1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 876 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 876 2(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
1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 876 2(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: 1.
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;
  • 1 ÷ 2 = 0 + 1;

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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.414 213 562 373 095 048 801 688 724 209 698 078 569 671 876 2.

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) 0.414 213 562 373 095 048 801 688 724 209 698 078 569 671 876 2 × 2 = 0 + 0.828 427 124 746 190 097 603 377 448 419 396 157 139 343 752 4;
  • 2) 0.828 427 124 746 190 097 603 377 448 419 396 157 139 343 752 4 × 2 = 1 + 0.656 854 249 492 380 195 206 754 896 838 792 314 278 687 504 8;
  • 3) 0.656 854 249 492 380 195 206 754 896 838 792 314 278 687 504 8 × 2 = 1 + 0.313 708 498 984 760 390 413 509 793 677 584 628 557 375 009 6;
  • 4) 0.313 708 498 984 760 390 413 509 793 677 584 628 557 375 009 6 × 2 = 0 + 0.627 416 997 969 520 780 827 019 587 355 169 257 114 750 019 2;
  • 5) 0.627 416 997 969 520 780 827 019 587 355 169 257 114 750 019 2 × 2 = 1 + 0.254 833 995 939 041 561 654 039 174 710 338 514 229 500 038 4;
  • 6) 0.254 833 995 939 041 561 654 039 174 710 338 514 229 500 038 4 × 2 = 0 + 0.509 667 991 878 083 123 308 078 349 420 677 028 459 000 076 8;
  • 7) 0.509 667 991 878 083 123 308 078 349 420 677 028 459 000 076 8 × 2 = 1 + 0.019 335 983 756 166 246 616 156 698 841 354 056 918 000 153 6;
  • 8) 0.019 335 983 756 166 246 616 156 698 841 354 056 918 000 153 6 × 2 = 0 + 0.038 671 967 512 332 493 232 313 397 682 708 113 836 000 307 2;
  • 9) 0.038 671 967 512 332 493 232 313 397 682 708 113 836 000 307 2 × 2 = 0 + 0.077 343 935 024 664 986 464 626 795 365 416 227 672 000 614 4;
  • 10) 0.077 343 935 024 664 986 464 626 795 365 416 227 672 000 614 4 × 2 = 0 + 0.154 687 870 049 329 972 929 253 590 730 832 455 344 001 228 8;
  • 11) 0.154 687 870 049 329 972 929 253 590 730 832 455 344 001 228 8 × 2 = 0 + 0.309 375 740 098 659 945 858 507 181 461 664 910 688 002 457 6;
  • 12) 0.309 375 740 098 659 945 858 507 181 461 664 910 688 002 457 6 × 2 = 0 + 0.618 751 480 197 319 891 717 014 362 923 329 821 376 004 915 2;
  • 13) 0.618 751 480 197 319 891 717 014 362 923 329 821 376 004 915 2 × 2 = 1 + 0.237 502 960 394 639 783 434 028 725 846 659 642 752 009 830 4;
  • 14) 0.237 502 960 394 639 783 434 028 725 846 659 642 752 009 830 4 × 2 = 0 + 0.475 005 920 789 279 566 868 057 451 693 319 285 504 019 660 8;
  • 15) 0.475 005 920 789 279 566 868 057 451 693 319 285 504 019 660 8 × 2 = 0 + 0.950 011 841 578 559 133 736 114 903 386 638 571 008 039 321 6;
  • 16) 0.950 011 841 578 559 133 736 114 903 386 638 571 008 039 321 6 × 2 = 1 + 0.900 023 683 157 118 267 472 229 806 773 277 142 016 078 643 2;
  • 17) 0.900 023 683 157 118 267 472 229 806 773 277 142 016 078 643 2 × 2 = 1 + 0.800 047 366 314 236 534 944 459 613 546 554 284 032 157 286 4;
  • 18) 0.800 047 366 314 236 534 944 459 613 546 554 284 032 157 286 4 × 2 = 1 + 0.600 094 732 628 473 069 888 919 227 093 108 568 064 314 572 8;
