2.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 080 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 2.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 080 3(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
2.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 080 3(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: 2.
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;
  • 2 ÷ 2 = 1 + 0;
  • 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.

2(10) =

10(2)


3. Convert to binary (base 2) the fractional part: 0.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 080 3.

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) 0.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 080 3 × 2 = 1 + 0.436 563 656 918 090 470 720 574 942 705 324 995 514 494 187 399 919 149 933 935 255 448 160 6;
  • 2) 0.436 563 656 918 090 470 720 574 942 705 324 995 514 494 187 399 919 149 933 935 255 448 160 6 × 2 = 0 + 0.873 127 313 836 180 941 441 149 885 410 649 991 028 988 374 799 838 299 867 870 510 896 321 2;
  • 3) 0.873 127 313 836 180 941 441 149 885 410 649 991 028 988 374 799 838 299 867 870 510 896 321 2 × 2 = 1 + 0.746 254 627 672 361 882 882 299 770 821 299 982 057 976 749 599 676 599 735 741 021 792 642 4;
  • 4) 0.746 254 627 672 361 882 882 299 770 821 299 982 057 976 749 599 676 599 735 741 021 792 642 4 × 2 = 1 + 0.492 509 255 344 723 765 764 599 541 642 599 964 115 953 499 199 353 199 471 482 043 585 284 8;
  • 5) 0.492 509 255 344 723 765 764 599 541 642 599 964 115 953 499 199 353 199 471 482 043 585 284 8 × 2 = 0 + 0.985 018 510 689 447 531 529 199 083 285 199 928 231 906 998 398 706 398 942 964 087 170 569 6;
  • 6) 0.985 018 510 689 447 531 529 199 083 285 199 928 231 906 998 398 706 398 942 964 087 170 569 6 × 2 = 1 + 0.970 037 021 378 895 063 058 398 166 570 399 856 463 813 996 797 412 797 885 928 174 341 139 2;
  • 7) 0.970 037 021 378 895 063 058 398 166 570 399 856 463 813 996 797 412 797 885 928 174 341 139 2 × 2 = 1 + 0.940 074 042 757 790 126 116 796 333 140 799 712 927 627 993 594 825 595 771 856 348 682 278 4;
  • 8) 0.940 074 042 757 790 126 116 796 333 140 799 712 927 627 993 594 825 595 771 856 348 682 278 4 × 2 = 1 + 0.880 148 085 515 580 252 233 592 666 281 599 425 855 255 987 189 651 191 543 712 697 364 556 8;
  • 9) 0.880 148 085 515 580 252 233 592 666 281 599 425 855 255 987 189 651 191 543 712 697 364 556 8 × 2 = 1 + 0.760 296 171 031 160 504 467 185 332 563 198 851 710 511 974 379 302 383 087 425 394 729 113 6;
  • 10) 0.760 296 171 031 160 504 467 185 332 563 198 851 710 511 974 379 302 383 087 425 394 729 113 6 × 2 = 1 + 0.520 592 342 062 321 008 934 370 665 126 397 703 421 023 948 758 604 766 174 850 789 458 227 2;
  • 11) 0.520 592 342 062 321 008 934 370 665 126 397 703 421 023 948 758 604 766 174 850 789 458 227 2 × 2 = 1 + 0.041 184 684 124 642 017 868 741 330 252 795 406 842 047 897 517 209 532 349 701 578 916 454 4;
  • 12) 0.041 184 684 124 642 017 868 741 330 252 795 406 842 047 897 517 209 532 349 701 578 916 454 4 × 2 = 0 + 0.082 369 368 249 284 035 737 482 660 505 590 813 684 095 795 034 419 064 699 403 157 832 908 8;
  • 13) 0.082 369 368 249 284 035 737 482 660 505 590 813 684 095 795 034 419 064 699 403 157 832 908 8 × 2 = 0 + 0.164 738 736 498 568 071 474 965 321 011 181 627 368 191 590 068 838 129 398 806 315 665 817 6;
