0.000 020 830 729 321 671 205 134 999 154 509 660 617 429 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 429(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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 429(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 429.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 429 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 234 858;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 234 858 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 469 716;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 469 716 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 939 432;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 939 432 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 878 864;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 878 864 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 757 728;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 757 728 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 515 456;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 515 456 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 030 912;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 030 912 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 061 824;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 061 824 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 123 648;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 123 648 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 247 296;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 247 296 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 944 494 592;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 944 494 592 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 888 989 184;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 888 989 184 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 777 978 368;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 777 978 368 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 555 956 736;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 555 956 736 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 111 913 472;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 111 913 472 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 223 826 944;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 223 826 944 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 447 653 888;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 447 653 888 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 895 307 776;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 895 307 776 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 790 615 552;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 790 615 552 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 891 581 231 104;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 891 581 231 104 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 783 162 462 208;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 783 162 462 208 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 566 324 924 416;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 566 324 924 416 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 132 649 848 832;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 132 649 848 832 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 265 299 697 664;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 265 299 697 664 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 530 599 395 328;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 530 599 395 328 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 061 198 790 656;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 061 198 790 656 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 122 397 581 312;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 122 397 581 312 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 244 795 162 624;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 244 795 162 624 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 489 590 325 248;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 489 590 325 248 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 816 979 180 650 496;
  • 31) 0.825 297 101 522 529 932 158 401 660 816 979 180 650 496 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 633 958 361 300 992;
  • 32) 0.650 594 203 045 059 864 316 803 321 633 958 361 300 992 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 267 916 722 601 984;
  • 33) 0.301 188 406 090 119 728 633 606 643 267 916 722 601 984 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 535 833 445 203 968;
  • 34) 0.602 376 812 180 239 457 267 213 286 535 833 445 203 968 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 071 666 890 407 936;
  • 35) 0.204 753 624 360 478 914 534 426 573 071 666 890 407 936 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 143 333 780 815 872;
