0.000 020 830 729 321 671 205 134 999 154 509 727 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 727(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
0.000 020 830 729 321 671 205 134 999 154 509 727(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: 0.000 020 830 729 321 671 205 134 999 154 509 727.

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) 0.000 020 830 729 321 671 205 134 999 154 509 727 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 454;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 454 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 908;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 908 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 816;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 816 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 155 632;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 155 632 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 311 264;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 311 264 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 622 528;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 622 528 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 245 056;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 245 056 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 490 112;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 490 112 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 980 224;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 980 224 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 960 448;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 960 448 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 920 896;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 920 896 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 841 792;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 841 792 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 683 584;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 683 584 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 487 367 168;
  • 15) 0.341 290 669 206 261 024 931 826 147 487 367 168 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 974 734 336;
  • 16) 0.682 581 338 412 522 049 863 652 294 974 734 336 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 949 468 672;
  • 17) 0.365 162 676 825 044 099 727 304 589 949 468 672 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 898 937 344;
  • 18) 0.730 325 353 650 088 199 454 609 179 898 937 344 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 797 874 688;
  • 19) 0.460 650 707 300 176 398 909 218 359 797 874 688 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 595 749 376;
  • 20) 0.921 301 414 600 352 797 818 436 719 595 749 376 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 191 498 752;
  • 21) 0.842 602 829 200 705 595 636 873 439 191 498 752 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 382 997 504;
  • 22) 0.685 205 658 401 411 191 273 746 878 382 997 504 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 765 995 008;
  • 23) 0.370 411 316 802 822 382 547 493 756 765 995 008 × 2 = 0 + 0.740 822 633 605 644 765 094 987 513 531 990 016;
  • 24) 0.740 822 633 605 644 765 094 987 513 531 990 016 × 2 = 1 + 0.481 645 267 211 289 530 189 975 027 063 980 032;
  • 25) 0.481 645 267 211 289 530 189 975 027 063 980 032 × 2 = 0 + 0.963 290 534 422 579 060 379 950 054 127 960 064;
  • 26) 0.963 290 534 422 579 060 379 950 054 127 960 064 × 2 = 1 + 0.926 581 068 845 158 120 759 900 108 255 920 128;
  • 27) 0.926 581 068 845 158 120 759 900 108 255 920 128 × 2 = 1 + 0.853 162 137 690 316 241 519 800 216 511 840 256;
  • 28) 0.853 162 137 690 316 241 519 800 216 511 840 256 × 2 = 1 + 0.706 324 275 380 632 483 039 600 433 023 680 512;
  • 29) 0.706 324 275 380 632 483 039 600 433 023 680 512 × 2 = 1 + 0.412 648 550 761 264 966 079 200 866 047 361 024;
  • 30) 0.412 648 550 761 264 966 079 200 866 047 361 024 × 2 = 0 + 0.825 297 101 522 529 932 158 401 732 094 722 048;
  • 31) 0.825 297 101 522 529 932 158 401 732 094 722 048 × 2 = 1 + 0.650 594 203 045 059 864 316 803 464 189 444 096;
  • 32) 0.650 594 203 045 059 864 316 803 464 189 444 096 × 2 = 1 + 0.301 188 406 090 119 728 633 606 928 378 888 192;
  • 33) 0.301 188 406 090 119 728 633 606 928 378 888 192 × 2 = 0 + 0.602 376 812 180 239 457 267 213 856 757 776 384;
  • 34) 0.602 376 812 180 239 457 267 213 856 757 776 384 × 2 = 1 + 0.204 753 624 360 478 914 534 427 713 515 552 768;
