64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 0.314 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 51 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 0.314 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 51(10) converted and written in 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: 0.314 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 51.

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) 0.314 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 51 × 2 = 0 + 0.628 318 530 717 958 647 692 528 676 655 900 576 839 433 879 875 02;
  • 2) 0.628 318 530 717 958 647 692 528 676 655 900 576 839 433 879 875 02 × 2 = 1 + 0.256 637 061 435 917 295 385 057 353 311 801 153 678 867 759 750 04;
  • 3) 0.256 637 061 435 917 295 385 057 353 311 801 153 678 867 759 750 04 × 2 = 0 + 0.513 274 122 871 834 590 770 114 706 623 602 307 357 735 519 500 08;
  • 4) 0.513 274 122 871 834 590 770 114 706 623 602 307 357 735 519 500 08 × 2 = 1 + 0.026 548 245 743 669 181 540 229 413 247 204 614 715 471 039 000 16;
  • 5) 0.026 548 245 743 669 181 540 229 413 247 204 614 715 471 039 000 16 × 2 = 0 + 0.053 096 491 487 338 363 080 458 826 494 409 229 430 942 078 000 32;
  • 6) 0.053 096 491 487 338 363 080 458 826 494 409 229 430 942 078 000 32 × 2 = 0 + 0.106 192 982 974 676 726 160 917 652 988 818 458 861 884 156 000 64;
  • 7) 0.106 192 982 974 676 726 160 917 652 988 818 458 861 884 156 000 64 × 2 = 0 + 0.212 385 965 949 353 452 321 835 305 977 636 917 723 768 312 001 28;
  • 8) 0.212 385 965 949 353 452 321 835 305 977 636 917 723 768 312 001 28 × 2 = 0 + 0.424 771 931 898 706 904 643 670 611 955 273 835 447 536 624 002 56;
  • 9) 0.424 771 931 898 706 904 643 670 611 955 273 835 447 536 624 002 56 × 2 = 0 + 0.849 543 863 797 413 809 287 341 223 910 547 670 895 073 248 005 12;
  • 10) 0.849 543 863 797 413 809 287 341 223 910 547 670 895 073 248 005 12 × 2 = 1 + 0.699 087 727 594 827 618 574 682 447 821 095 341 790 146 496 010 24;
  • 11) 0.699 087 727 594 827 618 574 682 447 821 095 341 790 146 496 010 24 × 2 = 1 + 0.398 175 455 189 655 237 149 364 895 642 190 683 580 292 992 020 48;
  • 12) 0.398 175 455 189 655 237 149 364 895 642 190 683 580 292 992 020 48 × 2 = 0 + 0.796 350 910 379 310 474 298 729 791 284 381 367 160 585 984 040 96;
  • 13) 0.796 350 910 379 310 474 298 729 791 284 381 367 160 585 984 040 96 × 2 = 1 + 0.592 701 820 758 620 948 597 459 582 568 762 734 321 171 968 081 92;
  • 14) 0.592 701 820 758 620 948 597 459 582 568 762 734 321 171 968 081 92 × 2 = 1 + 0.185 403 641 517 241 897 194 919 165 137 525 468 642 343 936 163 84;
  • 15) 0.185 403 641 517 241 897 194 919 165 137 525 468 642 343 936 163 84 × 2 = 0 + 0.370 807 283 034 483 794 389 838 330 275 050 937 284 687 872 327 68;
  • 16) 0.370 807 283 034 483 794 389 838 330 275 050 937 284 687 872 327 68 × 2 = 0 + 0.741 614 566 068 967 588 779 676 660 550 101 874 569 375 744 655 36;
  • 17) 0.741 614 566 068 967 588 779 676 660 550 101 874 569 375 744 655 36 × 2 = 1 + 0.483 229 132 137 935 177 559 353 321 100 203 749 138 751 489 310 72;
  • 18) 0.483 229 132 137 935 177 559 353 321 100 203 749 138 751 489 310 72 × 2 = 0 + 0.966 458 264 275 870 355 118 706 642 200 407 498 277 502 978 621 44;
  • 19) 0.966 458 264 275 870 355 118 706 642 200 407 498 277 502 978 621 44 × 2 = 1 + 0.932 916 528 551 740 710 237 413 284 400 814 996 555 005 957 242 88;
  • 20) 0.932 916 528 551 740 710 237 413 284 400 814 996 555 005 957 242 88 × 2 = 1 + 0.865 833 057 103 481 420 474 826 568 801 629 993 110 011 914 485 76;
  • 21) 0.865 833 057 103 481 420 474 826 568 801 629 993 110 011 914 485 76 × 2 = 1 + 0.731 666 114 206 962 840 949 653 137 603 259 986 220 023 828 971 52;
  • 22) 0.731 666 114 206 962 840 949 653 137 603 259 986 220 023 828 971 52 × 2 = 1 + 0.463 332 228 413 925 681 899 306 275 206 519 972 440 047 657 943 04;
  • 23) 0.463 332 228 413 925 681 899 306 275 206 519 972 440 047 657 943 04 × 2 = 0 + 0.926 664 456 827 851 363 798 612 550 413 039 944 880 095 315 886 08;
  • 24) 0.926 664 456 827 851 363 798 612 550 413 039 944 880 095 315 886 08 × 2 = 1 + 0.853 328 913 655 702 727 597 225 100 826 079 889 760 190 631 772 16;
  • 25) 0.853 328 913 655 702 727 597 225 100 826 079 889 760 190 631 772 16 × 2 = 1 + 0.706 657 827 311 405 455 194 450 201 652 159 779 520 381 263 544 32;
  • 26) 0.706 657 827 311 405 455 194 450 201 652 159 779 520 381 263 544 32 × 2 = 1 + 0.413 315 654 622 810 910 388 900 403 304 319 559 040 762 527 088 64;
