0.000 020 830 729 321 671 205 134 999 154 509 777 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 777(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
0.000 020 830 729 321 671 205 134 999 154 509 777(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: 0.000 020 830 729 321 671 205 134 999 154 509 777.

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) 0.000 020 830 729 321 671 205 134 999 154 509 777 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 554;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 554 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 039 108;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 039 108 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 078 216;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 078 216 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 156 432;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 156 432 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 312 864;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 312 864 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 625 728;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 625 728 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 251 456;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 251 456 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 502 912;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 502 912 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 109 005 824;
  • 10) 0.010 665 333 412 695 657 029 119 567 109 005 824 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 218 011 648;
  • 11) 0.021 330 666 825 391 314 058 239 134 218 011 648 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 436 023 296;
  • 12) 0.042 661 333 650 782 628 116 478 268 436 023 296 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 872 046 592;
  • 13) 0.085 322 667 301 565 256 232 956 536 872 046 592 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 744 093 184;
  • 14) 0.170 645 334 603 130 512 465 913 073 744 093 184 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 488 186 368;
  • 15) 0.341 290 669 206 261 024 931 826 147 488 186 368 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 976 372 736;
  • 16) 0.682 581 338 412 522 049 863 652 294 976 372 736 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 952 745 472;
  • 17) 0.365 162 676 825 044 099 727 304 589 952 745 472 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 905 490 944;
  • 18) 0.730 325 353 650 088 199 454 609 179 905 490 944 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 810 981 888;
  • 19) 0.460 650 707 300 176 398 909 218 359 810 981 888 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 621 963 776;
  • 20) 0.921 301 414 600 352 797 818 436 719 621 963 776 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 243 927 552;
  • 21) 0.842 602 829 200 705 595 636 873 439 243 927 552 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 487 855 104;
  • 22) 0.685 205 658 401 411 191 273 746 878 487 855 104 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 975 710 208;
  • 23) 0.370 411 316 802 822 382 547 493 756 975 710 208 × 2 = 0 + 0.740 822 633 605 644 765 094 987 513 951 420 416;
  • 24) 0.740 822 633 605 644 765 094 987 513 951 420 416 × 2 = 1 + 0.481 645 267 211 289 530 189 975 027 902 840 832;
  • 25) 0.481 645 267 211 289 530 189 975 027 902 840 832 × 2 = 0 + 0.963 290 534 422 579 060 379 950 055 805 681 664;
  • 26) 0.963 290 534 422 579 060 379 950 055 805 681 664 × 2 = 1 + 0.926 581 068 845 158 120 759 900 111 611 363 328;
  • 27) 0.926 581 068 845 158 120 759 900 111 611 363 328 × 2 = 1 + 0.853 162 137 690 316 241 519 800 223 222 726 656;
  • 28) 0.853 162 137 690 316 241 519 800 223 222 726 656 × 2 = 1 + 0.706 324 275 380 632 483 039 600 446 445 453 312;
  • 29) 0.706 324 275 380 632 483 039 600 446 445 453 312 × 2 = 1 + 0.412 648 550 761 264 966 079 200 892 890 906 624;
  • 30) 0.412 648 550 761 264 966 079 200 892 890 906 624 × 2 = 0 + 0.825 297 101 522 529 932 158 401 785 781 813 248;
  • 31) 0.825 297 101 522 529 932 158 401 785 781 813 248 × 2 = 1 + 0.650 594 203 045 059 864 316 803 571 563 626 496;
  • 32) 0.650 594 203 045 059 864 316 803 571 563 626 496 × 2 = 1 + 0.301 188 406 090 119 728 633 607 143 127 252 992;
  • 33) 0.301 188 406 090 119 728 633 607 143 127 252 992 × 2 = 0 + 0.602 376 812 180 239 457 267 214 286 254 505 984;
  • 34) 0.602 376 812 180 239 457 267 214 286 254 505 984 × 2 = 1 + 0.204 753 624 360 478 914 534 428 572 509 011 968;
  • 35) 0.204 753 624 360 478 914 534 428 572 509 011 968 × 2 = 0 + 0.409 507 248 720 957 829 068 857 145 018 023 936;
  • 36) 0.409 507 248 720 957 829 068 857 145 018 023 936 × 2 = 0 + 0.819 014 497 441 915 658 137 714 290 036 047 872;