  • 14) 0.164 738 736 498 568 071 474 965 321 011 181 627 368 191 590 068 838 129 398 806 315 665 817 6 × 2 = 0 + 0.329 477 472 997 136 142 949 930 642 022 363 254 736 383 180 137 676 258 797 612 631 331 635 2;
  • 15) 0.329 477 472 997 136 142 949 930 642 022 363 254 736 383 180 137 676 258 797 612 631 331 635 2 × 2 = 0 + 0.658 954 945 994 272 285 899 861 284 044 726 509 472 766 360 275 352 517 595 225 262 663 270 4;
  • 16) 0.658 954 945 994 272 285 899 861 284 044 726 509 472 766 360 275 352 517 595 225 262 663 270 4 × 2 = 1 + 0.317 909 891 988 544 571 799 722 568 089 453 018 945 532 720 550 705 035 190 450 525 326 540 8;
  • 17) 0.317 909 891 988 544 571 799 722 568 089 453 018 945 532 720 550 705 035 190 450 525 326 540 8 × 2 = 0 + 0.635 819 783 977 089 143 599 445 136 178 906 037 891 065 441 101 410 070 380 901 050 653 081 6;
  • 18) 0.635 819 783 977 089 143 599 445 136 178 906 037 891 065 441 101 410 070 380 901 050 653 081 6 × 2 = 1 + 0.271 639 567 954 178 287 198 890 272 357 812 075 782 130 882 202 820 140 761 802 101 306 163 2;
  • 19) 0.271 639 567 954 178 287 198 890 272 357 812 075 782 130 882 202 820 140 761 802 101 306 163 2 × 2 = 0 + 0.543 279 135 908 356 574 397 780 544 715 624 151 564 261 764 405 640 281 523 604 202 612 326 4;
  • 20) 0.543 279 135 908 356 574 397 780 544 715 624 151 564 261 764 405 640 281 523 604 202 612 326 4 × 2 = 1 + 0.086 558 271 816 713 148 795 561 089 431 248 303 128 523 528 811 280 563 047 208 405 224 652 8;
  • 21) 0.086 558 271 816 713 148 795 561 089 431 248 303 128 523 528 811 280 563 047 208 405 224 652 8 × 2 = 0 + 0.173 116 543 633 426 297 591 122 178 862 496 606 257 047 057 622 561 126 094 416 810 449 305 6;
  • 22) 0.173 116 543 633 426 297 591 122 178 862 496 606 257 047 057 622 561 126 094 416 810 449 305 6 × 2 = 0 + 0.346 233 087 266 852 595 182 244 357 724 993 212 514 094 115 245 122 252 188 833 620 898 611 2;
  • 23) 0.346 233 087 266 852 595 182 244 357 724 993 212 514 094 115 245 122 252 188 833 620 898 611 2 × 2 = 0 + 0.692 466 174 533 705 190 364 488 715 449 986 425 028 188 230 490 244 504 377 667 241 797 222 4;
  • 24) 0.692 466 174 533 705 190 364 488 715 449 986 425 028 188 230 490 244 504 377 667 241 797 222 4 × 2 = 1 + 0.384 932 349 067 410 380 728 977 430 899 972 850 056 376 460 980 489 008 755 334 483 594 444 8;
  • 25) 0.384 932 349 067 410 380 728 977 430 899 972 850 056 376 460 980 489 008 755 334 483 594 444 8 × 2 = 0 + 0.769 864 698 134 820 761 457 954 861 799 945 700 112 752 921 960 978 017 510 668 967 188 889 6;
  • 26) 0.769 864 698 134 820 761 457 954 861 799 945 700 112 752 921 960 978 017 510 668 967 188 889 6 × 2 = 1 + 0.539 729 396 269 641 522 915 909 723 599 891 400 225 505 843 921 956 035 021 337 934 377 779 2;
  • 27) 0.539 729 396 269 641 522 915 909 723 599 891 400 225 505 843 921 956 035 021 337 934 377 779 2 × 2 = 1 + 0.079 458 792 539 283 045 831 819 447 199 782 800 451 011 687 843 912 070 042 675 868 755 558 4;
  • 28) 0.079 458 792 539 283 045 831 819 447 199 782 800 451 011 687 843 912 070 042 675 868 755 558 4 × 2 = 0 + 0.158 917 585 078 566 091 663 638 894 399 565 600 902 023 375 687 824 140 085 351 737 511 116 8;
  • 29) 0.158 917 585 078 566 091 663 638 894 399 565 600 902 023 375 687 824 140 085 351 737 511 116 8 × 2 = 0 + 0.317 835 170 157 132 183 327 277 788 799 131 201 804 046 751 375 648 280 170 703 475 022 233 6;
