0.000 020 830 729 321 671 205 134 999 154 509 670 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 670 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
0.000 020 830 729 321 671 205 134 999 154 509 670 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: 0.000 020 830 729 321 671 205 134 999 154 509 670 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) 0.000 020 830 729 321 671 205 134 999 154 509 670 3 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 340 6;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 340 6 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 681 2;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 681 2 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 362 4;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 362 4 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 724 8;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 724 8 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 449 6;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 449 6 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 899 2;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 899 2 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 237 798 4;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 237 798 4 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 475 596 8;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 475 596 8 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 951 193 6;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 951 193 6 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 902 387 2;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 902 387 2 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 804 774 4;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 804 774 4 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 609 548 8;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 609 548 8 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 219 097 6;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 219 097 6 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 438 195 2;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 438 195 2 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 876 390 4;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 876 390 4 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 752 780 8;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 752 780 8 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 891 505 561 6;
  • 18) 0.730 325 353 650 088 199 454 609 179 891 505 561 6 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 783 011 123 2;
  • 19) 0.460 650 707 300 176 398 909 218 359 783 011 123 2 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 566 022 246 4;
  • 20) 0.921 301 414 600 352 797 818 436 719 566 022 246 4 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 132 044 492 8;
  • 21) 0.842 602 829 200 705 595 636 873 439 132 044 492 8 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 264 088 985 6;
  • 22) 0.685 205 658 401 411 191 273 746 878 264 088 985 6 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 528 177 971 2;
  • 23) 0.370 411 316 802 822 382 547 493 756 528 177 971 2 × 2 = 0 + 0.740 822 633 605 644 765 094 987 513 056 355 942 4;
  • 24) 0.740 822 633 605 644 765 094 987 513 056 355 942 4 × 2 = 1 + 0.481 645 267 211 289 530 189 975 026 112 711 884 8;
  • 25) 0.481 645 267 211 289 530 189 975 026 112 711 884 8 × 2 = 0 + 0.963 290 534 422 579 060 379 950 052 225 423 769 6;
  • 26) 0.963 290 534 422 579 060 379 950 052 225 423 769 6 × 2 = 1 + 0.926 581 068 845 158 120 759 900 104 450 847 539 2;
  • 27) 0.926 581 068 845 158 120 759 900 104 450 847 539 2 × 2 = 1 + 0.853 162 137 690 316 241 519 800 208 901 695 078 4;
  • 28) 0.853 162 137 690 316 241 519 800 208 901 695 078 4 × 2 = 1 + 0.706 324 275 380 632 483 039 600 417 803 390 156 8;
  • 29) 0.706 324 275 380 632 483 039 600 417 803 390 156 8 × 2 = 1 + 0.412 648 550 761 264 966 079 200 835 606 780 313 6;
  • 30) 0.412 648 550 761 264 966 079 200 835 606 780 313 6 × 2 = 0 + 0.825 297 101 522 529 932 158 401 671 213 560 627 2;
  • 31) 0.825 297 101 522 529 932 158 401 671 213 560 627 2 × 2 = 1 + 0.650 594 203 045 059 864 316 803 342 427 121 254 4;
  • 32) 0.650 594 203 045 059 864 316 803 342 427 121 254 4 × 2 = 1 + 0.301 188 406 090 119 728 633 606 684 854 242 508 8;
  • 33) 0.301 188 406 090 119 728 633 606 684 854 242 508 8 × 2 = 0 + 0.602 376 812 180 239 457 267 213 369 708 485 017 6;
  • 34) 0.602 376 812 180 239 457 267 213 369 708 485 017 6 × 2 = 1 + 0.204 753 624 360 478 914 534 426 739 416 970 035 2;
