0.000 020 830 729 321 671 205 134 999 154 509 660 617 302 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 617 302(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
0.000 020 830 729 321 671 205 134 999 154 509 660 617 302(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 617 302.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 617 302 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 234 604;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 234 604 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 469 208;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 469 208 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 938 416;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 938 416 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 876 832;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 876 832 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 753 664;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 753 664 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 507 328;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 507 328 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 014 656;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 014 656 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 029 312;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 029 312 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 058 624;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 058 624 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 117 248;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 117 248 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 944 234 496;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 944 234 496 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 888 468 992;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 888 468 992 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 776 937 984;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 776 937 984 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 553 875 968;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 553 875 968 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 107 751 936;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 107 751 936 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 215 503 872;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 215 503 872 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 431 007 744;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 431 007 744 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 862 015 488;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 862 015 488 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 724 030 976;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 724 030 976 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 891 448 061 952;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 891 448 061 952 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 782 896 123 904;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 782 896 123 904 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 565 792 247 808;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 565 792 247 808 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 131 584 495 616;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 131 584 495 616 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 263 168 991 232;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 263 168 991 232 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 526 337 982 464;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 526 337 982 464 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 052 675 964 928;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 052 675 964 928 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 105 351 929 856;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 105 351 929 856 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 210 703 859 712;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 210 703 859 712 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 421 407 719 424;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 421 407 719 424 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 816 842 815 438 848;
  • 31) 0.825 297 101 522 529 932 158 401 660 816 842 815 438 848 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 633 685 630 877 696;
  • 32) 0.650 594 203 045 059 864 316 803 321 633 685 630 877 696 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 267 371 261 755 392;
  • 33) 0.301 188 406 090 119 728 633 606 643 267 371 261 755 392 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 534 742 523 510 784;
  • 34) 0.602 376 812 180 239 457 267 213 286 534 742 523 510 784 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 069 485 047 021 568;
  • 35) 0.204 753 624 360 478 914 534 426 573 069 485 047 021 568 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 138 970 094 043 136;