  • 30) 0.317 835 170 157 132 183 327 277 788 799 131 201 804 046 751 375 648 280 170 703 475 022 233 6 × 2 = 0 + 0.635 670 340 314 264 366 654 555 577 598 262 403 608 093 502 751 296 560 341 406 950 044 467 2;
  • 31) 0.635 670 340 314 264 366 654 555 577 598 262 403 608 093 502 751 296 560 341 406 950 044 467 2 × 2 = 1 + 0.271 340 680 628 528 733 309 111 155 196 524 807 216 187 005 502 593 120 682 813 900 088 934 4;
  • 32) 0.271 340 680 628 528 733 309 111 155 196 524 807 216 187 005 502 593 120 682 813 900 088 934 4 × 2 = 0 + 0.542 681 361 257 057 466 618 222 310 393 049 614 432 374 011 005 186 241 365 627 800 177 868 8;
  • 33) 0.542 681 361 257 057 466 618 222 310 393 049 614 432 374 011 005 186 241 365 627 800 177 868 8 × 2 = 1 + 0.085 362 722 514 114 933 236 444 620 786 099 228 864 748 022 010 372 482 731 255 600 355 737 6;
  • 34) 0.085 362 722 514 114 933 236 444 620 786 099 228 864 748 022 010 372 482 731 255 600 355 737 6 × 2 = 0 + 0.170 725 445 028 229 866 472 889 241 572 198 457 729 496 044 020 744 965 462 511 200 711 475 2;
  • 35) 0.170 725 445 028 229 866 472 889 241 572 198 457 729 496 044 020 744 965 462 511 200 711 475 2 × 2 = 0 + 0.341 450 890 056 459 732 945 778 483 144 396 915 458 992 088 041 489 930 925 022 401 422 950 4;
  • 36) 0.341 450 890 056 459 732 945 778 483 144 396 915 458 992 088 041 489 930 925 022 401 422 950 4 × 2 = 0 + 0.682 901 780 112 919 465 891 556 966 288 793 830 917 984 176 082 979 861 850 044 802 845 900 8;
  • 37) 0.682 901 780 112 919 465 891 556 966 288 793 830 917 984 176 082 979 861 850 044 802 845 900 8 × 2 = 1 + 0.365 803 560 225 838 931 783 113 932 577 587 661 835 968 352 165 959 723 700 089 605 691 801 6;
  • 38) 0.365 803 560 225 838 931 783 113 932 577 587 661 835 968 352 165 959 723 700 089 605 691 801 6 × 2 = 0 + 0.731 607 120 451 677 863 566 227 865 155 175 323 671 936 704 331 919 447 400 179 211 383 603 2;
  • 39) 0.731 607 120 451 677 863 566 227 865 155 175 323 671 936 704 331 919 447 400 179 211 383 603 2 × 2 = 1 + 0.463 214 240 903 355 727 132 455 730 310 350 647 343 873 408 663 838 894 800 358 422 767 206 4;
  • 40) 0.463 214 240 903 355 727 132 455 730 310 350 647 343 873 408 663 838 894 800 358 422 767 206 4 × 2 = 0 + 0.926 428 481 806 711 454 264 911 460 620 701 294 687 746 817 327 677 789 600 716 845 534 412 8;
  • 41) 0.926 428 481 806 711 454 264 911 460 620 701 294 687 746 817 327 677 789 600 716 845 534 412 8 × 2 = 1 + 0.852 856 963 613 422 908 529 822 921 241 402 589 375 493 634 655 355 579 201 433 691 068 825 6;
  • 42) 0.852 856 963 613 422 908 529 822 921 241 402 589 375 493 634 655 355 579 201 433 691 068 825 6 × 2 = 1 + 0.705 713 927 226 845 817 059 645 842 482 805 178 750 987 269 310 711 158 402 867 382 137 651 2;
  • 43) 0.705 713 927 226 845 817 059 645 842 482 805 178 750 987 269 310 711 158 402 867 382 137 651 2 × 2 = 1 + 0.411 427 854 453 691 634 119 291 684 965 610 357 501 974 538 621 422 316 805 734 764 275 302 4;
  • 44) 0.411 427 854 453 691 634 119 291 684 965 610 357 501 974 538 621 422 316 805 734 764 275 302 4 × 2 = 0 + 0.822 855 708 907 383 268 238 583 369 931 220 715 003 949 077 242 844 633 611 469 528 550 604 8;