  • 27) 0.413 315 654 622 810 910 388 900 403 304 319 559 040 762 527 088 64 × 2 = 0 + 0.826 631 309 245 621 820 777 800 806 608 639 118 081 525 054 177 28;
  • 28) 0.826 631 309 245 621 820 777 800 806 608 639 118 081 525 054 177 28 × 2 = 1 + 0.653 262 618 491 243 641 555 601 613 217 278 236 163 050 108 354 56;
  • 29) 0.653 262 618 491 243 641 555 601 613 217 278 236 163 050 108 354 56 × 2 = 1 + 0.306 525 236 982 487 283 111 203 226 434 556 472 326 100 216 709 12;
  • 30) 0.306 525 236 982 487 283 111 203 226 434 556 472 326 100 216 709 12 × 2 = 0 + 0.613 050 473 964 974 566 222 406 452 869 112 944 652 200 433 418 24;
  • 31) 0.613 050 473 964 974 566 222 406 452 869 112 944 652 200 433 418 24 × 2 = 1 + 0.226 100 947 929 949 132 444 812 905 738 225 889 304 400 866 836 48;
  • 32) 0.226 100 947 929 949 132 444 812 905 738 225 889 304 400 866 836 48 × 2 = 0 + 0.452 201 895 859 898 264 889 625 811 476 451 778 608 801 733 672 96;
  • 33) 0.452 201 895 859 898 264 889 625 811 476 451 778 608 801 733 672 96 × 2 = 0 + 0.904 403 791 719 796 529 779 251 622 952 903 557 217 603 467 345 92;
  • 34) 0.904 403 791 719 796 529 779 251 622 952 903 557 217 603 467 345 92 × 2 = 1 + 0.808 807 583 439 593 059 558 503 245 905 807 114 435 206 934 691 84;
  • 35) 0.808 807 583 439 593 059 558 503 245 905 807 114 435 206 934 691 84 × 2 = 1 + 0.617 615 166 879 186 119 117 006 491 811 614 228 870 413 869 383 68;
  • 36) 0.617 615 166 879 186 119 117 006 491 811 614 228 870 413 869 383 68 × 2 = 1 + 0.235 230 333 758 372 238 234 012 983 623 228 457 740 827 738 767 36;
  • 37) 0.235 230 333 758 372 238 234 012 983 623 228 457 740 827 738 767 36 × 2 = 0 + 0.470 460 667 516 744 476 468 025 967 246 456 915 481 655 477 534 72;
  • 38) 0.470 460 667 516 744 476 468 025 967 246 456 915 481 655 477 534 72 × 2 = 0 + 0.940 921 335 033 488 952 936 051 934 492 913 830 963 310 955 069 44;
  • 39) 0.940 921 335 033 488 952 936 051 934 492 913 830 963 310 955 069 44 × 2 = 1 + 0.881 842 670 066 977 905 872 103 868 985 827 661 926 621 910 138 88;
  • 40) 0.881 842 670 066 977 905 872 103 868 985 827 661 926 621 910 138 88 × 2 = 1 + 0.763 685 340 133 955 811 744 207 737 971 655 323 853 243 820 277 76;
  • 41) 0.763 685 340 133 955 811 744 207 737 971 655 323 853 243 820 277 76 × 2 = 1 + 0.527 370 680 267 911 623 488 415 475 943 310 647 706 487 640 555 52;
  • 42) 0.527 370 680 267 911 623 488 415 475 943 310 647 706 487 640 555 52 × 2 = 1 + 0.054 741 360 535 823 246 976 830 951 886 621 295 412 975 281 111 04;
  • 43) 0.054 741 360 535 823 246 976 830 951 886 621 295 412 975 281 111 04 × 2 = 0 + 0.109 482 721 071 646 493 953 661 903 773 242 590 825 950 562 222 08;
  • 44) 0.109 482 721 071 646 493 953 661 903 773 242 590 825 950 562 222 08 × 2 = 0 + 0.218 965 442 143 292 987 907 323 807 546 485 181 651 901 124 444 16;
  • 45) 0.218 965 442 143 292 987 907 323 807 546 485 181 651 901 124 444 16 × 2 = 0 + 0.437 930 884 286 585 975 814 647 615 092 970 363 303 802 248 888 32;
  • 46) 0.437 930 884 286 585 975 814 647 615 092 970 363 303 802 248 888 32 × 2 = 0 + 0.875 861 768 573 171 951 629 295 230 185 940 726 607 604 497 776 64;
  • 47) 0.875 861 768 573 171 951 629 295 230 185 940 726 607 604 497 776 64 × 2 = 1 + 0.751 723 537 146 343 903 258 590 460 371 881 453 215 208 995 553 28;
  • 48) 0.751 723 537 146 343 903 258 590 460 371 881 453 215 208 995 553 28 × 2 = 1 + 0.503 447 074 292 687 806 517 180 920 743 762 906 430 417 991 106 56;
  • 49) 0.503 447 074 292 687 806 517 180 920 743 762 906 430 417 991 106 56 × 2 = 1 + 0.006 894 148 585 375 613 034 361 841 487 525 812 860 835 982 213 12;
  • 50) 0.006 894 148 585 375 613 034 361 841 487 525 812 860 835 982 213 12 × 2 = 0 + 0.013 788 297 170 751 226 068 723 682 975 051 625 721 671 964 426 24;
  • 51) 0.013 788 297 170 751 226 068 723 682 975 051 625 721 671 964 426 24 × 2 = 0 + 0.027 576 594 341 502 452 137 447 365 950 103 251 443 343 928 852 48;
  • 52) 0.027 576 594 341 502 452 137 447 365 950 103 251 443 343 928 852 48 × 2 = 0 + 0.055 153 188 683 004 904 274 894 731 900 206 502 886 687 857 704 96;
  • 53) 0.055 153 188 683 004 904 274 894 731 900 206 502 886 687 857 704 96 × 2 = 0 + 0.110 306 377 366 009 808 549 789 463 800 413 005 773 375 715 409 92;
  • 54) 0.110 306 377 366 009 808 549 789 463 800 413 005 773 375 715 409 92 × 2 = 0 + 0.220 612 754 732 019 617 099 578 927 600 826 011 546 751 430 819 84;