  • 36) 0.409 507 248 720 957 829 068 853 146 138 970 094 043 136 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 277 940 188 086 272;
  • 37) 0.819 014 497 441 915 658 137 706 292 277 940 188 086 272 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 555 880 376 172 544;
  • 38) 0.638 028 994 883 831 316 275 412 584 555 880 376 172 544 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 111 760 752 345 088;
  • 39) 0.276 057 989 767 662 632 550 825 169 111 760 752 345 088 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 223 521 504 690 176;
  • 40) 0.552 115 979 535 325 265 101 650 338 223 521 504 690 176 × 2 = 1 + 0.104 231 959 070 650 530 203 300 676 447 043 009 380 352;
  • 41) 0.104 231 959 070 650 530 203 300 676 447 043 009 380 352 × 2 = 0 + 0.208 463 918 141 301 060 406 601 352 894 086 018 760 704;
  • 42) 0.208 463 918 141 301 060 406 601 352 894 086 018 760 704 × 2 = 0 + 0.416 927 836 282 602 120 813 202 705 788 172 037 521 408;
  • 43) 0.416 927 836 282 602 120 813 202 705 788 172 037 521 408 × 2 = 0 + 0.833 855 672 565 204 241 626 405 411 576 344 075 042 816;
  • 44) 0.833 855 672 565 204 241 626 405 411 576 344 075 042 816 × 2 = 1 + 0.667 711 345 130 408 483 252 810 823 152 688 150 085 632;
  • 45) 0.667 711 345 130 408 483 252 810 823 152 688 150 085 632 × 2 = 1 + 0.335 422 690 260 816 966 505 621 646 305 376 300 171 264;
  • 46) 0.335 422 690 260 816 966 505 621 646 305 376 300 171 264 × 2 = 0 + 0.670 845 380 521 633 933 011 243 292 610 752 600 342 528;
  • 47) 0.670 845 380 521 633 933 011 243 292 610 752 600 342 528 × 2 = 1 + 0.341 690 761 043 267 866 022 486 585 221 505 200 685 056;
  • 48) 0.341 690 761 043 267 866 022 486 585 221 505 200 685 056 × 2 = 0 + 0.683 381 522 086 535 732 044 973 170 443 010 401 370 112;
  • 49) 0.683 381 522 086 535 732 044 973 170 443 010 401 370 112 × 2 = 1 + 0.366 763 044 173 071 464 089 946 340 886 020 802 740 224;
  • 50) 0.366 763 044 173 071 464 089 946 340 886 020 802 740 224 × 2 = 0 + 0.733 526 088 346 142 928 179 892 681 772 041 605 480 448;
  • 51) 0.733 526 088 346 142 928 179 892 681 772 041 605 480 448 × 2 = 1 + 0.467 052 176 692 285 856 359 785 363 544 083 210 960 896;
  • 52) 0.467 052 176 692 285 856 359 785 363 544 083 210 960 896 × 2 = 0 + 0.934 104 353 384 571 712 719 570 727 088 166 421 921 792;
  • 53) 0.934 104 353 384 571 712 719 570 727 088 166 421 921 792 × 2 = 1 + 0.868 208 706 769 143 425 439 141 454 176 332 843 843 584;
  • 54) 0.868 208 706 769 143 425 439 141 454 176 332 843 843 584 × 2 = 1 + 0.736 417 413 538 286 850 878 282 908 352 665 687 687 168;
  • 55) 0.736 417 413 538 286 850 878 282 908 352 665 687 687 168 × 2 = 1 + 0.472 834 827 076 573 701 756 565 816 705 331 375 374 336;
  • 56) 0.472 834 827 076 573 701 756 565 816 705 331 375 374 336 × 2 = 0 + 0.945 669 654 153 147 403 513 131 633 410 662 750 748 672;
  • 57) 0.945 669 654 153 147 403 513 131 633 410 662 750 748 672 × 2 = 1 + 0.891 339 308 306 294 807 026 263 266 821 325 501 497 344;
  • 58) 0.891 339 308 306 294 807 026 263 266 821 325 501 497 344 × 2 = 1 + 0.782 678 616 612 589 614 052 526 533 642 651 002 994 688;
  • 59) 0.782 678 616 612 589 614 052 526 533 642 651 002 994 688 × 2 = 1 + 0.565 357 233 225 179 228 105 053 067 285 302 005 989 376;
  • 60) 0.565 357 233 225 179 228 105 053 067 285 302 005 989 376 × 2 = 1 + 0.130 714 466 450 358 456 210 106 134 570 604 011 978 752;
  • 61) 0.130 714 466 450 358 456 210 106 134 570 604 011 978 752 × 2 = 0 + 0.261 428 932 900 716 912 420 212 269 141 208 023 957 504;
  • 62) 0.261 428 932 900 716 912 420 212 269 141 208 023 957 504 × 2 = 0 + 0.522 857 865 801 433 824 840 424 538 282 416 047 915 008;
  • 63) 0.522 857 865 801 433 824 840 424 538 282 416 047 915 008 × 2 = 1 + 0.045 715 731 602 867 649 680 849 076 564 832 095 830 016;
  • 64) 0.045 715 731 602 867 649 680 849 076 564 832 095 830 016 × 2 = 0 + 0.091 431 463 205 735 299 361 698 153 129 664 191 660 032;
  • 65) 0.091 431 463 205 735 299 361 698 153 129 664 191 660 032 × 2 = 0 + 0.182 862 926 411 470 598 723 396 306 259 328 383 320 064;
  • 66) 0.182 862 926 411 470 598 723 396 306 259 328 383 320 064 × 2 = 0 + 0.365 725 852 822 941 197 446 792 612 518 656 766 640 128;
  • 67) 0.365 725 852 822 941 197 446 792 612 518 656 766 640 128 × 2 = 0 + 0.731 451 705 645 882 394 893 585 225 037 313 533 280 256;
  • 68) 0.731 451 705 645 882 394 893 585 225 037 313 533 280 256 × 2 = 1 + 0.462 903 411 291 764 789 787 170 450 074 627 066 560 512;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 302(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 617 302(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 617 302(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 0.000 020 830 729 321 671 205 134 999 154 509 660 617 302 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