Decimal to 64 Bit IEEE 754 Binary: Convert Number 0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 174 24 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number 0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 174 24(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation standard (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: 0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 174 24.

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) 0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 174 24 × 2 = 1 + 0.570 860 049 649 700 895 268 369 897 652 419 700 540 091 025 730 348 48;
  • 2) 0.570 860 049 649 700 895 268 369 897 652 419 700 540 091 025 730 348 48 × 2 = 1 + 0.141 720 099 299 401 790 536 739 795 304 839 401 080 182 051 460 696 96;
  • 3) 0.141 720 099 299 401 790 536 739 795 304 839 401 080 182 051 460 696 96 × 2 = 0 + 0.283 440 198 598 803 581 073 479 590 609 678 802 160 364 102 921 393 92;
  • 4) 0.283 440 198 598 803 581 073 479 590 609 678 802 160 364 102 921 393 92 × 2 = 0 + 0.566 880 397 197 607 162 146 959 181 219 357 604 320 728 205 842 787 84;
  • 5) 0.566 880 397 197 607 162 146 959 181 219 357 604 320 728 205 842 787 84 × 2 = 1 + 0.133 760 794 395 214 324 293 918 362 438 715 208 641 456 411 685 575 68;
  • 6) 0.133 760 794 395 214 324 293 918 362 438 715 208 641 456 411 685 575 68 × 2 = 0 + 0.267 521 588 790 428 648 587 836 724 877 430 417 282 912 823 371 151 36;
  • 7) 0.267 521 588 790 428 648 587 836 724 877 430 417 282 912 823 371 151 36 × 2 = 0 + 0.535 043 177 580 857 297 175 673 449 754 860 834 565 825 646 742 302 72;
  • 8) 0.535 043 177 580 857 297 175 673 449 754 860 834 565 825 646 742 302 72 × 2 = 1 + 0.070 086 355 161 714 594 351 346 899 509 721 669 131 651 293 484 605 44;
  • 9) 0.070 086 355 161 714 594 351 346 899 509 721 669 131 651 293 484 605 44 × 2 = 0 + 0.140 172 710 323 429 188 702 693 799 019 443 338 263 302 586 969 210 88;
  • 10) 0.140 172 710 323 429 188 702 693 799 019 443 338 263 302 586 969 210 88 × 2 = 0 + 0.280 345 420 646 858 377 405 387 598 038 886 676 526 605 173 938 421 76;
  • 11) 0.280 345 420 646 858 377 405 387 598 038 886 676 526 605 173 938 421 76 × 2 = 0 + 0.560 690 841 293 716 754 810 775 196 077 773 353 053 210 347 876 843 52;
  • 12) 0.560 690 841 293 716 754 810 775 196 077 773 353 053 210 347 876 843 52 × 2 = 1 + 0.121 381 682 587 433 509 621 550 392 155 546 706 106 420 695 753 687 04;
  • 13) 0.121 381 682 587 433 509 621 550 392 155 546 706 106 420 695 753 687 04 × 2 = 0 + 0.242 763 365 174 867 019 243 100 784 311 093 412 212 841 391 507 374 08;
  • 14) 0.242 763 365 174 867 019 243 100 784 311 093 412 212 841 391 507 374 08 × 2 = 0 + 0.485 526 730 349 734 038 486 201 568 622 186 824 425 682 783 014 748 16;
  • 15) 0.485 526 730 349 734 038 486 201 568 622 186 824 425 682 783 014 748 16 × 2 = 0 + 0.971 053 460 699 468 076 972 403 137 244 373 648 851 365 566 029 496 32;
  • 16) 0.971 053 460 699 468 076 972 403 137 244 373 648 851 365 566 029 496 32 × 2 = 1 + 0.942 106 921 398 936 153 944 806 274 488 747 297 702 731 132 058 992 64;
  • 17) 0.942 106 921 398 936 153 944 806 274 488 747 297 702 731 132 058 992 64 × 2 = 1 + 0.884 213 842 797 872 307 889 612 548 977 494 595 405 462 264 117 985 28;
  • 18) 0.884 213 842 797 872 307 889 612 548 977 494 595 405 462 264 117 985 28 × 2 = 1 + 0.768 427 685 595 744 615 779 225 097 954 989 190 810 924 528 235 970 56;
  • 19) 0.768 427 685 595 744 615 779 225 097 954 989 190 810 924 528 235 970 56 × 2 = 1 + 0.536 855 371 191 489 231 558 450 195 909 978 381 621 849 056 471 941 12;
  • 20) 0.536 855 371 191 489 231 558 450 195 909 978 381 621 849 056 471 941 12 × 2 = 1 + 0.073 710 742 382 978 463 116 900 391 819 956 763 243 698 112 943 882 24;
