0.000 020 830 729 321 671 205 134 999 154 509 394 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 394(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
0.000 020 830 729 321 671 205 134 999 154 509 394(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: 0.000 020 830 729 321 671 205 134 999 154 509 394.

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) 0.000 020 830 729 321 671 205 134 999 154 509 394 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 018 788;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 018 788 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 037 576;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 037 576 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 075 152;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 075 152 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 150 304;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 150 304 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 300 608;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 300 608 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 601 216;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 601 216 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 202 432;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 202 432 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 404 864;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 404 864 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 809 728;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 809 728 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 619 456;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 619 456 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 238 912;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 238 912 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 870 477 824;
  • 13) 0.085 322 667 301 565 256 232 956 536 870 477 824 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 740 955 648;
  • 14) 0.170 645 334 603 130 512 465 913 073 740 955 648 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 481 911 296;
  • 15) 0.341 290 669 206 261 024 931 826 147 481 911 296 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 963 822 592;
  • 16) 0.682 581 338 412 522 049 863 652 294 963 822 592 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 927 645 184;
  • 17) 0.365 162 676 825 044 099 727 304 589 927 645 184 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 855 290 368;
  • 18) 0.730 325 353 650 088 199 454 609 179 855 290 368 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 710 580 736;
  • 19) 0.460 650 707 300 176 398 909 218 359 710 580 736 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 421 161 472;
  • 20) 0.921 301 414 600 352 797 818 436 719 421 161 472 × 2 = 1 + 0.842 602 829 200 705 595 636 873 438 842 322 944;
  • 21) 0.842 602 829 200 705 595 636 873 438 842 322 944 × 2 = 1 + 0.685 205 658 401 411 191 273 746 877 684 645 888;
  • 22) 0.685 205 658 401 411 191 273 746 877 684 645 888 × 2 = 1 + 0.370 411 316 802 822 382 547 493 755 369 291 776;
  • 23) 0.370 411 316 802 822 382 547 493 755 369 291 776 × 2 = 0 + 0.740 822 633 605 644 765 094 987 510 738 583 552;
  • 24) 0.740 822 633 605 644 765 094 987 510 738 583 552 × 2 = 1 + 0.481 645 267 211 289 530 189 975 021 477 167 104;
  • 25) 0.481 645 267 211 289 530 189 975 021 477 167 104 × 2 = 0 + 0.963 290 534 422 579 060 379 950 042 954 334 208;
  • 26) 0.963 290 534 422 579 060 379 950 042 954 334 208 × 2 = 1 + 0.926 581 068 845 158 120 759 900 085 908 668 416;
  • 27) 0.926 581 068 845 158 120 759 900 085 908 668 416 × 2 = 1 + 0.853 162 137 690 316 241 519 800 171 817 336 832;
  • 28) 0.853 162 137 690 316 241 519 800 171 817 336 832 × 2 = 1 + 0.706 324 275 380 632 483 039 600 343 634 673 664;
  • 29) 0.706 324 275 380 632 483 039 600 343 634 673 664 × 2 = 1 + 0.412 648 550 761 264 966 079 200 687 269 347 328;
  • 30) 0.412 648 550 761 264 966 079 200 687 269 347 328 × 2 = 0 + 0.825 297 101 522 529 932 158 401 374 538 694 656;
