0.000 020 830 729 321 671 205 134 999 154 509 748 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 748(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
0.000 020 830 729 321 671 205 134 999 154 509 748(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: 0.000 020 830 729 321 671 205 134 999 154 509 748.

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) 0.000 020 830 729 321 671 205 134 999 154 509 748 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 496;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 496 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 992;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 992 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 984;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 984 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 155 968;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 155 968 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 311 936;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 311 936 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 623 872;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 623 872 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 247 744;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 247 744 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 495 488;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 495 488 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 990 976;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 990 976 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 981 952;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 981 952 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 963 904;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 963 904 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 927 808;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 927 808 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 855 616;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 855 616 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 487 711 232;
  • 15) 0.341 290 669 206 261 024 931 826 147 487 711 232 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 975 422 464;
  • 16) 0.682 581 338 412 522 049 863 652 294 975 422 464 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 950 844 928;
  • 17) 0.365 162 676 825 044 099 727 304 589 950 844 928 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 901 689 856;
  • 18) 0.730 325 353 650 088 199 454 609 179 901 689 856 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 803 379 712;
  • 19) 0.460 650 707 300 176 398 909 218 359 803 379 712 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 606 759 424;
  • 20) 0.921 301 414 600 352 797 818 436 719 606 759 424 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 213 518 848;
  • 21) 0.842 602 829 200 705 595 636 873 439 213 518 848 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 427 037 696;
  • 22) 0.685 205 658 401 411 191 273 746 878 427 037 696 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 854 075 392;
  • 23) 0.370 411 316 802 822 382 547 493 756 854 075 392 × 2 = 0 + 0.740 822 633 605 644 765 094 987 513 708 150 784;
  • 24) 0.740 822 633 605 644 765 094 987 513 708 150 784 × 2 = 1 + 0.481 645 267 211 289 530 189 975 027 416 301 568;
  • 25) 0.481 645 267 211 289 530 189 975 027 416 301 568 × 2 = 0 + 0.963 290 534 422 579 060 379 950 054 832 603 136;
  • 26) 0.963 290 534 422 579 060 379 950 054 832 603 136 × 2 = 1 + 0.926 581 068 845 158 120 759 900 109 665 206 272;
  • 27) 0.926 581 068 845 158 120 759 900 109 665 206 272 × 2 = 1 + 0.853 162 137 690 316 241 519 800 219 330 412 544;
  • 28) 0.853 162 137 690 316 241 519 800 219 330 412 544 × 2 = 1 + 0.706 324 275 380 632 483 039 600 438 660 825 088;
  • 29) 0.706 324 275 380 632 483 039 600 438 660 825 088 × 2 = 1 + 0.412 648 550 761 264 966 079 200 877 321 650 176;
  • 30) 0.412 648 550 761 264 966 079 200 877 321 650 176 × 2 = 0 + 0.825 297 101 522 529 932 158 401 754 643 300 352;
  • 31) 0.825 297 101 522 529 932 158 401 754 643 300 352 × 2 = 1 + 0.650 594 203 045 059 864 316 803 509 286 600 704;
  • 32) 0.650 594 203 045 059 864 316 803 509 286 600 704 × 2 = 1 + 0.301 188 406 090 119 728 633 607 018 573 201 408;
  • 33) 0.301 188 406 090 119 728 633 607 018 573 201 408 × 2 = 0 + 0.602 376 812 180 239 457 267 214 037 146 402 816;
  • 34) 0.602 376 812 180 239 457 267 214 037 146 402 816 × 2 = 1 + 0.204 753 624 360 478 914 534 428 074 292 805 632;