  • 35) 0.204 753 624 360 478 914 534 427 713 515 552 768 × 2 = 0 + 0.409 507 248 720 957 829 068 855 427 031 105 536;
  • 36) 0.409 507 248 720 957 829 068 855 427 031 105 536 × 2 = 0 + 0.819 014 497 441 915 658 137 710 854 062 211 072;
  • 37) 0.819 014 497 441 915 658 137 710 854 062 211 072 × 2 = 1 + 0.638 028 994 883 831 316 275 421 708 124 422 144;
  • 38) 0.638 028 994 883 831 316 275 421 708 124 422 144 × 2 = 1 + 0.276 057 989 767 662 632 550 843 416 248 844 288;
  • 39) 0.276 057 989 767 662 632 550 843 416 248 844 288 × 2 = 0 + 0.552 115 979 535 325 265 101 686 832 497 688 576;
  • 40) 0.552 115 979 535 325 265 101 686 832 497 688 576 × 2 = 1 + 0.104 231 959 070 650 530 203 373 664 995 377 152;
  • 41) 0.104 231 959 070 650 530 203 373 664 995 377 152 × 2 = 0 + 0.208 463 918 141 301 060 406 747 329 990 754 304;
  • 42) 0.208 463 918 141 301 060 406 747 329 990 754 304 × 2 = 0 + 0.416 927 836 282 602 120 813 494 659 981 508 608;
  • 43) 0.416 927 836 282 602 120 813 494 659 981 508 608 × 2 = 0 + 0.833 855 672 565 204 241 626 989 319 963 017 216;
  • 44) 0.833 855 672 565 204 241 626 989 319 963 017 216 × 2 = 1 + 0.667 711 345 130 408 483 253 978 639 926 034 432;
  • 45) 0.667 711 345 130 408 483 253 978 639 926 034 432 × 2 = 1 + 0.335 422 690 260 816 966 507 957 279 852 068 864;
  • 46) 0.335 422 690 260 816 966 507 957 279 852 068 864 × 2 = 0 + 0.670 845 380 521 633 933 015 914 559 704 137 728;
  • 47) 0.670 845 380 521 633 933 015 914 559 704 137 728 × 2 = 1 + 0.341 690 761 043 267 866 031 829 119 408 275 456;
  • 48) 0.341 690 761 043 267 866 031 829 119 408 275 456 × 2 = 0 + 0.683 381 522 086 535 732 063 658 238 816 550 912;
  • 49) 0.683 381 522 086 535 732 063 658 238 816 550 912 × 2 = 1 + 0.366 763 044 173 071 464 127 316 477 633 101 824;
  • 50) 0.366 763 044 173 071 464 127 316 477 633 101 824 × 2 = 0 + 0.733 526 088 346 142 928 254 632 955 266 203 648;
  • 51) 0.733 526 088 346 142 928 254 632 955 266 203 648 × 2 = 1 + 0.467 052 176 692 285 856 509 265 910 532 407 296;
  • 52) 0.467 052 176 692 285 856 509 265 910 532 407 296 × 2 = 0 + 0.934 104 353 384 571 713 018 531 821 064 814 592;
  • 53) 0.934 104 353 384 571 713 018 531 821 064 814 592 × 2 = 1 + 0.868 208 706 769 143 426 037 063 642 129 629 184;
  • 54) 0.868 208 706 769 143 426 037 063 642 129 629 184 × 2 = 1 + 0.736 417 413 538 286 852 074 127 284 259 258 368;
  • 55) 0.736 417 413 538 286 852 074 127 284 259 258 368 × 2 = 1 + 0.472 834 827 076 573 704 148 254 568 518 516 736;
  • 56) 0.472 834 827 076 573 704 148 254 568 518 516 736 × 2 = 0 + 0.945 669 654 153 147 408 296 509 137 037 033 472;
  • 57) 0.945 669 654 153 147 408 296 509 137 037 033 472 × 2 = 1 + 0.891 339 308 306 294 816 593 018 274 074 066 944;
  • 58) 0.891 339 308 306 294 816 593 018 274 074 066 944 × 2 = 1 + 0.782 678 616 612 589 633 186 036 548 148 133 888;
  • 59) 0.782 678 616 612 589 633 186 036 548 148 133 888 × 2 = 1 + 0.565 357 233 225 179 266 372 073 096 296 267 776;
  • 60) 0.565 357 233 225 179 266 372 073 096 296 267 776 × 2 = 1 + 0.130 714 466 450 358 532 744 146 192 592 535 552;
  • 61) 0.130 714 466 450 358 532 744 146 192 592 535 552 × 2 = 0 + 0.261 428 932 900 717 065 488 292 385 185 071 104;
  • 62) 0.261 428 932 900 717 065 488 292 385 185 071 104 × 2 = 0 + 0.522 857 865 801 434 130 976 584 770 370 142 208;
  • 63) 0.522 857 865 801 434 130 976 584 770 370 142 208 × 2 = 1 + 0.045 715 731 602 868 261 953 169 540 740 284 416;
  • 64) 0.045 715 731 602 868 261 953 169 540 740 284 416 × 2 = 0 + 0.091 431 463 205 736 523 906 339 081 480 568 832;
  • 65) 0.091 431 463 205 736 523 906 339 081 480 568 832 × 2 = 0 + 0.182 862 926 411 473 047 812 678 162 961 137 664;
  • 66) 0.182 862 926 411 473 047 812 678 162 961 137 664 × 2 = 0 + 0.365 725 852 822 946 095 625 356 325 922 275 328;
  • 67) 0.365 725 852 822 946 095 625 356 325 922 275 328 × 2 = 0 + 0.731 451 705 645 892 191 250 712 651 844 550 656;
  • 68) 0.731 451 705 645 892 191 250 712 651 844 550 656 × 2 = 1 + 0.462 903 411 291 784 382 501 425 303 689 101 312;

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:


0.000 020 830 729 321 671 205 134 999 154 509 727(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:

0.000 020 830 729 321 671 205 134 999 154 509 727(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:


0.000 020 830 729 321 671 205 134 999 154 509 727(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 0.000 020 830 729 321 671 205 134 999 154 509 727 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