0.000 000 053 024 325 753 813 271 891 148 8 × 2 = 0 + 0.000 000 106 048 651 507 626 543 782 297 6;
  • 34) 0.000 000 106 048 651 507 626 543 782 297 6 × 2 = 0 + 0.000 000 212 097 303 015 253 087 564 595 2;
  • 35) 0.000 000 212 097 303 015 253 087 564 595 2 × 2 = 0 + 0.000 000 424 194 606 030 506 175 129 190 4;
  • 36) 0.000 000 424 194 606 030 506 175 129 190 4 × 2 = 0 + 0.000 000 848 389 212 061 012 350 258 380 8;
  • 37) 0.000 000 848 389 212 061 012 350 258 380 8 × 2 = 0 + 0.000 001 696 778 424 122 024 700 516 761 6;
  • 38) 0.000 001 696 778 424 122 024 700 516 761 6 × 2 = 0 + 0.000 003 393 556 848 244 049 401 033 523 2;
  • 39) 0.000 003 393 556 848 244 049 401 033 523 2 × 2 = 0 + 0.000 006 787 113 696 488 098 802 067 046 4;
  • 40) 0.000 006 787 113 696 488 098 802 067 046 4 × 2 = 0 + 0.000 013 574 227 392 976 197 604 134 092 8;
  • 41) 0.000 013 574 227 392 976 197 604 134 092 8 × 2 = 0 + 0.000 027 148 454 785 952 395 208 268 185 6;
  • 42) 0.000 027 148 454 785 952 395 208 268 185 6 × 2 = 0 + 0.000 054 296 909 571 904 790 416 536 371 2;
  • 43) 0.000 054 296 909 571 904 790 416 536 371 2 × 2 = 0 + 0.000 108 593 819 143 809 580 833 072 742 4;
  • 44) 0.000 108 593 819 143 809 580 833 072 742 4 × 2 = 0 + 0.000 217 187 638 287 619 161 666 145 484 8;
  • 45) 0.000 217 187 638 287 619 161 666 145 484 8 × 2 = 0 + 0.000 434 375 276 575 238 323 332 290 969 6;
  • 46) 0.000 434 375 276 575 238 323 332 290 969 6 × 2 = 0 + 0.000 868 750 553 150 476 646 664 581 939 2;
  • 47) 0.000 868 750 553 150 476 646 664 581 939 2 × 2 = 0 + 0.001 737 501 106 300 953 293 329 163 878 4;
  • 48) 0.001 737 501 106 300 953 293 329 163 878 4 × 2 = 0 + 0.003 475 002 212 601 906 586 658 327 756 8;
  • 49) 0.003 475 002 212 601 906 586 658 327 756 8 × 2 = 0 + 0.006 950 004 425 203 813 173 316 655 513 6;
  • 50) 0.006 950 004 425 203 813 173 316 655 513 6 × 2 = 0 + 0.013 900 008 850 407 626 346 633 311 027 2;
  • 51) 0.013 900 008 850 407 626 346 633 311 027 2 × 2 = 0 + 0.027 800 017 700 815 252 693 266 622 054 4;
  • 52) 0.027 800 017 700 815 252 693 266 622 054 4 × 2 = 0 + 0.055 600 035 401 630 505 386 533 244 108 8;
  • 53) 0.055 600 035 401 630 505 386 533 244 108 8 × 2 = 0 + 0.111 200 070 803 261 010 773 066 488 217 6;
  • 54) 0.111 200 070 803 261 010 773 066 488 217 6 × 2 = 0 + 0.222 400 141 606 522 021 546 132 976 435 2;
  • 55) 0.222 400 141 606 522 021 546 132 976 435 2 × 2 = 0 + 0.444 800 283 213 044 043 092 265 952 870 4;
  • 56) 0.444 800 283 213 044 043 092 265 952 870 4 × 2 = 0 + 0.889 600 566 426 088 086 184 531 905 740 8;
  • 57) 0.889 600 566 426 088 086 184 531 905 740 8 × 2 = 1 + 0.779 201 132 852 176 172 369 063 811 481 6;
  • 58) 0.779 201 132 852 176 172 369 063 811 481 6 × 2 = 1 + 0.558 402 265 704 352 344 738 127 622 963 2;
  • 59) 0.558 402 265 704 352 344 738 127 622 963 2 × 2 = 1 + 0.116 804 531 408 704 689 476 255 245 926 4;
  • 60) 0.116 804 531 408 704 689 476 255 245 926 4 × 2 = 0 + 0.233 609 062 817 409 378 952 510 491 852 8;
  • 61) 0.233 609 062 817 409 378 952 510 491 852 8 × 2 = 0 + 0.467 218 125 634 818 757 905 020 983 705 6;
  • 62) 0.467 218 125 634 818 757 905 020 983 705 6 × 2 = 0 + 0.934 436 251 269 637 515 810 041 967 411 2;
  • 63) 0.934 436 251 269 637 515 810 041 967 411 2 × 2 = 1 + 0.868 872 502 539 275 031 620 083 934 822 4;
