0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 319 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 319 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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 319 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: 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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 319 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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 319 3 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 234 618 868 638 6;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 234 618 868 638 6 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 469 237 737 277 2;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 469 237 737 277 2 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 938 475 474 554 4;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 938 475 474 554 4 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 876 950 949 108 8;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 876 950 949 108 8 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 753 901 898 217 6;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 753 901 898 217 6 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 507 803 796 435 2;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 507 803 796 435 2 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 015 607 592 870 4;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 015 607 592 870 4 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 031 215 185 740 8;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 031 215 185 740 8 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 062 430 371 481 6;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 062 430 371 481 6 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 124 860 742 963 2;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 124 860 742 963 2 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 944 249 721 485 926 4;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 944 249 721 485 926 4 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 888 499 442 971 852 8;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 888 499 442 971 852 8 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 776 998 885 943 705 6;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 776 998 885 943 705 6 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 553 997 771 887 411 2;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 553 997 771 887 411 2 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 107 995 543 774 822 4;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 107 995 543 774 822 4 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 215 991 087 549 644 8;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 215 991 087 549 644 8 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 431 982 175 099 289 6;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 431 982 175 099 289 6 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 863 964 350 198 579 2;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 863 964 350 198 579 2 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 727 928 700 397 158 4;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 727 928 700 397 158 4 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 891 455 857 400 794 316 8;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 891 455 857 400 794 316 8 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 782 911 714 801 588 633 6;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 782 911 714 801 588 633 6 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 565 823 429 603 177 267 2;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 565 823 429 603 177 267 2 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 131 646 859 206 354 534 4;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 131 646 859 206 354 534 4 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 263 293 718 412 709 068 8;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 263 293 718 412 709 068 8 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 526 587 436 825 418 137 6;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 526 587 436 825 418 137 6 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 053 174 873 650 836 275 2;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 053 174 873 650 836 275 2 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 106 349 747 301 672 550 4;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 106 349 747 301 672 550 4 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 212 699 494 603 345 100 8;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 212 699 494 603 345 100 8 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 425 398 989 206 690 201 6;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 425 398 989 206 690 201 6 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 816 850 797 978 413 380 403 2;
  • 31) 0.825 297 101 522 529 932 158 401 660 816 850 797 978 413 380 403 2 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 633 701 595 956 826 760 806 4;
  • 32) 0.650 594 203 045 059 864 316 803 321 633 701 595 956 826 760 806 4 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 267 403 191 913 653 521 612 8;
  • 33) 0.301 188 406 090 119 728 633 606 643 267 403 191 913 653 521 612 8 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 534 806 383 827 307 043 225 6;
  • 34) 0.602 376 812 180 239 457 267 213 286 534 806 383 827 307 043 225 6 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 069 612 767 654 614 086 451 2;
  • 35) 0.204 753 624 360 478 914 534 426 573 069 612 767 654 614 086 451 2 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 139 225 535 309 228 172 902 4;
  • 36) 0.409 507 248 720 957 829 068 853 146 139 225 535 309 228 172 902 4 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 278 451 070 618 456 345 804 8;
  • 37) 0.819 014 497 441 915 658 137 706 292 278 451 070 618 456 345 804 8 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 556 902 141 236 912 691 609 6;
  • 38) 0.638 028 994 883 831 316 275 412 584 556 902 141 236 912 691 609 6 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 113 804 282 473 825 383 219 2;