  • 36) 0.409 507 248 720 957 829 068 853 146 143 333 780 815 872 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 286 667 561 631 744;
  • 37) 0.819 014 497 441 915 658 137 706 292 286 667 561 631 744 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 573 335 123 263 488;
  • 38) 0.638 028 994 883 831 316 275 412 584 573 335 123 263 488 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 146 670 246 526 976;
  • 39) 0.276 057 989 767 662 632 550 825 169 146 670 246 526 976 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 293 340 493 053 952;
  • 40) 0.552 115 979 535 325 265 101 650 338 293 340 493 053 952 × 2 = 1 + 0.104 231 959 070 650 530 203 300 676 586 680 986 107 904;
  • 41) 0.104 231 959 070 650 530 203 300 676 586 680 986 107 904 × 2 = 0 + 0.208 463 918 141 301 060 406 601 353 173 361 972 215 808;
  • 42) 0.208 463 918 141 301 060 406 601 353 173 361 972 215 808 × 2 = 0 + 0.416 927 836 282 602 120 813 202 706 346 723 944 431 616;
  • 43) 0.416 927 836 282 602 120 813 202 706 346 723 944 431 616 × 2 = 0 + 0.833 855 672 565 204 241 626 405 412 693 447 888 863 232;
  • 44) 0.833 855 672 565 204 241 626 405 412 693 447 888 863 232 × 2 = 1 + 0.667 711 345 130 408 483 252 810 825 386 895 777 726 464;
  • 45) 0.667 711 345 130 408 483 252 810 825 386 895 777 726 464 × 2 = 1 + 0.335 422 690 260 816 966 505 621 650 773 791 555 452 928;
  • 46) 0.335 422 690 260 816 966 505 621 650 773 791 555 452 928 × 2 = 0 + 0.670 845 380 521 633 933 011 243 301 547 583 110 905 856;
  • 47) 0.670 845 380 521 633 933 011 243 301 547 583 110 905 856 × 2 = 1 + 0.341 690 761 043 267 866 022 486 603 095 166 221 811 712;
  • 48) 0.341 690 761 043 267 866 022 486 603 095 166 221 811 712 × 2 = 0 + 0.683 381 522 086 535 732 044 973 206 190 332 443 623 424;
  • 49) 0.683 381 522 086 535 732 044 973 206 190 332 443 623 424 × 2 = 1 + 0.366 763 044 173 071 464 089 946 412 380 664 887 246 848;
  • 50) 0.366 763 044 173 071 464 089 946 412 380 664 887 246 848 × 2 = 0 + 0.733 526 088 346 142 928 179 892 824 761 329 774 493 696;
  • 51) 0.733 526 088 346 142 928 179 892 824 761 329 774 493 696 × 2 = 1 + 0.467 052 176 692 285 856 359 785 649 522 659 548 987 392;
  • 52) 0.467 052 176 692 285 856 359 785 649 522 659 548 987 392 × 2 = 0 + 0.934 104 353 384 571 712 719 571 299 045 319 097 974 784;
  • 53) 0.934 104 353 384 571 712 719 571 299 045 319 097 974 784 × 2 = 1 + 0.868 208 706 769 143 425 439 142 598 090 638 195 949 568;
  • 54) 0.868 208 706 769 143 425 439 142 598 090 638 195 949 568 × 2 = 1 + 0.736 417 413 538 286 850 878 285 196 181 276 391 899 136;
  • 55) 0.736 417 413 538 286 850 878 285 196 181 276 391 899 136 × 2 = 1 + 0.472 834 827 076 573 701 756 570 392 362 552 783 798 272;
  • 56) 0.472 834 827 076 573 701 756 570 392 362 552 783 798 272 × 2 = 0 + 0.945 669 654 153 147 403 513 140 784 725 105 567 596 544;
  • 57) 0.945 669 654 153 147 403 513 140 784 725 105 567 596 544 × 2 = 1 + 0.891 339 308 306 294 807 026 281 569 450 211 135 193 088;
  • 58) 0.891 339 308 306 294 807 026 281 569 450 211 135 193 088 × 2 = 1 + 0.782 678 616 612 589 614 052 563 138 900 422 270 386 176;
  • 59) 0.782 678 616 612 589 614 052 563 138 900 422 270 386 176 × 2 = 1 + 0.565 357 233 225 179 228 105 126 277 800 844 540 772 352;
  • 60) 0.565 357 233 225 179 228 105 126 277 800 844 540 772 352 × 2 = 1 + 0.130 714 466 450 358 456 210 252 555 601 689 081 544 704;
  • 61) 0.130 714 466 450 358 456 210 252 555 601 689 081 544 704 × 2 = 0 + 0.261 428 932 900 716 912 420 505 111 203 378 163 089 408;
  • 62) 0.261 428 932 900 716 912 420 505 111 203 378 163 089 408 × 2 = 0 + 0.522 857 865 801 433 824 841 010 222 406 756 326 178 816;
  • 63) 0.522 857 865 801 433 824 841 010 222 406 756 326 178 816 × 2 = 1 + 0.045 715 731 602 867 649 682 020 444 813 512 652 357 632;
  • 64) 0.045 715 731 602 867 649 682 020 444 813 512 652 357 632 × 2 = 0 + 0.091 431 463 205 735 299 364 040 889 627 025 304 715 264;
  • 65) 0.091 431 463 205 735 299 364 040 889 627 025 304 715 264 × 2 = 0 + 0.182 862 926 411 470 598 728 081 779 254 050 609 430 528;
  • 66) 0.182 862 926 411 470 598 728 081 779 254 050 609 430 528 × 2 = 0 + 0.365 725 852 822 941 197 456 163 558 508 101 218 861 056;
  • 67) 0.365 725 852 822 941 197 456 163 558 508 101 218 861 056 × 2 = 0 + 0.731 451 705 645 882 394 912 327 117 016 202 437 722 112;
  • 68) 0.731 451 705 645 882 394 912 327 117 016 202 437 722 112 × 2 = 1 + 0.462 903 411 291 764 789 824 654 234 032 404 875 444 224;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 429(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 429(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 429(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 0.000 020 830 729 321 671 205 134 999 154 509 660 617 429 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