0.000 020 830 729 321 671 205 134 999 154 509 397 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 397(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
0.000 020 830 729 321 671 205 134 999 154 509 397(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: 0.000 020 830 729 321 671 205 134 999 154 509 397.

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) 0.000 020 830 729 321 671 205 134 999 154 509 397 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 018 794;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 018 794 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 037 588;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 037 588 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 075 176;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 075 176 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 150 352;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 150 352 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 300 704;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 300 704 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 601 408;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 601 408 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 202 816;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 202 816 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 405 632;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 405 632 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 811 264;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 811 264 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 622 528;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 622 528 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 245 056;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 245 056 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 870 490 112;
  • 13) 0.085 322 667 301 565 256 232 956 536 870 490 112 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 740 980 224;
  • 14) 0.170 645 334 603 130 512 465 913 073 740 980 224 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 481 960 448;
  • 15) 0.341 290 669 206 261 024 931 826 147 481 960 448 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 963 920 896;
  • 16) 0.682 581 338 412 522 049 863 652 294 963 920 896 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 927 841 792;
  • 17) 0.365 162 676 825 044 099 727 304 589 927 841 792 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 855 683 584;
  • 18) 0.730 325 353 650 088 199 454 609 179 855 683 584 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 711 367 168;
  • 19) 0.460 650 707 300 176 398 909 218 359 711 367 168 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 422 734 336;
  • 20) 0.921 301 414 600 352 797 818 436 719 422 734 336 × 2 = 1 + 0.842 602 829 200 705 595 636 873 438 845 468 672;
  • 21) 0.842 602 829 200 705 595 636 873 438 845 468 672 × 2 = 1 + 0.685 205 658 401 411 191 273 746 877 690 937 344;
  • 22) 0.685 205 658 401 411 191 273 746 877 690 937 344 × 2 = 1 + 0.370 411 316 802 822 382 547 493 755 381 874 688;
  • 23) 0.370 411 316 802 822 382 547 493 755 381 874 688 × 2 = 0 + 0.740 822 633 605 644 765 094 987 510 763 749 376;
  • 24) 0.740 822 633 605 644 765 094 987 510 763 749 376 × 2 = 1 + 0.481 645 267 211 289 530 189 975 021 527 498 752;
  • 25) 0.481 645 267 211 289 530 189 975 021 527 498 752 × 2 = 0 + 0.963 290 534 422 579 060 379 950 043 054 997 504;
  • 26) 0.963 290 534 422 579 060 379 950 043 054 997 504 × 2 = 1 + 0.926 581 068 845 158 120 759 900 086 109 995 008;
  • 27) 0.926 581 068 845 158 120 759 900 086 109 995 008 × 2 = 1 + 0.853 162 137 690 316 241 519 800 172 219 990 016;
  • 28) 0.853 162 137 690 316 241 519 800 172 219 990 016 × 2 = 1 + 0.706 324 275 380 632 483 039 600 344 439 980 032;
  • 29) 0.706 324 275 380 632 483 039 600 344 439 980 032 × 2 = 1 + 0.412 648 550 761 264 966 079 200 688 879 960 064;
  • 30) 0.412 648 550 761 264 966 079 200 688 879 960 064 × 2 = 0 + 0.825 297 101 522 529 932 158 401 377 759 920 128;
  • 31) 0.825 297 101 522 529 932 158 401 377 759 920 128 × 2 = 1 + 0.650 594 203 045 059 864 316 802 755 519 840 256;
  • 32) 0.650 594 203 045 059 864 316 802 755 519 840 256 × 2 = 1 + 0.301 188 406 090 119 728 633 605 511 039 680 512;
  • 33) 0.301 188 406 090 119 728 633 605 511 039 680 512 × 2 = 0 + 0.602 376 812 180 239 457 267 211 022 079 361 024;
  • 34) 0.602 376 812 180 239 457 267 211 022 079 361 024 × 2 = 1 + 0.204 753 624 360 478 914 534 422 044 158 722 048;
