0.000 020 830 729 321 671 205 134 999 154 509 646 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 646(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
0.000 020 830 729 321 671 205 134 999 154 509 646(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: 0.000 020 830 729 321 671 205 134 999 154 509 646.

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) 0.000 020 830 729 321 671 205 134 999 154 509 646 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 292;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 292 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 584;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 584 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 168;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 168 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 336;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 336 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 308 672;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 308 672 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 617 344;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 617 344 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 234 688;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 234 688 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 469 376;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 469 376 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 938 752;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 938 752 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 877 504;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 877 504 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 755 008;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 755 008 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 510 016;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 510 016 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 020 032;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 020 032 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 040 064;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 040 064 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 080 128;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 080 128 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 944 160 256;
  • 17) 0.365 162 676 825 044 099 727 304 589 944 160 256 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 888 320 512;
  • 18) 0.730 325 353 650 088 199 454 609 179 888 320 512 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 776 641 024;
  • 19) 0.460 650 707 300 176 398 909 218 359 776 641 024 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 553 282 048;
  • 20) 0.921 301 414 600 352 797 818 436 719 553 282 048 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 106 564 096;
  • 21) 0.842 602 829 200 705 595 636 873 439 106 564 096 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 213 128 192;
  • 22) 0.685 205 658 401 411 191 273 746 878 213 128 192 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 426 256 384;
  • 23) 0.370 411 316 802 822 382 547 493 756 426 256 384 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 852 512 768;
  • 24) 0.740 822 633 605 644 765 094 987 512 852 512 768 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 705 025 536;
  • 25) 0.481 645 267 211 289 530 189 975 025 705 025 536 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 410 051 072;
  • 26) 0.963 290 534 422 579 060 379 950 051 410 051 072 × 2 = 1 + 0.926 581 068 845 158 120 759 900 102 820 102 144;
  • 27) 0.926 581 068 845 158 120 759 900 102 820 102 144 × 2 = 1 + 0.853 162 137 690 316 241 519 800 205 640 204 288;
  • 28) 0.853 162 137 690 316 241 519 800 205 640 204 288 × 2 = 1 + 0.706 324 275 380 632 483 039 600 411 280 408 576;
  • 29) 0.706 324 275 380 632 483 039 600 411 280 408 576 × 2 = 1 + 0.412 648 550 761 264 966 079 200 822 560 817 152;
  • 30) 0.412 648 550 761 264 966 079 200 822 560 817 152 × 2 = 0 + 0.825 297 101 522 529 932 158 401 645 121 634 304;
  • 31) 0.825 297 101 522 529 932 158 401 645 121 634 304 × 2 = 1 + 0.650 594 203 045 059 864 316 803 290 243 268 608;
  • 32) 0.650 594 203 045 059 864 316 803 290 243 268 608 × 2 = 1 + 0.301 188 406 090 119 728 633 606 580 486 537 216;
  • 33) 0.301 188 406 090 119 728 633 606 580 486 537 216 × 2 = 0 + 0.602 376 812 180 239 457 267 213 160 973 074 432;
  • 34) 0.602 376 812 180 239 457 267 213 160 973 074 432 × 2 = 1 + 0.204 753 624 360 478 914 534 426 321 946 148 864;