  • 37) 0.819 014 497 441 915 658 137 714 290 036 047 872 × 2 = 1 + 0.638 028 994 883 831 316 275 428 580 072 095 744;
  • 38) 0.638 028 994 883 831 316 275 428 580 072 095 744 × 2 = 1 + 0.276 057 989 767 662 632 550 857 160 144 191 488;
  • 39) 0.276 057 989 767 662 632 550 857 160 144 191 488 × 2 = 0 + 0.552 115 979 535 325 265 101 714 320 288 382 976;
  • 40) 0.552 115 979 535 325 265 101 714 320 288 382 976 × 2 = 1 + 0.104 231 959 070 650 530 203 428 640 576 765 952;
  • 41) 0.104 231 959 070 650 530 203 428 640 576 765 952 × 2 = 0 + 0.208 463 918 141 301 060 406 857 281 153 531 904;
  • 42) 0.208 463 918 141 301 060 406 857 281 153 531 904 × 2 = 0 + 0.416 927 836 282 602 120 813 714 562 307 063 808;
  • 43) 0.416 927 836 282 602 120 813 714 562 307 063 808 × 2 = 0 + 0.833 855 672 565 204 241 627 429 124 614 127 616;
  • 44) 0.833 855 672 565 204 241 627 429 124 614 127 616 × 2 = 1 + 0.667 711 345 130 408 483 254 858 249 228 255 232;
  • 45) 0.667 711 345 130 408 483 254 858 249 228 255 232 × 2 = 1 + 0.335 422 690 260 816 966 509 716 498 456 510 464;
  • 46) 0.335 422 690 260 816 966 509 716 498 456 510 464 × 2 = 0 + 0.670 845 380 521 633 933 019 432 996 913 020 928;
  • 47) 0.670 845 380 521 633 933 019 432 996 913 020 928 × 2 = 1 + 0.341 690 761 043 267 866 038 865 993 826 041 856;
  • 48) 0.341 690 761 043 267 866 038 865 993 826 041 856 × 2 = 0 + 0.683 381 522 086 535 732 077 731 987 652 083 712;
  • 49) 0.683 381 522 086 535 732 077 731 987 652 083 712 × 2 = 1 + 0.366 763 044 173 071 464 155 463 975 304 167 424;
  • 50) 0.366 763 044 173 071 464 155 463 975 304 167 424 × 2 = 0 + 0.733 526 088 346 142 928 310 927 950 608 334 848;
  • 51) 0.733 526 088 346 142 928 310 927 950 608 334 848 × 2 = 1 + 0.467 052 176 692 285 856 621 855 901 216 669 696;
  • 52) 0.467 052 176 692 285 856 621 855 901 216 669 696 × 2 = 0 + 0.934 104 353 384 571 713 243 711 802 433 339 392;
  • 53) 0.934 104 353 384 571 713 243 711 802 433 339 392 × 2 = 1 + 0.868 208 706 769 143 426 487 423 604 866 678 784;
  • 54) 0.868 208 706 769 143 426 487 423 604 866 678 784 × 2 = 1 + 0.736 417 413 538 286 852 974 847 209 733 357 568;
  • 55) 0.736 417 413 538 286 852 974 847 209 733 357 568 × 2 = 1 + 0.472 834 827 076 573 705 949 694 419 466 715 136;
  • 56) 0.472 834 827 076 573 705 949 694 419 466 715 136 × 2 = 0 + 0.945 669 654 153 147 411 899 388 838 933 430 272;
  • 57) 0.945 669 654 153 147 411 899 388 838 933 430 272 × 2 = 1 + 0.891 339 308 306 294 823 798 777 677 866 860 544;
  • 58) 0.891 339 308 306 294 823 798 777 677 866 860 544 × 2 = 1 + 0.782 678 616 612 589 647 597 555 355 733 721 088;
  • 59) 0.782 678 616 612 589 647 597 555 355 733 721 088 × 2 = 1 + 0.565 357 233 225 179 295 195 110 711 467 442 176;
  • 60) 0.565 357 233 225 179 295 195 110 711 467 442 176 × 2 = 1 + 0.130 714 466 450 358 590 390 221 422 934 884 352;
  • 61) 0.130 714 466 450 358 590 390 221 422 934 884 352 × 2 = 0 + 0.261 428 932 900 717 180 780 442 845 869 768 704;
  • 62) 0.261 428 932 900 717 180 780 442 845 869 768 704 × 2 = 0 + 0.522 857 865 801 434 361 560 885 691 739 537 408;
  • 63) 0.522 857 865 801 434 361 560 885 691 739 537 408 × 2 = 1 + 0.045 715 731 602 868 723 121 771 383 479 074 816;
  • 64) 0.045 715 731 602 868 723 121 771 383 479 074 816 × 2 = 0 + 0.091 431 463 205 737 446 243 542 766 958 149 632;
  • 65) 0.091 431 463 205 737 446 243 542 766 958 149 632 × 2 = 0 + 0.182 862 926 411 474 892 487 085 533 916 299 264;
  • 66) 0.182 862 926 411 474 892 487 085 533 916 299 264 × 2 = 0 + 0.365 725 852 822 949 784 974 171 067 832 598 528;
  • 67) 0.365 725 852 822 949 784 974 171 067 832 598 528 × 2 = 0 + 0.731 451 705 645 899 569 948 342 135 665 197 056;
  • 68) 0.731 451 705 645 899 569 948 342 135 665 197 056 × 2 = 1 + 0.462 903 411 291 799 139 896 684 271 330 394 112;

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:


0.000 020 830 729 321 671 205 134 999 154 509 777(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:

0.000 020 830 729 321 671 205 134 999 154 509 777(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:


0.000 020 830 729 321 671 205 134 999 154 509 777(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 0.000 020 830 729 321 671 205 134 999 154 509 777 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

Share

» Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 817 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 08, 2026 [August]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: August - by our visitors

» Improve your social skills: The Hidden Requirement in Every Single Job Description. Gemini Notebook YouTube Video of 8.15 minutes

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