  • 64) 0.868 872 502 539 275 031 620 083 934 822 4 × 2 = 1 + 0.737 745 005 078 550 063 240 167 869 644 8;
  • 65) 0.737 745 005 078 550 063 240 167 869 644 8 × 2 = 1 + 0.475 490 010 157 100 126 480 335 739 289 6;
  • 66) 0.475 490 010 157 100 126 480 335 739 289 6 × 2 = 0 + 0.950 980 020 314 200 252 960 671 478 579 2;
  • 67) 0.950 980 020 314 200 252 960 671 478 579 2 × 2 = 1 + 0.901 960 040 628 400 505 921 342 957 158 4;
  • 68) 0.901 960 040 628 400 505 921 342 957 158 4 × 2 = 1 + 0.803 920 081 256 801 011 842 685 914 316 8;
  • 69) 0.803 920 081 256 801 011 842 685 914 316 8 × 2 = 1 + 0.607 840 162 513 602 023 685 371 828 633 6;
  • 70) 0.607 840 162 513 602 023 685 371 828 633 6 × 2 = 1 + 0.215 680 325 027 204 047 370 743 657 267 2;
  • 71) 0.215 680 325 027 204 047 370 743 657 267 2 × 2 = 0 + 0.431 360 650 054 408 094 741 487 314 534 4;
  • 72) 0.431 360 650 054 408 094 741 487 314 534 4 × 2 = 0 + 0.862 721 300 108 816 189 482 974 629 068 8;
  • 73) 0.862 721 300 108 816 189 482 974 629 068 8 × 2 = 1 + 0.725 442 600 217 632 378 965 949 258 137 6;
  • 74) 0.725 442 600 217 632 378 965 949 258 137 6 × 2 = 1 + 0.450 885 200 435 264 757 931 898 516 275 2;
  • 75) 0.450 885 200 435 264 757 931 898 516 275 2 × 2 = 0 + 0.901 770 400 870 529 515 863 797 032 550 4;
  • 76) 0.901 770 400 870 529 515 863 797 032 550 4 × 2 = 1 + 0.803 540 801 741 059 031 727 594 065 100 8;
  • 77) 0.803 540 801 741 059 031 727 594 065 100 8 × 2 = 1 + 0.607 081 603 482 118 063 455 188 130 201 6;
  • 78) 0.607 081 603 482 118 063 455 188 130 201 6 × 2 = 1 + 0.214 163 206 964 236 126 910 376 260 403 2;
  • 79) 0.214 163 206 964 236 126 910 376 260 403 2 × 2 = 0 + 0.428 326 413 928 472 253 820 752 520 806 4;
  • 80) 0.428 326 413 928 472 253 820 752 520 806 4 × 2 = 0 + 0.856 652 827 856 944 507 641 505 041 612 8;
  • 81) 0.856 652 827 856 944 507 641 505 041 612 8 × 2 = 1 + 0.713 305 655 713 889 015 283 010 083 225 6;
  • 82) 0.713 305 655 713 889 015 283 010 083 225 6 × 2 = 1 + 0.426 611 311 427 778 030 566 020 166 451 2;
  • 83) 0.426 611 311 427 778 030 566 020 166 451 2 × 2 = 0 + 0.853 222 622 855 556 061 132 040 332 902 4;
  • 84) 0.853 222 622 855 556 061 132 040 332 902 4 × 2 = 1 + 0.706 445 245 711 112 122 264 080 665 804 8;
  • 85) 0.706 445 245 711 112 122 264 080 665 804 8 × 2 = 1 + 0.412 890 491 422 224 244 528 161 331 609 6;
  • 86) 0.412 890 491 422 224 244 528 161 331 609 6 × 2 = 0 + 0.825 780 982 844 448 489 056 322 663 219 2;
  • 87) 0.825 780 982 844 448 489 056 322 663 219 2 × 2 = 1 + 0.651 561 965 688 896 978 112 645 326 438 4;
  • 88) 0.651 561 965 688 896 978 112 645 326 438 4 × 2 = 1 + 0.303 123 931 377 793 956 225 290 652 876 8;
  • 89) 0.303 123 931 377 793 956 225 290 652 876 8 × 2 = 0 + 0.606 247 862 755 587 912 450 581 305 753 6;
  • 90) 0.606 247 862 755 587 912 450 581 305 753 6 × 2 = 1 + 0.212 495 725 511 175 824 901 162 611 507 2;
  • 91) 0.212 495 725 511 175 824 901 162 611 507 2 × 2 = 0 + 0.424 991 451 022 351 649 802 325 223 014 4;
  • 92) 0.424 991 451 022 351 649 802 325 223 014 4 × 2 = 0 + 0.849 982 902 044 703 299 604 650 446 028 8;
  • 93) 0.849 982 902 044 703 299 604 650 446 028 8 × 2 = 1 + 0.699 965 804 089 406 599 209 300 892 057 6;
  • 94) 0.699 965 804 089 406 599 209 300 892 057 6 × 2 = 1 + 0.399 931 608 178 813 198 418 601 784 115 2;
  • 95) 0.399 931 608 178 813 198 418 601 784 115 2 × 2 = 0 + 0.799 863 216 357 626 396 837 203 568 230 4;