  • 35) 0.204 753 624 360 478 914 534 426 321 946 148 864 × 2 = 0 + 0.409 507 248 720 957 829 068 852 643 892 297 728;
  • 36) 0.409 507 248 720 957 829 068 852 643 892 297 728 × 2 = 0 + 0.819 014 497 441 915 658 137 705 287 784 595 456;
  • 37) 0.819 014 497 441 915 658 137 705 287 784 595 456 × 2 = 1 + 0.638 028 994 883 831 316 275 410 575 569 190 912;
  • 38) 0.638 028 994 883 831 316 275 410 575 569 190 912 × 2 = 1 + 0.276 057 989 767 662 632 550 821 151 138 381 824;
  • 39) 0.276 057 989 767 662 632 550 821 151 138 381 824 × 2 = 0 + 0.552 115 979 535 325 265 101 642 302 276 763 648;
  • 40) 0.552 115 979 535 325 265 101 642 302 276 763 648 × 2 = 1 + 0.104 231 959 070 650 530 203 284 604 553 527 296;
  • 41) 0.104 231 959 070 650 530 203 284 604 553 527 296 × 2 = 0 + 0.208 463 918 141 301 060 406 569 209 107 054 592;
  • 42) 0.208 463 918 141 301 060 406 569 209 107 054 592 × 2 = 0 + 0.416 927 836 282 602 120 813 138 418 214 109 184;
  • 43) 0.416 927 836 282 602 120 813 138 418 214 109 184 × 2 = 0 + 0.833 855 672 565 204 241 626 276 836 428 218 368;
  • 44) 0.833 855 672 565 204 241 626 276 836 428 218 368 × 2 = 1 + 0.667 711 345 130 408 483 252 553 672 856 436 736;
  • 45) 0.667 711 345 130 408 483 252 553 672 856 436 736 × 2 = 1 + 0.335 422 690 260 816 966 505 107 345 712 873 472;
  • 46) 0.335 422 690 260 816 966 505 107 345 712 873 472 × 2 = 0 + 0.670 845 380 521 633 933 010 214 691 425 746 944;
  • 47) 0.670 845 380 521 633 933 010 214 691 425 746 944 × 2 = 1 + 0.341 690 761 043 267 866 020 429 382 851 493 888;
  • 48) 0.341 690 761 043 267 866 020 429 382 851 493 888 × 2 = 0 + 0.683 381 522 086 535 732 040 858 765 702 987 776;
  • 49) 0.683 381 522 086 535 732 040 858 765 702 987 776 × 2 = 1 + 0.366 763 044 173 071 464 081 717 531 405 975 552;
  • 50) 0.366 763 044 173 071 464 081 717 531 405 975 552 × 2 = 0 + 0.733 526 088 346 142 928 163 435 062 811 951 104;
  • 51) 0.733 526 088 346 142 928 163 435 062 811 951 104 × 2 = 1 + 0.467 052 176 692 285 856 326 870 125 623 902 208;
  • 52) 0.467 052 176 692 285 856 326 870 125 623 902 208 × 2 = 0 + 0.934 104 353 384 571 712 653 740 251 247 804 416;
  • 53) 0.934 104 353 384 571 712 653 740 251 247 804 416 × 2 = 1 + 0.868 208 706 769 143 425 307 480 502 495 608 832;
  • 54) 0.868 208 706 769 143 425 307 480 502 495 608 832 × 2 = 1 + 0.736 417 413 538 286 850 614 961 004 991 217 664;
  • 55) 0.736 417 413 538 286 850 614 961 004 991 217 664 × 2 = 1 + 0.472 834 827 076 573 701 229 922 009 982 435 328;
  • 56) 0.472 834 827 076 573 701 229 922 009 982 435 328 × 2 = 0 + 0.945 669 654 153 147 402 459 844 019 964 870 656;
  • 57) 0.945 669 654 153 147 402 459 844 019 964 870 656 × 2 = 1 + 0.891 339 308 306 294 804 919 688 039 929 741 312;
  • 58) 0.891 339 308 306 294 804 919 688 039 929 741 312 × 2 = 1 + 0.782 678 616 612 589 609 839 376 079 859 482 624;
  • 59) 0.782 678 616 612 589 609 839 376 079 859 482 624 × 2 = 1 + 0.565 357 233 225 179 219 678 752 159 718 965 248;
  • 60) 0.565 357 233 225 179 219 678 752 159 718 965 248 × 2 = 1 + 0.130 714 466 450 358 439 357 504 319 437 930 496;
  • 61) 0.130 714 466 450 358 439 357 504 319 437 930 496 × 2 = 0 + 0.261 428 932 900 716 878 715 008 638 875 860 992;
  • 62) 0.261 428 932 900 716 878 715 008 638 875 860 992 × 2 = 0 + 0.522 857 865 801 433 757 430 017 277 751 721 984;
  • 63) 0.522 857 865 801 433 757 430 017 277 751 721 984 × 2 = 1 + 0.045 715 731 602 867 514 860 034 555 503 443 968;
  • 64) 0.045 715 731 602 867 514 860 034 555 503 443 968 × 2 = 0 + 0.091 431 463 205 735 029 720 069 111 006 887 936;
  • 65) 0.091 431 463 205 735 029 720 069 111 006 887 936 × 2 = 0 + 0.182 862 926 411 470 059 440 138 222 013 775 872;
  • 66) 0.182 862 926 411 470 059 440 138 222 013 775 872 × 2 = 0 + 0.365 725 852 822 940 118 880 276 444 027 551 744;
  • 67) 0.365 725 852 822 940 118 880 276 444 027 551 744 × 2 = 0 + 0.731 451 705 645 880 237 760 552 888 055 103 488;
  • 68) 0.731 451 705 645 880 237 760 552 888 055 103 488 × 2 = 1 + 0.462 903 411 291 760 475 521 105 776 110 206 976;

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:


0.000 020 830 729 321 671 205 134 999 154 509 646(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:

0.000 020 830 729 321 671 205 134 999 154 509 646(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:


0.000 020 830 729 321 671 205 134 999 154 509 646(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 0.000 020 830 729 321 671 205 134 999 154 509 646 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