  • 19) 0.600 094 732 628 473 069 888 919 227 093 108 568 064 314 572 8 × 2 = 1 + 0.200 189 465 256 946 139 777 838 454 186 217 136 128 629 145 6;
  • 20) 0.200 189 465 256 946 139 777 838 454 186 217 136 128 629 145 6 × 2 = 0 + 0.400 378 930 513 892 279 555 676 908 372 434 272 257 258 291 2;
  • 21) 0.400 378 930 513 892 279 555 676 908 372 434 272 257 258 291 2 × 2 = 0 + 0.800 757 861 027 784 559 111 353 816 744 868 544 514 516 582 4;
  • 22) 0.800 757 861 027 784 559 111 353 816 744 868 544 514 516 582 4 × 2 = 1 + 0.601 515 722 055 569 118 222 707 633 489 737 089 029 033 164 8;
  • 23) 0.601 515 722 055 569 118 222 707 633 489 737 089 029 033 164 8 × 2 = 1 + 0.203 031 444 111 138 236 445 415 266 979 474 178 058 066 329 6;
  • 24) 0.203 031 444 111 138 236 445 415 266 979 474 178 058 066 329 6 × 2 = 0 + 0.406 062 888 222 276 472 890 830 533 958 948 356 116 132 659 2;
  • 25) 0.406 062 888 222 276 472 890 830 533 958 948 356 116 132 659 2 × 2 = 0 + 0.812 125 776 444 552 945 781 661 067 917 896 712 232 265 318 4;
  • 26) 0.812 125 776 444 552 945 781 661 067 917 896 712 232 265 318 4 × 2 = 1 + 0.624 251 552 889 105 891 563 322 135 835 793 424 464 530 636 8;
  • 27) 0.624 251 552 889 105 891 563 322 135 835 793 424 464 530 636 8 × 2 = 1 + 0.248 503 105 778 211 783 126 644 271 671 586 848 929 061 273 6;
  • 28) 0.248 503 105 778 211 783 126 644 271 671 586 848 929 061 273 6 × 2 = 0 + 0.497 006 211 556 423 566 253 288 543 343 173 697 858 122 547 2;
  • 29) 0.497 006 211 556 423 566 253 288 543 343 173 697 858 122 547 2 × 2 = 0 + 0.994 012 423 112 847 132 506 577 086 686 347 395 716 245 094 4;
  • 30) 0.994 012 423 112 847 132 506 577 086 686 347 395 716 245 094 4 × 2 = 1 + 0.988 024 846 225 694 265 013 154 173 372 694 791 432 490 188 8;
  • 31) 0.988 024 846 225 694 265 013 154 173 372 694 791 432 490 188 8 × 2 = 1 + 0.976 049 692 451 388 530 026 308 346 745 389 582 864 980 377 6;
  • 32) 0.976 049 692 451 388 530 026 308 346 745 389 582 864 980 377 6 × 2 = 1 + 0.952 099 384 902 777 060 052 616 693 490 779 165 729 960 755 2;
  • 33) 0.952 099 384 902 777 060 052 616 693 490 779 165 729 960 755 2 × 2 = 1 + 0.904 198 769 805 554 120 105 233 386 981 558 331 459 921 510 4;
  • 34) 0.904 198 769 805 554 120 105 233 386 981 558 331 459 921 510 4 × 2 = 1 + 0.808 397 539 611 108 240 210 466 773 963 116 662 919 843 020 8;
  • 35) 0.808 397 539 611 108 240 210 466 773 963 116 662 919 843 020 8 × 2 = 1 + 0.616 795 079 222 216 480 420 933 547 926 233 325 839 686 041 6;
  • 36) 0.616 795 079 222 216 480 420 933 547 926 233 325 839 686 041 6 × 2 = 1 + 0.233 590 158 444 432 960 841 867 095 852 466 651 679 372 083 2;
  • 37) 0.233 590 158 444 432 960 841 867 095 852 466 651 679 372 083 2 × 2 = 0 + 0.467 180 316 888 865 921 683 734 191 704 933 303 358 744 166 4;
  • 38) 0.467 180 316 888 865 921 683 734 191 704 933 303 358 744 166 4 × 2 = 0 + 0.934 360 633 777 731 843 367 468 383 409 866 606 717 488 332 8;
  • 39) 0.934 360 633 777 731 843 367 468 383 409 866 606 717 488 332 8 × 2 = 1 + 0.868 721 267 555 463 686 734 936 766 819 733 213 434 976 665 6;
  • 40) 0.868 721 267 555 463 686 734 936 766 819 733 213 434 976 665 6 × 2 = 1 + 0.737 442 535 110 927 373 469 873 533 639 466 426 869 953 331 2;
