0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 888 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 888(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
0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 888(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: 0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 888.

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) 0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 888 × 2 = 0 + 0.062 831 853 071 795 864 769 252 867 665 590 057 683 943 387 987 776;
  • 2) 0.062 831 853 071 795 864 769 252 867 665 590 057 683 943 387 987 776 × 2 = 0 + 0.125 663 706 143 591 729 538 505 735 331 180 115 367 886 775 975 552;
  • 3) 0.125 663 706 143 591 729 538 505 735 331 180 115 367 886 775 975 552 × 2 = 0 + 0.251 327 412 287 183 459 077 011 470 662 360 230 735 773 551 951 104;
  • 4) 0.251 327 412 287 183 459 077 011 470 662 360 230 735 773 551 951 104 × 2 = 0 + 0.502 654 824 574 366 918 154 022 941 324 720 461 471 547 103 902 208;
  • 5) 0.502 654 824 574 366 918 154 022 941 324 720 461 471 547 103 902 208 × 2 = 1 + 0.005 309 649 148 733 836 308 045 882 649 440 922 943 094 207 804 416;
  • 6) 0.005 309 649 148 733 836 308 045 882 649 440 922 943 094 207 804 416 × 2 = 0 + 0.010 619 298 297 467 672 616 091 765 298 881 845 886 188 415 608 832;
  • 7) 0.010 619 298 297 467 672 616 091 765 298 881 845 886 188 415 608 832 × 2 = 0 + 0.021 238 596 594 935 345 232 183 530 597 763 691 772 376 831 217 664;
  • 8) 0.021 238 596 594 935 345 232 183 530 597 763 691 772 376 831 217 664 × 2 = 0 + 0.042 477 193 189 870 690 464 367 061 195 527 383 544 753 662 435 328;
  • 9) 0.042 477 193 189 870 690 464 367 061 195 527 383 544 753 662 435 328 × 2 = 0 + 0.084 954 386 379 741 380 928 734 122 391 054 767 089 507 324 870 656;
  • 10) 0.084 954 386 379 741 380 928 734 122 391 054 767 089 507 324 870 656 × 2 = 0 + 0.169 908 772 759 482 761 857 468 244 782 109 534 179 014 649 741 312;
  • 11) 0.169 908 772 759 482 761 857 468 244 782 109 534 179 014 649 741 312 × 2 = 0 + 0.339 817 545 518 965 523 714 936 489 564 219 068 358 029 299 482 624;
  • 12) 0.339 817 545 518 965 523 714 936 489 564 219 068 358 029 299 482 624 × 2 = 0 + 0.679 635 091 037 931 047 429 872 979 128 438 136 716 058 598 965 248;
  • 13) 0.679 635 091 037 931 047 429 872 979 128 438 136 716 058 598 965 248 × 2 = 1 + 0.359 270 182 075 862 094 859 745 958 256 876 273 432 117 197 930 496;
  • 14) 0.359 270 182 075 862 094 859 745 958 256 876 273 432 117 197 930 496 × 2 = 0 + 0.718 540 364 151 724 189 719 491 916 513 752 546 864 234 395 860 992;
  • 15) 0.718 540 364 151 724 189 719 491 916 513 752 546 864 234 395 860 992 × 2 = 1 + 0.437 080 728 303 448 379 438 983 833 027 505 093 728 468 791 721 984;
  • 16) 0.437 080 728 303 448 379 438 983 833 027 505 093 728 468 791 721 984 × 2 = 0 + 0.874 161 456 606 896 758 877 967 666 055 010 187 456 937 583 443 968;
  • 17) 0.874 161 456 606 896 758 877 967 666 055 010 187 456 937 583 443 968 × 2 = 1 + 0.748 322 913 213 793 517 755 935 332 110 020 374 913 875 166 887 936;
  • 18) 0.748 322 913 213 793 517 755 935 332 110 020 374 913 875 166 887 936 × 2 = 1 + 0.496 645 826 427 587 035 511 870 664 220 040 749 827 750 333 775 872;
  • 19) 0.496 645 826 427 587 035 511 870 664 220 040 749 827 750 333 775 872 × 2 = 0 + 0.993 291 652 855 174 071 023 741 328 440 081 499 655 500 667 551 744;
  • 20) 0.993 291 652 855 174 071 023 741 328 440 081 499 655 500 667 551 744 × 2 = 1 + 0.986 583 305 710 348 142 047 482 656 880 162 999 311 001 335 103 488;
  • 21) 0.986 583 305 710 348 142 047 482 656 880 162 999 311 001 335 103 488 × 2 = 1 + 0.973 166 611 420 696 284 094 965 313 760 325 998 622 002 670 206 976;
  • 22) 0.973 166 611 420 696 284 094 965 313 760 325 998 622 002 670 206 976 × 2 = 1 + 0.946 333 222 841 392 568 189 930 627 520 651 997 244 005 340 413 952;
  • 23) 0.946 333 222 841 392 568 189 930 627 520 651 997 244 005 340 413 952 × 2 = 1 + 0.892 666 445 682 785 136 379 861 255 041 303 994 488 010 680 827 904;
  • 24) 0.892 666 445 682 785 136 379 861 255 041 303 994 488 010 680 827 904 × 2 = 1 + 0.785 332 891 365 570 272 759 722 510 082 607 988 976 021 361 655 808;
  • 25) 0.785 332 891 365 570 272 759 722 510 082 607 988 976 021 361 655 808 × 2 = 1 + 0.570 665 782 731 140 545 519 445 020 165 215 977 952 042 723 311 616;
  • 26) 0.570 665 782 731 140 545 519 445 020 165 215 977 952 042 723 311 616 × 2 = 1 + 0.141 331 565 462 281 091 038 890 040 330 431 955 904 085 446 623 232;
  • 27) 0.141 331 565 462 281 091 038 890 040 330 431 955 904 085 446 623 232 × 2 = 0 + 0.282 663 130 924 562 182 077 780 080 660 863 911 808 170 893 246 464;
