0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 313 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 313 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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 313 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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 313 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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 313 3 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 234 618 868 626 6;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 234 618 868 626 6 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 469 237 737 253 2;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 469 237 737 253 2 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 938 475 474 506 4;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 938 475 474 506 4 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 876 950 949 012 8;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 876 950 949 012 8 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 753 901 898 025 6;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 753 901 898 025 6 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 507 803 796 051 2;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 507 803 796 051 2 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 015 607 592 102 4;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 015 607 592 102 4 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 031 215 184 204 8;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 031 215 184 204 8 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 062 430 368 409 6;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 062 430 368 409 6 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 124 860 736 819 2;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 124 860 736 819 2 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 944 249 721 473 638 4;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 944 249 721 473 638 4 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 888 499 442 947 276 8;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 888 499 442 947 276 8 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 776 998 885 894 553 6;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 776 998 885 894 553 6 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 553 997 771 789 107 2;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 553 997 771 789 107 2 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 107 995 543 578 214 4;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 107 995 543 578 214 4 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 215 991 087 156 428 8;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 215 991 087 156 428 8 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 431 982 174 312 857 6;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 431 982 174 312 857 6 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 863 964 348 625 715 2;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 863 964 348 625 715 2 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 727 928 697 251 430 4;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 727 928 697 251 430 4 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 891 455 857 394 502 860 8;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 891 455 857 394 502 860 8 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 782 911 714 789 005 721 6;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 782 911 714 789 005 721 6 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 565 823 429 578 011 443 2;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 565 823 429 578 011 443 2 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 131 646 859 156 022 886 4;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 131 646 859 156 022 886 4 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 263 293 718 312 045 772 8;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 263 293 718 312 045 772 8 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 526 587 436 624 091 545 6;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 526 587 436 624 091 545 6 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 053 174 873 248 183 091 2;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 053 174 873 248 183 091 2 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 106 349 746 496 366 182 4;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 106 349 746 496 366 182 4 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 212 699 492 992 732 364 8;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 212 699 492 992 732 364 8 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 425 398 985 985 464 729 6;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 425 398 985 985 464 729 6 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 816 850 797 971 970 929 459 2;
  • 31) 0.825 297 101 522 529 932 158 401 660 816 850 797 971 970 929 459 2 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 633 701 595 943 941 858 918 4;
  • 32) 0.650 594 203 045 059 864 316 803 321 633 701 595 943 941 858 918 4 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 267 403 191 887 883 717 836 8;
  • 33) 0.301 188 406 090 119 728 633 606 643 267 403 191 887 883 717 836 8 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 534 806 383 775 767 435 673 6;
  • 34) 0.602 376 812 180 239 457 267 213 286 534 806 383 775 767 435 673 6 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 069 612 767 551 534 871 347 2;
  • 35) 0.204 753 624 360 478 914 534 426 573 069 612 767 551 534 871 347 2 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 139 225 535 103 069 742 694 4;