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...)


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:


0.314 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 51(10) =

0.0101 0000 0110 1100 1011 1101 1101 1010 0111 0011 1100 0011 1000 00(2)


5. Positive number before normalization:

0.314 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 51(10) =

0.0101 0000 0110 1100 1011 1101 1101 1010 0111 0011 1100 0011 1000 00(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 2 positions to the right, so that only one non zero digit remains to the left of it:


0.314 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 51(10) =

0.0101 0000 0110 1100 1011 1101 1101 1010 0111 0011 1100 0011 1000 00(2) =

0.0101 0000 0110 1100 1011 1101 1101 1010 0111 0011 1100 0011 1000 00(2) × 20 =

1.0100 0001 1011 0010 1111 0111 0110 1001 1100 1111 0000 1110 0000(2) × 2-2


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): -2

Mantissa (not normalized):
1.0100 0001 1011 0010 1111 0111 0110 1001 1100 1111 0000 1110 0000


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-2 + 2(11-1) - 1 =

(-2 + 1 023)(10) =

1 021(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;
  • 1 021 ÷ 2 = 510 + 1;
  • 510 ÷ 2 = 255 + 0;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 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) =

1021(10) =

011 1111 1101(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. 0100 0001 1011 0010 1111 0111 0110 1001 1100 1111 0000 1110 0000 =

0100 0001 1011 0010 1111 0111 0110 1001 1100 1111 0000 1110 0000


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 1111 1101

Mantissa (52 bits) =
0100 0001 1011 0010 1111 0111 0110 1001 1100 1111 0000 1110 0000


The base ten decimal number 0.314 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 51 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 011 1111 1101 - 0100 0001 1011 0010 1111 0111 0110 1001 1100 1111 0000 1110 0000

More operations with decimal numbers converted to 64 bit double precision IEEE 754 binary floating point representation:

» Number 0.314 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 5 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

» Number 0.314 159 265 358 979 323 846 264 338 327 950 288 419 716 939 937 52 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

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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