  • 28) 0.282 663 130 924 562 182 077 780 080 660 863 911 808 170 893 246 464 × 2 = 0 + 0.565 326 261 849 124 364 155 560 161 321 727 823 616 341 786 492 928;
  • 29) 0.565 326 261 849 124 364 155 560 161 321 727 823 616 341 786 492 928 × 2 = 1 + 0.130 652 523 698 248 728 311 120 322 643 455 647 232 683 572 985 856;
  • 30) 0.130 652 523 698 248 728 311 120 322 643 455 647 232 683 572 985 856 × 2 = 0 + 0.261 305 047 396 497 456 622 240 645 286 911 294 465 367 145 971 712;
  • 31) 0.261 305 047 396 497 456 622 240 645 286 911 294 465 367 145 971 712 × 2 = 0 + 0.522 610 094 792 994 913 244 481 290 573 822 588 930 734 291 943 424;
  • 32) 0.522 610 094 792 994 913 244 481 290 573 822 588 930 734 291 943 424 × 2 = 1 + 0.045 220 189 585 989 826 488 962 581 147 645 177 861 468 583 886 848;
  • 33) 0.045 220 189 585 989 826 488 962 581 147 645 177 861 468 583 886 848 × 2 = 0 + 0.090 440 379 171 979 652 977 925 162 295 290 355 722 937 167 773 696;
  • 34) 0.090 440 379 171 979 652 977 925 162 295 290 355 722 937 167 773 696 × 2 = 0 + 0.180 880 758 343 959 305 955 850 324 590 580 711 445 874 335 547 392;
  • 35) 0.180 880 758 343 959 305 955 850 324 590 580 711 445 874 335 547 392 × 2 = 0 + 0.361 761 516 687 918 611 911 700 649 181 161 422 891 748 671 094 784;
  • 36) 0.361 761 516 687 918 611 911 700 649 181 161 422 891 748 671 094 784 × 2 = 0 + 0.723 523 033 375 837 223 823 401 298 362 322 845 783 497 342 189 568;
  • 37) 0.723 523 033 375 837 223 823 401 298 362 322 845 783 497 342 189 568 × 2 = 1 + 0.447 046 066 751 674 447 646 802 596 724 645 691 566 994 684 379 136;
  • 38) 0.447 046 066 751 674 447 646 802 596 724 645 691 566 994 684 379 136 × 2 = 0 + 0.894 092 133 503 348 895 293 605 193 449 291 383 133 989 368 758 272;
  • 39) 0.894 092 133 503 348 895 293 605 193 449 291 383 133 989 368 758 272 × 2 = 1 + 0.788 184 267 006 697 790 587 210 386 898 582 766 267 978 737 516 544;
  • 40) 0.788 184 267 006 697 790 587 210 386 898 582 766 267 978 737 516 544 × 2 = 1 + 0.576 368 534 013 395 581 174 420 773 797 165 532 535 957 475 033 088;
  • 41) 0.576 368 534 013 395 581 174 420 773 797 165 532 535 957 475 033 088 × 2 = 1 + 0.152 737 068 026 791 162 348 841 547 594 331 065 071 914 950 066 176;
  • 42) 0.152 737 068 026 791 162 348 841 547 594 331 065 071 914 950 066 176 × 2 = 0 + 0.305 474 136 053 582 324 697 683 095 188 662 130 143 829 900 132 352;
  • 43) 0.305 474 136 053 582 324 697 683 095 188 662 130 143 829 900 132 352 × 2 = 0 + 0.610 948 272 107 164 649 395 366 190 377 324 260 287 659 800 264 704;
  • 44) 0.610 948 272 107 164 649 395 366 190 377 324 260 287 659 800 264 704 × 2 = 1 + 0.221 896 544 214 329 298 790 732 380 754 648 520 575 319 600 529 408;
  • 45) 0.221 896 544 214 329 298 790 732 380 754 648 520 575 319 600 529 408 × 2 = 0 + 0.443 793 088 428 658 597 581 464 761 509 297 041 150 639 201 058 816;
  • 46) 0.443 793 088 428 658 597 581 464 761 509 297 041 150 639 201 058 816 × 2 = 0 + 0.887 586 176 857 317 195 162 929 523 018 594 082 301 278 402 117 632;
  • 47) 0.887 586 176 857 317 195 162 929 523 018 594 082 301 278 402 117 632 × 2 = 1 + 0.775 172 353 714 634 390 325 859 046 037 188 164 602 556 804 235 264;
  • 48) 0.775 172 353 714 634 390 325 859 046 037 188 164 602 556 804 235 264 × 2 = 1 + 0.550 344 707 429 268 780 651 718 092 074 376 329 205 113 608 470 528;
  • 49) 0.550 344 707 429 268 780 651 718 092 074 376 329 205 113 608 470 528 × 2 = 1 + 0.100 689 414 858 537 561 303 436 184 148 752 658 410 227 216 941 056;
  • 50) 0.100 689 414 858 537 561 303 436 184 148 752 658 410 227 216 941 056 × 2 = 0 + 0.201 378 829 717 075 122 606 872 368 297 505 316 820 454 433 882 112;
  • 51) 0.201 378 829 717 075 122 606 872 368 297 505 316 820 454 433 882 112 × 2 = 0 + 0.402 757 659 434 150 245 213 744 736 595 010 633 640 908 867 764 224;
  • 52) 0.402 757 659 434 150 245 213 744 736 595 010 633 640 908 867 764 224 × 2 = 0 + 0.805 515 318 868 300 490 427 489 473 190 021 267 281 817 735 528 448;
  • 53) 0.805 515 318 868 300 490 427 489 473 190 021 267 281 817 735 528 448 × 2 = 1 + 0.611 030 637 736 600 980 854 978 946 380 042 534 563 635 471 056 896;
  • 54) 0.611 030 637 736 600 980 854 978 946 380 042 534 563 635 471 056 896 × 2 = 1 + 0.222 061 275 473 201 961 709 957 892 760 085 069 127 270 942 113 792;
  • 55) 0.222 061 275 473 201 961 709 957 892 760 085 069 127 270 942 113 792 × 2 = 0 + 0.444 122 550 946 403 923 419 915 785 520 170 138 254 541 884 227 584;
  • 56) 0.444 122 550 946 403 923 419 915 785 520 170 138 254 541 884 227 584 × 2 = 0 + 0.888 245 101 892 807 846 839 831 571 040 340 276 509 083 768 455 168;
  • 57) 0.888 245 101 892 807 846 839 831 571 040 340 276 509 083 768 455 168 × 2 = 1 + 0.776 490 203 785 615 693 679 663 142 080 680 553 018 167 536 910 336;