  • 39) 0.276 057 989 767 662 632 550 825 169 113 804 282 473 825 383 219 2 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 227 608 564 947 650 766 438 4;
  • 40) 0.552 115 979 535 325 265 101 650 338 227 608 564 947 650 766 438 4 × 2 = 1 + 0.104 231 959 070 650 530 203 300 676 455 217 129 895 301 532 876 8;
  • 41) 0.104 231 959 070 650 530 203 300 676 455 217 129 895 301 532 876 8 × 2 = 0 + 0.208 463 918 141 301 060 406 601 352 910 434 259 790 603 065 753 6;
  • 42) 0.208 463 918 141 301 060 406 601 352 910 434 259 790 603 065 753 6 × 2 = 0 + 0.416 927 836 282 602 120 813 202 705 820 868 519 581 206 131 507 2;
  • 43) 0.416 927 836 282 602 120 813 202 705 820 868 519 581 206 131 507 2 × 2 = 0 + 0.833 855 672 565 204 241 626 405 411 641 737 039 162 412 263 014 4;
  • 44) 0.833 855 672 565 204 241 626 405 411 641 737 039 162 412 263 014 4 × 2 = 1 + 0.667 711 345 130 408 483 252 810 823 283 474 078 324 824 526 028 8;
  • 45) 0.667 711 345 130 408 483 252 810 823 283 474 078 324 824 526 028 8 × 2 = 1 + 0.335 422 690 260 816 966 505 621 646 566 948 156 649 649 052 057 6;
  • 46) 0.335 422 690 260 816 966 505 621 646 566 948 156 649 649 052 057 6 × 2 = 0 + 0.670 845 380 521 633 933 011 243 293 133 896 313 299 298 104 115 2;
  • 47) 0.670 845 380 521 633 933 011 243 293 133 896 313 299 298 104 115 2 × 2 = 1 + 0.341 690 761 043 267 866 022 486 586 267 792 626 598 596 208 230 4;
  • 48) 0.341 690 761 043 267 866 022 486 586 267 792 626 598 596 208 230 4 × 2 = 0 + 0.683 381 522 086 535 732 044 973 172 535 585 253 197 192 416 460 8;
  • 49) 0.683 381 522 086 535 732 044 973 172 535 585 253 197 192 416 460 8 × 2 = 1 + 0.366 763 044 173 071 464 089 946 345 071 170 506 394 384 832 921 6;
  • 50) 0.366 763 044 173 071 464 089 946 345 071 170 506 394 384 832 921 6 × 2 = 0 + 0.733 526 088 346 142 928 179 892 690 142 341 012 788 769 665 843 2;
  • 51) 0.733 526 088 346 142 928 179 892 690 142 341 012 788 769 665 843 2 × 2 = 1 + 0.467 052 176 692 285 856 359 785 380 284 682 025 577 539 331 686 4;
  • 52) 0.467 052 176 692 285 856 359 785 380 284 682 025 577 539 331 686 4 × 2 = 0 + 0.934 104 353 384 571 712 719 570 760 569 364 051 155 078 663 372 8;
  • 53) 0.934 104 353 384 571 712 719 570 760 569 364 051 155 078 663 372 8 × 2 = 1 + 0.868 208 706 769 143 425 439 141 521 138 728 102 310 157 326 745 6;
  • 54) 0.868 208 706 769 143 425 439 141 521 138 728 102 310 157 326 745 6 × 2 = 1 + 0.736 417 413 538 286 850 878 283 042 277 456 204 620 314 653 491 2;
  • 55) 0.736 417 413 538 286 850 878 283 042 277 456 204 620 314 653 491 2 × 2 = 1 + 0.472 834 827 076 573 701 756 566 084 554 912 409 240 629 306 982 4;
  • 56) 0.472 834 827 076 573 701 756 566 084 554 912 409 240 629 306 982 4 × 2 = 0 + 0.945 669 654 153 147 403 513 132 169 109 824 818 481 258 613 964 8;
  • 57) 0.945 669 654 153 147 403 513 132 169 109 824 818 481 258 613 964 8 × 2 = 1 + 0.891 339 308 306 294 807 026 264 338 219 649 636 962 517 227 929 6;
  • 58) 0.891 339 308 306 294 807 026 264 338 219 649 636 962 517 227 929 6 × 2 = 1 + 0.782 678 616 612 589 614 052 528 676 439 299 273 925 034 455 859 2;
  • 59) 0.782 678 616 612 589 614 052 528 676 439 299 273 925 034 455 859 2 × 2 = 1 + 0.565 357 233 225 179 228 105 057 352 878 598 547 850 068 911 718 4;
  • 60) 0.565 357 233 225 179 228 105 057 352 878 598 547 850 068 911 718 4 × 2 = 1 + 0.130 714 466 450 358 456 210 114 705 757 197 095 700 137 823 436 8;
  • 61) 0.130 714 466 450 358 456 210 114 705 757 197 095 700 137 823 436 8 × 2 = 0 + 0.261 428 932 900 716 912 420 229 411 514 394 191 400 275 646 873 6;
  • 62) 0.261 428 932 900 716 912 420 229 411 514 394 191 400 275 646 873 6 × 2 = 0 + 0.522 857 865 801 433 824 840 458 823 028 788 382 800 551 293 747 2;
  • 63) 0.522 857 865 801 433 824 840 458 823 028 788 382 800 551 293 747 2 × 2 = 1 + 0.045 715 731 602 867 649 680 917 646 057 576 765 601 102 587 494 4;
  • 64) 0.045 715 731 602 867 649 680 917 646 057 576 765 601 102 587 494 4 × 2 = 0 + 0.091 431 463 205 735 299 361 835 292 115 153 531 202 205 174 988 8;
  • 65) 0.091 431 463 205 735 299 361 835 292 115 153 531 202 205 174 988 8 × 2 = 0 + 0.182 862 926 411 470 598 723 670 584 230 307 062 404 410 349 977 6;
  • 66) 0.182 862 926 411 470 598 723 670 584 230 307 062 404 410 349 977 6 × 2 = 0 + 0.365 725 852 822 941 197 447 341 168 460 614 124 808 820 699 955 2;
  • 67) 0.365 725 852 822 941 197 447 341 168 460 614 124 808 820 699 955 2 × 2 = 0 + 0.731 451 705 645 882 394 894 682 336 921 228 249 617 641 399 910 4;
  • 68) 0.731 451 705 645 882 394 894 682 336 921 228 249 617 641 399 910 4 × 2 = 1 + 0.462 903 411 291 764 789 789 364 673 842 456 499 235 282 799 820 8;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 319 3(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

5. Positive number before normalization:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 319 3(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 16 positions to the right, so that only one non zero digit remains to the left of it:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 319 3(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 20 =

1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 2-16


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): -16

Mantissa (not normalized):
1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-16 + 2(11-1) - 1 =

(-16 + 1 023)(10) =

1 007(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;
  • 1 007 ÷ 2 = 503 + 1;
  • 503 ÷ 2 = 251 + 1;
  • 251 ÷ 2 = 125 + 1;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 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) =

1007(10) =

011 1110 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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001 =

0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


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 1110 1111

Mantissa (52 bits) =
0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


Decimal number 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 319 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1110 1111 - 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001

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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