  • 45) 0.822 855 708 907 383 268 238 583 369 931 220 715 003 949 077 242 844 633 611 469 528 550 604 8 × 2 = 1 + 0.645 711 417 814 766 536 477 166 739 862 441 430 007 898 154 485 689 267 222 939 057 101 209 6;
  • 46) 0.645 711 417 814 766 536 477 166 739 862 441 430 007 898 154 485 689 267 222 939 057 101 209 6 × 2 = 1 + 0.291 422 835 629 533 072 954 333 479 724 882 860 015 796 308 971 378 534 445 878 114 202 419 2;
  • 47) 0.291 422 835 629 533 072 954 333 479 724 882 860 015 796 308 971 378 534 445 878 114 202 419 2 × 2 = 0 + 0.582 845 671 259 066 145 908 666 959 449 765 720 031 592 617 942 757 068 891 756 228 404 838 4;
  • 48) 0.582 845 671 259 066 145 908 666 959 449 765 720 031 592 617 942 757 068 891 756 228 404 838 4 × 2 = 1 + 0.165 691 342 518 132 291 817 333 918 899 531 440 063 185 235 885 514 137 783 512 456 809 676 8;
  • 49) 0.165 691 342 518 132 291 817 333 918 899 531 440 063 185 235 885 514 137 783 512 456 809 676 8 × 2 = 0 + 0.331 382 685 036 264 583 634 667 837 799 062 880 126 370 471 771 028 275 567 024 913 619 353 6;
  • 50) 0.331 382 685 036 264 583 634 667 837 799 062 880 126 370 471 771 028 275 567 024 913 619 353 6 × 2 = 0 + 0.662 765 370 072 529 167 269 335 675 598 125 760 252 740 943 542 056 551 134 049 827 238 707 2;
  • 51) 0.662 765 370 072 529 167 269 335 675 598 125 760 252 740 943 542 056 551 134 049 827 238 707 2 × 2 = 1 + 0.325 530 740 145 058 334 538 671 351 196 251 520 505 481 887 084 113 102 268 099 654 477 414 4;
  • 52) 0.325 530 740 145 058 334 538 671 351 196 251 520 505 481 887 084 113 102 268 099 654 477 414 4 × 2 = 0 + 0.651 061 480 290 116 669 077 342 702 392 503 041 010 963 774 168 226 204 536 199 308 954 828 8;
  • 53) 0.651 061 480 290 116 669 077 342 702 392 503 041 010 963 774 168 226 204 536 199 308 954 828 8 × 2 = 1 + 0.302 122 960 580 233 338 154 685 404 785 006 082 021 927 548 336 452 409 072 398 617 909 657 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:


0.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 080 3(10) =

0.1011 0111 1110 0001 0101 0001 0110 0010 1000 1010 1110 1101 0010 1(2)

5. Positive number before normalization:

2.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 080 3(10) =

10.1011 0111 1110 0001 0101 0001 0110 0010 1000 1010 1110 1101 0010 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:


2.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 080 3(10) =

10.1011 0111 1110 0001 0101 0001 0110 0010 1000 1010 1110 1101 0010 1(2) =

10.1011 0111 1110 0001 0101 0001 0110 0010 1000 1010 1110 1101 0010 1(2) × 20 =

1.0101 1011 1111 0000 1010 1000 1011 0001 0100 0101 0111 0110 1001 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):
1.0101 1011 1111 0000 1010 1000 1011 0001 0100 0101 0111 0110 1001 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. 0101 1011 1111 0000 1010 1000 1011 0001 0100 0101 0111 0110 1001 01 =

0101 1011 1111 0000 1010 1000 1011 0001 0100 0101 0111 0110 1001


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) =
0101 1011 1111 0000 1010 1000 1011 0001 0100 0101 0111 0110 1001


Decimal number 2.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 080 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 0101 1011 1111 0000 1010 1000 1011 0001 0100 0101 0111 0110 1001

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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