  • 21) 0.073 710 742 382 978 463 116 900 391 819 956 763 243 698 112 943 882 24 × 2 = 0 + 0.147 421 484 765 956 926 233 800 783 639 913 526 487 396 225 887 764 48;
  • 22) 0.147 421 484 765 956 926 233 800 783 639 913 526 487 396 225 887 764 48 × 2 = 0 + 0.294 842 969 531 913 852 467 601 567 279 827 052 974 792 451 775 528 96;
  • 23) 0.294 842 969 531 913 852 467 601 567 279 827 052 974 792 451 775 528 96 × 2 = 0 + 0.589 685 939 063 827 704 935 203 134 559 654 105 949 584 903 551 057 92;
  • 24) 0.589 685 939 063 827 704 935 203 134 559 654 105 949 584 903 551 057 92 × 2 = 1 + 0.179 371 878 127 655 409 870 406 269 119 308 211 899 169 807 102 115 84;
  • 25) 0.179 371 878 127 655 409 870 406 269 119 308 211 899 169 807 102 115 84 × 2 = 0 + 0.358 743 756 255 310 819 740 812 538 238 616 423 798 339 614 204 231 68;
  • 26) 0.358 743 756 255 310 819 740 812 538 238 616 423 798 339 614 204 231 68 × 2 = 0 + 0.717 487 512 510 621 639 481 625 076 477 232 847 596 679 228 408 463 36;
  • 27) 0.717 487 512 510 621 639 481 625 076 477 232 847 596 679 228 408 463 36 × 2 = 1 + 0.434 975 025 021 243 278 963 250 152 954 465 695 193 358 456 816 926 72;
  • 28) 0.434 975 025 021 243 278 963 250 152 954 465 695 193 358 456 816 926 72 × 2 = 0 + 0.869 950 050 042 486 557 926 500 305 908 931 390 386 716 913 633 853 44;
  • 29) 0.869 950 050 042 486 557 926 500 305 908 931 390 386 716 913 633 853 44 × 2 = 1 + 0.739 900 100 084 973 115 853 000 611 817 862 780 773 433 827 267 706 88;
  • 30) 0.739 900 100 084 973 115 853 000 611 817 862 780 773 433 827 267 706 88 × 2 = 1 + 0.479 800 200 169 946 231 706 001 223 635 725 561 546 867 654 535 413 76;
  • 31) 0.479 800 200 169 946 231 706 001 223 635 725 561 546 867 654 535 413 76 × 2 = 0 + 0.959 600 400 339 892 463 412 002 447 271 451 123 093 735 309 070 827 52;
  • 32) 0.959 600 400 339 892 463 412 002 447 271 451 123 093 735 309 070 827 52 × 2 = 1 + 0.919 200 800 679 784 926 824 004 894 542 902 246 187 470 618 141 655 04;
  • 33) 0.919 200 800 679 784 926 824 004 894 542 902 246 187 470 618 141 655 04 × 2 = 1 + 0.838 401 601 359 569 853 648 009 789 085 804 492 374 941 236 283 310 08;
  • 34) 0.838 401 601 359 569 853 648 009 789 085 804 492 374 941 236 283 310 08 × 2 = 1 + 0.676 803 202 719 139 707 296 019 578 171 608 984 749 882 472 566 620 16;
  • 35) 0.676 803 202 719 139 707 296 019 578 171 608 984 749 882 472 566 620 16 × 2 = 1 + 0.353 606 405 438 279 414 592 039 156 343 217 969 499 764 945 133 240 32;
  • 36) 0.353 606 405 438 279 414 592 039 156 343 217 969 499 764 945 133 240 32 × 2 = 0 + 0.707 212 810 876 558 829 184 078 312 686 435 938 999 529 890 266 480 64;
  • 37) 0.707 212 810 876 558 829 184 078 312 686 435 938 999 529 890 266 480 64 × 2 = 1 + 0.414 425 621 753 117 658 368 156 625 372 871 877 999 059 780 532 961 28;
  • 38) 0.414 425 621 753 117 658 368 156 625 372 871 877 999 059 780 532 961 28 × 2 = 0 + 0.828 851 243 506 235 316 736 313 250 745 743 755 998 119 561 065 922 56;
  • 39) 0.828 851 243 506 235 316 736 313 250 745 743 755 998 119 561 065 922 56 × 2 = 1 + 0.657 702 487 012 470 633 472 626 501 491 487 511 996 239 122 131 845 12;
  • 40) 0.657 702 487 012 470 633 472 626 501 491 487 511 996 239 122 131 845 12 × 2 = 1 + 0.315 404 974 024 941 266 945 253 002 982 975 023 992 478 244 263 690 24;
  • 41) 0.315 404 974 024 941 266 945 253 002 982 975 023 992 478 244 263 690 24 × 2 = 0 + 0.630 809 948 049 882 533 890 506 005 965 950 047 984 956 488 527 380 48;
  • 42) 0.630 809 948 049 882 533 890 506 005 965 950 047 984 956 488 527 380 48 × 2 = 1 + 0.261 619 896 099 765 067 781 012 011 931 900 095 969 912 977 054 760 96;