  • 35) 0.204 753 624 360 478 914 534 428 074 292 805 632 × 2 = 0 + 0.409 507 248 720 957 829 068 856 148 585 611 264;
  • 36) 0.409 507 248 720 957 829 068 856 148 585 611 264 × 2 = 0 + 0.819 014 497 441 915 658 137 712 297 171 222 528;
  • 37) 0.819 014 497 441 915 658 137 712 297 171 222 528 × 2 = 1 + 0.638 028 994 883 831 316 275 424 594 342 445 056;
  • 38) 0.638 028 994 883 831 316 275 424 594 342 445 056 × 2 = 1 + 0.276 057 989 767 662 632 550 849 188 684 890 112;
  • 39) 0.276 057 989 767 662 632 550 849 188 684 890 112 × 2 = 0 + 0.552 115 979 535 325 265 101 698 377 369 780 224;
  • 40) 0.552 115 979 535 325 265 101 698 377 369 780 224 × 2 = 1 + 0.104 231 959 070 650 530 203 396 754 739 560 448;
  • 41) 0.104 231 959 070 650 530 203 396 754 739 560 448 × 2 = 0 + 0.208 463 918 141 301 060 406 793 509 479 120 896;
  • 42) 0.208 463 918 141 301 060 406 793 509 479 120 896 × 2 = 0 + 0.416 927 836 282 602 120 813 587 018 958 241 792;
  • 43) 0.416 927 836 282 602 120 813 587 018 958 241 792 × 2 = 0 + 0.833 855 672 565 204 241 627 174 037 916 483 584;
  • 44) 0.833 855 672 565 204 241 627 174 037 916 483 584 × 2 = 1 + 0.667 711 345 130 408 483 254 348 075 832 967 168;
  • 45) 0.667 711 345 130 408 483 254 348 075 832 967 168 × 2 = 1 + 0.335 422 690 260 816 966 508 696 151 665 934 336;
  • 46) 0.335 422 690 260 816 966 508 696 151 665 934 336 × 2 = 0 + 0.670 845 380 521 633 933 017 392 303 331 868 672;
  • 47) 0.670 845 380 521 633 933 017 392 303 331 868 672 × 2 = 1 + 0.341 690 761 043 267 866 034 784 606 663 737 344;
  • 48) 0.341 690 761 043 267 866 034 784 606 663 737 344 × 2 = 0 + 0.683 381 522 086 535 732 069 569 213 327 474 688;
  • 49) 0.683 381 522 086 535 732 069 569 213 327 474 688 × 2 = 1 + 0.366 763 044 173 071 464 139 138 426 654 949 376;
  • 50) 0.366 763 044 173 071 464 139 138 426 654 949 376 × 2 = 0 + 0.733 526 088 346 142 928 278 276 853 309 898 752;
  • 51) 0.733 526 088 346 142 928 278 276 853 309 898 752 × 2 = 1 + 0.467 052 176 692 285 856 556 553 706 619 797 504;
  • 52) 0.467 052 176 692 285 856 556 553 706 619 797 504 × 2 = 0 + 0.934 104 353 384 571 713 113 107 413 239 595 008;
  • 53) 0.934 104 353 384 571 713 113 107 413 239 595 008 × 2 = 1 + 0.868 208 706 769 143 426 226 214 826 479 190 016;
  • 54) 0.868 208 706 769 143 426 226 214 826 479 190 016 × 2 = 1 + 0.736 417 413 538 286 852 452 429 652 958 380 032;
  • 55) 0.736 417 413 538 286 852 452 429 652 958 380 032 × 2 = 1 + 0.472 834 827 076 573 704 904 859 305 916 760 064;
  • 56) 0.472 834 827 076 573 704 904 859 305 916 760 064 × 2 = 0 + 0.945 669 654 153 147 409 809 718 611 833 520 128;
  • 57) 0.945 669 654 153 147 409 809 718 611 833 520 128 × 2 = 1 + 0.891 339 308 306 294 819 619 437 223 667 040 256;
  • 58) 0.891 339 308 306 294 819 619 437 223 667 040 256 × 2 = 1 + 0.782 678 616 612 589 639 238 874 447 334 080 512;
  • 59) 0.782 678 616 612 589 639 238 874 447 334 080 512 × 2 = 1 + 0.565 357 233 225 179 278 477 748 894 668 161 024;
  • 60) 0.565 357 233 225 179 278 477 748 894 668 161 024 × 2 = 1 + 0.130 714 466 450 358 556 955 497 789 336 322 048;
  • 61) 0.130 714 466 450 358 556 955 497 789 336 322 048 × 2 = 0 + 0.261 428 932 900 717 113 910 995 578 672 644 096;
  • 62) 0.261 428 932 900 717 113 910 995 578 672 644 096 × 2 = 0 + 0.522 857 865 801 434 227 821 991 157 345 288 192;
  • 63) 0.522 857 865 801 434 227 821 991 157 345 288 192 × 2 = 1 + 0.045 715 731 602 868 455 643 982 314 690 576 384;
  • 64) 0.045 715 731 602 868 455 643 982 314 690 576 384 × 2 = 0 + 0.091 431 463 205 736 911 287 964 629 381 152 768;
  • 65) 0.091 431 463 205 736 911 287 964 629 381 152 768 × 2 = 0 + 0.182 862 926 411 473 822 575 929 258 762 305 536;
  • 66) 0.182 862 926 411 473 822 575 929 258 762 305 536 × 2 = 0 + 0.365 725 852 822 947 645 151 858 517 524 611 072;
  • 67) 0.365 725 852 822 947 645 151 858 517 524 611 072 × 2 = 0 + 0.731 451 705 645 895 290 303 717 035 049 222 144;
  • 68) 0.731 451 705 645 895 290 303 717 035 049 222 144 × 2 = 1 + 0.462 903 411 291 790 580 607 434 070 098 444 288;

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:


0.000 020 830 729 321 671 205 134 999 154 509 748(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:

0.000 020 830 729 321 671 205 134 999 154 509 748(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:


0.000 020 830 729 321 671 205 134 999 154 509 748(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 0.000 020 830 729 321 671 205 134 999 154 509 748 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