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:


0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 888(10) =

0.0000 1000 0000 1010 1101 1111 1100 1001 0000 1011 1001 0011 1000 1100 1(2)

5. Positive number before normalization:

0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 888(10) =

0.0000 1000 0000 1010 1101 1111 1100 1001 0000 1011 1001 0011 1000 1100 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 5 positions to the right, so that only one non zero digit remains to the left of it:


0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 888(10) =

0.0000 1000 0000 1010 1101 1111 1100 1001 0000 1011 1001 0011 1000 1100 1(2) =

0.0000 1000 0000 1010 1101 1111 1100 1001 0000 1011 1001 0011 1000 1100 1(2) × 20 =

1.0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001(2) × 2-5


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): -5

Mantissa (not normalized):
1.0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-5 + 2(11-1) - 1 =

(-5 + 1 023)(10) =

1 018(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;
  • 1 018 ÷ 2 = 509 + 0;
  • 509 ÷ 2 = 254 + 1;
  • 254 ÷ 2 = 127 + 0;
  • 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) =

1018(10) =

011 1111 1010(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. 0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001 =

0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001


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 1010

Mantissa (52 bits) =
0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001


Decimal number 0.031 415 926 535 897 932 384 626 433 832 795 028 841 971 693 993 888 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1010 - 0000 0001 0101 1011 1111 1001 0010 0001 0111 0010 0111 0001 1001

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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