  • 96) 0.799 863 216 357 626 396 837 203 568 230 4 × 2 = 1 + 0.599 726 432 715 252 793 674 407 136 460 8;
  • 97) 0.599 726 432 715 252 793 674 407 136 460 8 × 2 = 1 + 0.199 452 865 430 505 587 348 814 272 921 6;
  • 98) 0.199 452 865 430 505 587 348 814 272 921 6 × 2 = 0 + 0.398 905 730 861 011 174 697 628 545 843 2;
  • 99) 0.398 905 730 861 011 174 697 628 545 843 2 × 2 = 0 + 0.797 811 461 722 022 349 395 257 091 686 4;
  • 100) 0.797 811 461 722 022 349 395 257 091 686 4 × 2 = 1 + 0.595 622 923 444 044 698 790 514 183 372 8;
  • 101) 0.595 622 923 444 044 698 790 514 183 372 8 × 2 = 1 + 0.191 245 846 888 089 397 581 028 366 745 6;
  • 102) 0.191 245 846 888 089 397 581 028 366 745 6 × 2 = 0 + 0.382 491 693 776 178 795 162 056 733 491 2;
  • 103) 0.382 491 693 776 178 795 162 056 733 491 2 × 2 = 0 + 0.764 983 387 552 357 590 324 113 466 982 4;
  • 104) 0.764 983 387 552 357 590 324 113 466 982 4 × 2 = 1 + 0.529 966 775 104 715 180 648 226 933 964 8;
  • 105) 0.529 966 775 104 715 180 648 226 933 964 8 × 2 = 1 + 0.059 933 550 209 430 361 296 453 867 929 6;
  • 106) 0.059 933 550 209 430 361 296 453 867 929 6 × 2 = 0 + 0.119 867 100 418 860 722 592 907 735 859 2;
  • 107) 0.119 867 100 418 860 722 592 907 735 859 2 × 2 = 0 + 0.239 734 200 837 721 445 185 815 471 718 4;
  • 108) 0.239 734 200 837 721 445 185 815 471 718 4 × 2 = 0 + 0.479 468 401 675 442 890 371 630 943 436 8;
  • 109) 0.479 468 401 675 442 890 371 630 943 436 8 × 2 = 0 + 0.958 936 803 350 885 780 743 261 886 873 6;

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:


0.000 000 000 000 000 012 345 687 894 572 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1101 1001 1001 1000 0(2)

5. Positive number before normalization:

0.000 000 000 000 000 012 345 687 894 572 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1101 1001 1001 1000 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 57 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 012 345 687 894 572 8(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1101 1001 1001 1000 0(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1101 1001 1001 1000 0(2) × 20 =

1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1011 0011 0011 0000(2) × 2-57


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): -57

Mantissa (not normalized):
1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1011 0011 0011 0000


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-57 + 2(11-1) - 1 =

(-57 + 1 023)(10) =

966(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;
  • 966 ÷ 2 = 483 + 0;
  • 483 ÷ 2 = 241 + 1;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

966(10) =

011 1100 0110(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. 1100 0111 0111 1001 1011 1001 1011 0110 1001 1011 0011 0011 0000 =

1100 0111 0111 1001 1011 1001 1011 0110 1001 1011 0011 0011 0000


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 1100 0110

Mantissa (52 bits) =
1100 0111 0111 1001 1011 1001 1011 0110 1001 1011 0011 0011 0000


Decimal number 0.000 000 000 000 000 012 345 687 894 572 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 0110 - 1100 0111 0111 1001 1011 1001 1011 0110 1001 1011 0011 0011 0000

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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