  • 36) 0.409 507 248 720 957 829 068 853 146 139 225 535 103 069 742 694 4 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 278 451 070 206 139 485 388 8;
  • 37) 0.819 014 497 441 915 658 137 706 292 278 451 070 206 139 485 388 8 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 556 902 140 412 278 970 777 6;
  • 38) 0.638 028 994 883 831 316 275 412 584 556 902 140 412 278 970 777 6 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 113 804 280 824 557 941 555 2;
  • 39) 0.276 057 989 767 662 632 550 825 169 113 804 280 824 557 941 555 2 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 227 608 561 649 115 883 110 4;
  • 40) 0.552 115 979 535 325 265 101 650 338 227 608 561 649 115 883 110 4 × 2 = 1 + 0.104 231 959 070 650 530 203 300 676 455 217 123 298 231 766 220 8;
  • 41) 0.104 231 959 070 650 530 203 300 676 455 217 123 298 231 766 220 8 × 2 = 0 + 0.208 463 918 141 301 060 406 601 352 910 434 246 596 463 532 441 6;
  • 42) 0.208 463 918 141 301 060 406 601 352 910 434 246 596 463 532 441 6 × 2 = 0 + 0.416 927 836 282 602 120 813 202 705 820 868 493 192 927 064 883 2;
  • 43) 0.416 927 836 282 602 120 813 202 705 820 868 493 192 927 064 883 2 × 2 = 0 + 0.833 855 672 565 204 241 626 405 411 641 736 986 385 854 129 766 4;
  • 44) 0.833 855 672 565 204 241 626 405 411 641 736 986 385 854 129 766 4 × 2 = 1 + 0.667 711 345 130 408 483 252 810 823 283 473 972 771 708 259 532 8;
  • 45) 0.667 711 345 130 408 483 252 810 823 283 473 972 771 708 259 532 8 × 2 = 1 + 0.335 422 690 260 816 966 505 621 646 566 947 945 543 416 519 065 6;
  • 46) 0.335 422 690 260 816 966 505 621 646 566 947 945 543 416 519 065 6 × 2 = 0 + 0.670 845 380 521 633 933 011 243 293 133 895 891 086 833 038 131 2;
  • 47) 0.670 845 380 521 633 933 011 243 293 133 895 891 086 833 038 131 2 × 2 = 1 + 0.341 690 761 043 267 866 022 486 586 267 791 782 173 666 076 262 4;
  • 48) 0.341 690 761 043 267 866 022 486 586 267 791 782 173 666 076 262 4 × 2 = 0 + 0.683 381 522 086 535 732 044 973 172 535 583 564 347 332 152 524 8;
  • 49) 0.683 381 522 086 535 732 044 973 172 535 583 564 347 332 152 524 8 × 2 = 1 + 0.366 763 044 173 071 464 089 946 345 071 167 128 694 664 305 049 6;
  • 50) 0.366 763 044 173 071 464 089 946 345 071 167 128 694 664 305 049 6 × 2 = 0 + 0.733 526 088 346 142 928 179 892 690 142 334 257 389 328 610 099 2;
  • 51) 0.733 526 088 346 142 928 179 892 690 142 334 257 389 328 610 099 2 × 2 = 1 + 0.467 052 176 692 285 856 359 785 380 284 668 514 778 657 220 198 4;
  • 52) 0.467 052 176 692 285 856 359 785 380 284 668 514 778 657 220 198 4 × 2 = 0 + 0.934 104 353 384 571 712 719 570 760 569 337 029 557 314 440 396 8;
  • 53) 0.934 104 353 384 571 712 719 570 760 569 337 029 557 314 440 396 8 × 2 = 1 + 0.868 208 706 769 143 425 439 141 521 138 674 059 114 628 880 793 6;
  • 54) 0.868 208 706 769 143 425 439 141 521 138 674 059 114 628 880 793 6 × 2 = 1 + 0.736 417 413 538 286 850 878 283 042 277 348 118 229 257 761 587 2;
  • 55) 0.736 417 413 538 286 850 878 283 042 277 348 118 229 257 761 587 2 × 2 = 1 + 0.472 834 827 076 573 701 756 566 084 554 696 236 458 515 523 174 4;
  • 56) 0.472 834 827 076 573 701 756 566 084 554 696 236 458 515 523 174 4 × 2 = 0 + 0.945 669 654 153 147 403 513 132 169 109 392 472 917 031 046 348 8;
  • 57) 0.945 669 654 153 147 403 513 132 169 109 392 472 917 031 046 348 8 × 2 = 1 + 0.891 339 308 306 294 807 026 264 338 218 784 945 834 062 092 697 6;
  • 58) 0.891 339 308 306 294 807 026 264 338 218 784 945 834 062 092 697 6 × 2 = 1 + 0.782 678 616 612 589 614 052 528 676 437 569 891 668 124 185 395 2;
  • 59) 0.782 678 616 612 589 614 052 528 676 437 569 891 668 124 185 395 2 × 2 = 1 + 0.565 357 233 225 179 228 105 057 352 875 139 783 336 248 370 790 4;
  • 60) 0.565 357 233 225 179 228 105 057 352 875 139 783 336 248 370 790 4 × 2 = 1 + 0.130 714 466 450 358 456 210 114 705 750 279 566 672 496 741 580 8;
  • 61) 0.130 714 466 450 358 456 210 114 705 750 279 566 672 496 741 580 8 × 2 = 0 + 0.261 428 932 900 716 912 420 229 411 500 559 133 344 993 483 161 6;
  • 62) 0.261 428 932 900 716 912 420 229 411 500 559 133 344 993 483 161 6 × 2 = 0 + 0.522 857 865 801 433 824 840 458 823 001 118 266 689 986 966 323 2;
  • 63) 0.522 857 865 801 433 824 840 458 823 001 118 266 689 986 966 323 2 × 2 = 1 + 0.045 715 731 602 867 649 680 917 646 002 236 533 379 973 932 646 4;
  • 64) 0.045 715 731 602 867 649 680 917 646 002 236 533 379 973 932 646 4 × 2 = 0 + 0.091 431 463 205 735 299 361 835 292 004 473 066 759 947 865 292 8;
  • 65) 0.091 431 463 205 735 299 361 835 292 004 473 066 759 947 865 292 8 × 2 = 0 + 0.182 862 926 411 470 598 723 670 584 008 946 133 519 895 730 585 6;
  • 66) 0.182 862 926 411 470 598 723 670 584 008 946 133 519 895 730 585 6 × 2 = 0 + 0.365 725 852 822 941 197 447 341 168 017 892 267 039 791 461 171 2;
  • 67) 0.365 725 852 822 941 197 447 341 168 017 892 267 039 791 461 171 2 × 2 = 0 + 0.731 451 705 645 882 394 894 682 336 035 784 534 079 582 922 342 4;
  • 68) 0.731 451 705 645 882 394 894 682 336 035 784 534 079 582 922 342 4 × 2 = 1 + 0.462 903 411 291 764 789 789 364 672 071 569 068 159 165 844 684 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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 313 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:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 313 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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 313 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 0.000 020 830 729 321 671 205 134 999 154 509 660 617 309 434 313 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