  • 41) 0.737 442 535 110 927 373 469 873 533 639 466 426 869 953 331 2 × 2 = 1 + 0.474 885 070 221 854 746 939 747 067 278 932 853 739 906 662 4;
  • 42) 0.474 885 070 221 854 746 939 747 067 278 932 853 739 906 662 4 × 2 = 0 + 0.949 770 140 443 709 493 879 494 134 557 865 707 479 813 324 8;
  • 43) 0.949 770 140 443 709 493 879 494 134 557 865 707 479 813 324 8 × 2 = 1 + 0.899 540 280 887 418 987 758 988 269 115 731 414 959 626 649 6;
  • 44) 0.899 540 280 887 418 987 758 988 269 115 731 414 959 626 649 6 × 2 = 1 + 0.799 080 561 774 837 975 517 976 538 231 462 829 919 253 299 2;
  • 45) 0.799 080 561 774 837 975 517 976 538 231 462 829 919 253 299 2 × 2 = 1 + 0.598 161 123 549 675 951 035 953 076 462 925 659 838 506 598 4;
  • 46) 0.598 161 123 549 675 951 035 953 076 462 925 659 838 506 598 4 × 2 = 1 + 0.196 322 247 099 351 902 071 906 152 925 851 319 677 013 196 8;
  • 47) 0.196 322 247 099 351 902 071 906 152 925 851 319 677 013 196 8 × 2 = 0 + 0.392 644 494 198 703 804 143 812 305 851 702 639 354 026 393 6;
  • 48) 0.392 644 494 198 703 804 143 812 305 851 702 639 354 026 393 6 × 2 = 0 + 0.785 288 988 397 407 608 287 624 611 703 405 278 708 052 787 2;
  • 49) 0.785 288 988 397 407 608 287 624 611 703 405 278 708 052 787 2 × 2 = 1 + 0.570 577 976 794 815 216 575 249 223 406 810 557 416 105 574 4;
  • 50) 0.570 577 976 794 815 216 575 249 223 406 810 557 416 105 574 4 × 2 = 1 + 0.141 155 953 589 630 433 150 498 446 813 621 114 832 211 148 8;
  • 51) 0.141 155 953 589 630 433 150 498 446 813 621 114 832 211 148 8 × 2 = 0 + 0.282 311 907 179 260 866 300 996 893 627 242 229 664 422 297 6;
  • 52) 0.282 311 907 179 260 866 300 996 893 627 242 229 664 422 297 6 × 2 = 0 + 0.564 623 814 358 521 732 601 993 787 254 484 459 328 844 595 2;
  • 53) 0.564 623 814 358 521 732 601 993 787 254 484 459 328 844 595 2 × 2 = 1 + 0.129 247 628 717 043 465 203 987 574 508 968 918 657 689 190 4;

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:


0.414 213 562 373 095 048 801 688 724 209 698 078 569 671 876 2(10) =

0.0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 1(2)

5. Positive number before normalization:

1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 876 2(10) =

1.0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 876 2(10) =

1.0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 1(2) =

1.0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 1(2) × 20


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): 0

Mantissa (not normalized):
1.0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 2 = 511 + 1;
  • 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) =

1023(10) =

011 1111 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100 1 =

0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100


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 1111

Mantissa (52 bits) =
0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100


Decimal number 1.414 213 562 373 095 048 801 688 724 209 698 078 569 671 876 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 0110 1010 0000 1001 1110 0110 0110 0111 1111 0011 1011 1100 1100

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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