  • 35) 0.204 753 624 360 478 914 534 426 739 416 970 035 2 × 2 = 0 + 0.409 507 248 720 957 829 068 853 478 833 940 070 4;
  • 36) 0.409 507 248 720 957 829 068 853 478 833 940 070 4 × 2 = 0 + 0.819 014 497 441 915 658 137 706 957 667 880 140 8;
  • 37) 0.819 014 497 441 915 658 137 706 957 667 880 140 8 × 2 = 1 + 0.638 028 994 883 831 316 275 413 915 335 760 281 6;
  • 38) 0.638 028 994 883 831 316 275 413 915 335 760 281 6 × 2 = 1 + 0.276 057 989 767 662 632 550 827 830 671 520 563 2;
  • 39) 0.276 057 989 767 662 632 550 827 830 671 520 563 2 × 2 = 0 + 0.552 115 979 535 325 265 101 655 661 343 041 126 4;
  • 40) 0.552 115 979 535 325 265 101 655 661 343 041 126 4 × 2 = 1 + 0.104 231 959 070 650 530 203 311 322 686 082 252 8;
  • 41) 0.104 231 959 070 650 530 203 311 322 686 082 252 8 × 2 = 0 + 0.208 463 918 141 301 060 406 622 645 372 164 505 6;
  • 42) 0.208 463 918 141 301 060 406 622 645 372 164 505 6 × 2 = 0 + 0.416 927 836 282 602 120 813 245 290 744 329 011 2;
  • 43) 0.416 927 836 282 602 120 813 245 290 744 329 011 2 × 2 = 0 + 0.833 855 672 565 204 241 626 490 581 488 658 022 4;
  • 44) 0.833 855 672 565 204 241 626 490 581 488 658 022 4 × 2 = 1 + 0.667 711 345 130 408 483 252 981 162 977 316 044 8;
  • 45) 0.667 711 345 130 408 483 252 981 162 977 316 044 8 × 2 = 1 + 0.335 422 690 260 816 966 505 962 325 954 632 089 6;
  • 46) 0.335 422 690 260 816 966 505 962 325 954 632 089 6 × 2 = 0 + 0.670 845 380 521 633 933 011 924 651 909 264 179 2;
  • 47) 0.670 845 380 521 633 933 011 924 651 909 264 179 2 × 2 = 1 + 0.341 690 761 043 267 866 023 849 303 818 528 358 4;
  • 48) 0.341 690 761 043 267 866 023 849 303 818 528 358 4 × 2 = 0 + 0.683 381 522 086 535 732 047 698 607 637 056 716 8;
  • 49) 0.683 381 522 086 535 732 047 698 607 637 056 716 8 × 2 = 1 + 0.366 763 044 173 071 464 095 397 215 274 113 433 6;
  • 50) 0.366 763 044 173 071 464 095 397 215 274 113 433 6 × 2 = 0 + 0.733 526 088 346 142 928 190 794 430 548 226 867 2;
  • 51) 0.733 526 088 346 142 928 190 794 430 548 226 867 2 × 2 = 1 + 0.467 052 176 692 285 856 381 588 861 096 453 734 4;
  • 52) 0.467 052 176 692 285 856 381 588 861 096 453 734 4 × 2 = 0 + 0.934 104 353 384 571 712 763 177 722 192 907 468 8;
  • 53) 0.934 104 353 384 571 712 763 177 722 192 907 468 8 × 2 = 1 + 0.868 208 706 769 143 425 526 355 444 385 814 937 6;
  • 54) 0.868 208 706 769 143 425 526 355 444 385 814 937 6 × 2 = 1 + 0.736 417 413 538 286 851 052 710 888 771 629 875 2;
  • 55) 0.736 417 413 538 286 851 052 710 888 771 629 875 2 × 2 = 1 + 0.472 834 827 076 573 702 105 421 777 543 259 750 4;
  • 56) 0.472 834 827 076 573 702 105 421 777 543 259 750 4 × 2 = 0 + 0.945 669 654 153 147 404 210 843 555 086 519 500 8;
  • 57) 0.945 669 654 153 147 404 210 843 555 086 519 500 8 × 2 = 1 + 0.891 339 308 306 294 808 421 687 110 173 039 001 6;
  • 58) 0.891 339 308 306 294 808 421 687 110 173 039 001 6 × 2 = 1 + 0.782 678 616 612 589 616 843 374 220 346 078 003 2;
  • 59) 0.782 678 616 612 589 616 843 374 220 346 078 003 2 × 2 = 1 + 0.565 357 233 225 179 233 686 748 440 692 156 006 4;
  • 60) 0.565 357 233 225 179 233 686 748 440 692 156 006 4 × 2 = 1 + 0.130 714 466 450 358 467 373 496 881 384 312 012 8;
  • 61) 0.130 714 466 450 358 467 373 496 881 384 312 012 8 × 2 = 0 + 0.261 428 932 900 716 934 746 993 762 768 624 025 6;
  • 62) 0.261 428 932 900 716 934 746 993 762 768 624 025 6 × 2 = 0 + 0.522 857 865 801 433 869 493 987 525 537 248 051 2;
  • 63) 0.522 857 865 801 433 869 493 987 525 537 248 051 2 × 2 = 1 + 0.045 715 731 602 867 738 987 975 051 074 496 102 4;
  • 64) 0.045 715 731 602 867 738 987 975 051 074 496 102 4 × 2 = 0 + 0.091 431 463 205 735 477 975 950 102 148 992 204 8;
  • 65) 0.091 431 463 205 735 477 975 950 102 148 992 204 8 × 2 = 0 + 0.182 862 926 411 470 955 951 900 204 297 984 409 6;
  • 66) 0.182 862 926 411 470 955 951 900 204 297 984 409 6 × 2 = 0 + 0.365 725 852 822 941 911 903 800 408 595 968 819 2;
  • 67) 0.365 725 852 822 941 911 903 800 408 595 968 819 2 × 2 = 0 + 0.731 451 705 645 883 823 807 600 817 191 937 638 4;
  • 68) 0.731 451 705 645 883 823 807 600 817 191 937 638 4 × 2 = 1 + 0.462 903 411 291 767 647 615 201 634 383 875 276 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:


0.000 020 830 729 321 671 205 134 999 154 509 670 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:

0.000 020 830 729 321 671 205 134 999 154 509 670 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:


0.000 020 830 729 321 671 205 134 999 154 509 670 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 0.000 020 830 729 321 671 205 134 999 154 509 670 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