  • 43) 0.261 619 896 099 765 067 781 012 011 931 900 095 969 912 977 054 760 96 × 2 = 0 + 0.523 239 792 199 530 135 562 024 023 863 800 191 939 825 954 109 521 92;
  • 44) 0.523 239 792 199 530 135 562 024 023 863 800 191 939 825 954 109 521 92 × 2 = 1 + 0.046 479 584 399 060 271 124 048 047 727 600 383 879 651 908 219 043 84;
  • 45) 0.046 479 584 399 060 271 124 048 047 727 600 383 879 651 908 219 043 84 × 2 = 0 + 0.092 959 168 798 120 542 248 096 095 455 200 767 759 303 816 438 087 68;
  • 46) 0.092 959 168 798 120 542 248 096 095 455 200 767 759 303 816 438 087 68 × 2 = 0 + 0.185 918 337 596 241 084 496 192 190 910 401 535 518 607 632 876 175 36;
  • 47) 0.185 918 337 596 241 084 496 192 190 910 401 535 518 607 632 876 175 36 × 2 = 0 + 0.371 836 675 192 482 168 992 384 381 820 803 071 037 215 265 752 350 72;
  • 48) 0.371 836 675 192 482 168 992 384 381 820 803 071 037 215 265 752 350 72 × 2 = 0 + 0.743 673 350 384 964 337 984 768 763 641 606 142 074 430 531 504 701 44;
  • 49) 0.743 673 350 384 964 337 984 768 763 641 606 142 074 430 531 504 701 44 × 2 = 1 + 0.487 346 700 769 928 675 969 537 527 283 212 284 148 861 063 009 402 88;
  • 50) 0.487 346 700 769 928 675 969 537 527 283 212 284 148 861 063 009 402 88 × 2 = 0 + 0.974 693 401 539 857 351 939 075 054 566 424 568 297 722 126 018 805 76;
  • 51) 0.974 693 401 539 857 351 939 075 054 566 424 568 297 722 126 018 805 76 × 2 = 1 + 0.949 386 803 079 714 703 878 150 109 132 849 136 595 444 252 037 611 52;
  • 52) 0.949 386 803 079 714 703 878 150 109 132 849 136 595 444 252 037 611 52 × 2 = 1 + 0.898 773 606 159 429 407 756 300 218 265 698 273 190 888 504 075 223 04;
  • 53) 0.898 773 606 159 429 407 756 300 218 265 698 273 190 888 504 075 223 04 × 2 = 1 + 0.797 547 212 318 858 815 512 600 436 531 396 546 381 777 008 150 446 08;

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:


0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 174 24(10) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2)

5. Positive number before normalization:

0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 174 24(10) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the right, so that only one non zero digit remains to the left of it:


0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 174 24(10) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2) =

0.1100 1001 0001 0001 1111 0001 0010 1101 1110 1011 0101 0000 1011 1(2) × 20 =

1.1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111(2) × 2-1


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): -1

Mantissa (not normalized):
1.1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-1 + 1 023)(10) =

1 022(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;
  • 1 022 ÷ 2 = 511 + 0;
  • 511 ÷ 2 = 255 + 1;
  • 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) =

1022(10) =

011 1111 1110(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. 1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111 =

1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111


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 1110

Mantissa (52 bits) =
1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111


The base ten decimal number 0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 174 24 converted and written in 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1110 - 1001 0010 0010 0011 1110 0010 0101 1011 1101 0110 1010 0001 0111

» Decimal number in base ten 0.785 430 024 824 850 447 634 184 948 826 209 850 270 045 512 865 173 74 converted and written as 64 bit double precision IEEE 754 binary floating point representation standard

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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