  • 35) 0.204 753 624 360 478 914 534 422 044 158 722 048 × 2 = 0 + 0.409 507 248 720 957 829 068 844 088 317 444 096;
  • 36) 0.409 507 248 720 957 829 068 844 088 317 444 096 × 2 = 0 + 0.819 014 497 441 915 658 137 688 176 634 888 192;
  • 37) 0.819 014 497 441 915 658 137 688 176 634 888 192 × 2 = 1 + 0.638 028 994 883 831 316 275 376 353 269 776 384;
  • 38) 0.638 028 994 883 831 316 275 376 353 269 776 384 × 2 = 1 + 0.276 057 989 767 662 632 550 752 706 539 552 768;
  • 39) 0.276 057 989 767 662 632 550 752 706 539 552 768 × 2 = 0 + 0.552 115 979 535 325 265 101 505 413 079 105 536;
  • 40) 0.552 115 979 535 325 265 101 505 413 079 105 536 × 2 = 1 + 0.104 231 959 070 650 530 203 010 826 158 211 072;
  • 41) 0.104 231 959 070 650 530 203 010 826 158 211 072 × 2 = 0 + 0.208 463 918 141 301 060 406 021 652 316 422 144;
  • 42) 0.208 463 918 141 301 060 406 021 652 316 422 144 × 2 = 0 + 0.416 927 836 282 602 120 812 043 304 632 844 288;
  • 43) 0.416 927 836 282 602 120 812 043 304 632 844 288 × 2 = 0 + 0.833 855 672 565 204 241 624 086 609 265 688 576;
  • 44) 0.833 855 672 565 204 241 624 086 609 265 688 576 × 2 = 1 + 0.667 711 345 130 408 483 248 173 218 531 377 152;
  • 45) 0.667 711 345 130 408 483 248 173 218 531 377 152 × 2 = 1 + 0.335 422 690 260 816 966 496 346 437 062 754 304;
  • 46) 0.335 422 690 260 816 966 496 346 437 062 754 304 × 2 = 0 + 0.670 845 380 521 633 932 992 692 874 125 508 608;
  • 47) 0.670 845 380 521 633 932 992 692 874 125 508 608 × 2 = 1 + 0.341 690 761 043 267 865 985 385 748 251 017 216;
  • 48) 0.341 690 761 043 267 865 985 385 748 251 017 216 × 2 = 0 + 0.683 381 522 086 535 731 970 771 496 502 034 432;
  • 49) 0.683 381 522 086 535 731 970 771 496 502 034 432 × 2 = 1 + 0.366 763 044 173 071 463 941 542 993 004 068 864;
  • 50) 0.366 763 044 173 071 463 941 542 993 004 068 864 × 2 = 0 + 0.733 526 088 346 142 927 883 085 986 008 137 728;
  • 51) 0.733 526 088 346 142 927 883 085 986 008 137 728 × 2 = 1 + 0.467 052 176 692 285 855 766 171 972 016 275 456;
  • 52) 0.467 052 176 692 285 855 766 171 972 016 275 456 × 2 = 0 + 0.934 104 353 384 571 711 532 343 944 032 550 912;
  • 53) 0.934 104 353 384 571 711 532 343 944 032 550 912 × 2 = 1 + 0.868 208 706 769 143 423 064 687 888 065 101 824;
  • 54) 0.868 208 706 769 143 423 064 687 888 065 101 824 × 2 = 1 + 0.736 417 413 538 286 846 129 375 776 130 203 648;
  • 55) 0.736 417 413 538 286 846 129 375 776 130 203 648 × 2 = 1 + 0.472 834 827 076 573 692 258 751 552 260 407 296;
  • 56) 0.472 834 827 076 573 692 258 751 552 260 407 296 × 2 = 0 + 0.945 669 654 153 147 384 517 503 104 520 814 592;
  • 57) 0.945 669 654 153 147 384 517 503 104 520 814 592 × 2 = 1 + 0.891 339 308 306 294 769 035 006 209 041 629 184;
  • 58) 0.891 339 308 306 294 769 035 006 209 041 629 184 × 2 = 1 + 0.782 678 616 612 589 538 070 012 418 083 258 368;
  • 59) 0.782 678 616 612 589 538 070 012 418 083 258 368 × 2 = 1 + 0.565 357 233 225 179 076 140 024 836 166 516 736;
  • 60) 0.565 357 233 225 179 076 140 024 836 166 516 736 × 2 = 1 + 0.130 714 466 450 358 152 280 049 672 333 033 472;
  • 61) 0.130 714 466 450 358 152 280 049 672 333 033 472 × 2 = 0 + 0.261 428 932 900 716 304 560 099 344 666 066 944;
  • 62) 0.261 428 932 900 716 304 560 099 344 666 066 944 × 2 = 0 + 0.522 857 865 801 432 609 120 198 689 332 133 888;
  • 63) 0.522 857 865 801 432 609 120 198 689 332 133 888 × 2 = 1 + 0.045 715 731 602 865 218 240 397 378 664 267 776;
  • 64) 0.045 715 731 602 865 218 240 397 378 664 267 776 × 2 = 0 + 0.091 431 463 205 730 436 480 794 757 328 535 552;
  • 65) 0.091 431 463 205 730 436 480 794 757 328 535 552 × 2 = 0 + 0.182 862 926 411 460 872 961 589 514 657 071 104;
  • 66) 0.182 862 926 411 460 872 961 589 514 657 071 104 × 2 = 0 + 0.365 725 852 822 921 745 923 179 029 314 142 208;
  • 67) 0.365 725 852 822 921 745 923 179 029 314 142 208 × 2 = 0 + 0.731 451 705 645 843 491 846 358 058 628 284 416;
  • 68) 0.731 451 705 645 843 491 846 358 058 628 284 416 × 2 = 1 + 0.462 903 411 291 686 983 692 716 117 256 568 832;

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:


0.000 020 830 729 321 671 205 134 999 154 509 397(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:

0.000 020 830 729 321 671 205 134 999 154 509 397(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:


0.000 020 830 729 321 671 205 134 999 154 509 397(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 0.000 020 830 729 321 671 205 134 999 154 509 397 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