  • 31) 0.825 297 101 522 529 932 158 401 374 538 694 656 × 2 = 1 + 0.650 594 203 045 059 864 316 802 749 077 389 312;
  • 32) 0.650 594 203 045 059 864 316 802 749 077 389 312 × 2 = 1 + 0.301 188 406 090 119 728 633 605 498 154 778 624;
  • 33) 0.301 188 406 090 119 728 633 605 498 154 778 624 × 2 = 0 + 0.602 376 812 180 239 457 267 210 996 309 557 248;
  • 34) 0.602 376 812 180 239 457 267 210 996 309 557 248 × 2 = 1 + 0.204 753 624 360 478 914 534 421 992 619 114 496;
  • 35) 0.204 753 624 360 478 914 534 421 992 619 114 496 × 2 = 0 + 0.409 507 248 720 957 829 068 843 985 238 228 992;
  • 36) 0.409 507 248 720 957 829 068 843 985 238 228 992 × 2 = 0 + 0.819 014 497 441 915 658 137 687 970 476 457 984;
  • 37) 0.819 014 497 441 915 658 137 687 970 476 457 984 × 2 = 1 + 0.638 028 994 883 831 316 275 375 940 952 915 968;
  • 38) 0.638 028 994 883 831 316 275 375 940 952 915 968 × 2 = 1 + 0.276 057 989 767 662 632 550 751 881 905 831 936;
  • 39) 0.276 057 989 767 662 632 550 751 881 905 831 936 × 2 = 0 + 0.552 115 979 535 325 265 101 503 763 811 663 872;
  • 40) 0.552 115 979 535 325 265 101 503 763 811 663 872 × 2 = 1 + 0.104 231 959 070 650 530 203 007 527 623 327 744;
  • 41) 0.104 231 959 070 650 530 203 007 527 623 327 744 × 2 = 0 + 0.208 463 918 141 301 060 406 015 055 246 655 488;
  • 42) 0.208 463 918 141 301 060 406 015 055 246 655 488 × 2 = 0 + 0.416 927 836 282 602 120 812 030 110 493 310 976;
  • 43) 0.416 927 836 282 602 120 812 030 110 493 310 976 × 2 = 0 + 0.833 855 672 565 204 241 624 060 220 986 621 952;
  • 44) 0.833 855 672 565 204 241 624 060 220 986 621 952 × 2 = 1 + 0.667 711 345 130 408 483 248 120 441 973 243 904;
  • 45) 0.667 711 345 130 408 483 248 120 441 973 243 904 × 2 = 1 + 0.335 422 690 260 816 966 496 240 883 946 487 808;
  • 46) 0.335 422 690 260 816 966 496 240 883 946 487 808 × 2 = 0 + 0.670 845 380 521 633 932 992 481 767 892 975 616;
  • 47) 0.670 845 380 521 633 932 992 481 767 892 975 616 × 2 = 1 + 0.341 690 761 043 267 865 984 963 535 785 951 232;
  • 48) 0.341 690 761 043 267 865 984 963 535 785 951 232 × 2 = 0 + 0.683 381 522 086 535 731 969 927 071 571 902 464;
  • 49) 0.683 381 522 086 535 731 969 927 071 571 902 464 × 2 = 1 + 0.366 763 044 173 071 463 939 854 143 143 804 928;
  • 50) 0.366 763 044 173 071 463 939 854 143 143 804 928 × 2 = 0 + 0.733 526 088 346 142 927 879 708 286 287 609 856;
  • 51) 0.733 526 088 346 142 927 879 708 286 287 609 856 × 2 = 1 + 0.467 052 176 692 285 855 759 416 572 575 219 712;
  • 52) 0.467 052 176 692 285 855 759 416 572 575 219 712 × 2 = 0 + 0.934 104 353 384 571 711 518 833 145 150 439 424;
  • 53) 0.934 104 353 384 571 711 518 833 145 150 439 424 × 2 = 1 + 0.868 208 706 769 143 423 037 666 290 300 878 848;
  • 54) 0.868 208 706 769 143 423 037 666 290 300 878 848 × 2 = 1 + 0.736 417 413 538 286 846 075 332 580 601 757 696;
  • 55) 0.736 417 413 538 286 846 075 332 580 601 757 696 × 2 = 1 + 0.472 834 827 076 573 692 150 665 161 203 515 392;
  • 56) 0.472 834 827 076 573 692 150 665 161 203 515 392 × 2 = 0 + 0.945 669 654 153 147 384 301 330 322 407 030 784;
  • 57) 0.945 669 654 153 147 384 301 330 322 407 030 784 × 2 = 1 + 0.891 339 308 306 294 768 602 660 644 814 061 568;
  • 58) 0.891 339 308 306 294 768 602 660 644 814 061 568 × 2 = 1 + 0.782 678 616 612 589 537 205 321 289 628 123 136;
  • 59) 0.782 678 616 612 589 537 205 321 289 628 123 136 × 2 = 1 + 0.565 357 233 225 179 074 410 642 579 256 246 272;
  • 60) 0.565 357 233 225 179 074 410 642 579 256 246 272 × 2 = 1 + 0.130 714 466 450 358 148 821 285 158 512 492 544;
  • 61) 0.130 714 466 450 358 148 821 285 158 512 492 544 × 2 = 0 + 0.261 428 932 900 716 297 642 570 317 024 985 088;
  • 62) 0.261 428 932 900 716 297 642 570 317 024 985 088 × 2 = 0 + 0.522 857 865 801 432 595 285 140 634 049 970 176;
  • 63) 0.522 857 865 801 432 595 285 140 634 049 970 176 × 2 = 1 + 0.045 715 731 602 865 190 570 281 268 099 940 352;
  • 64) 0.045 715 731 602 865 190 570 281 268 099 940 352 × 2 = 0 + 0.091 431 463 205 730 381 140 562 536 199 880 704;
  • 65) 0.091 431 463 205 730 381 140 562 536 199 880 704 × 2 = 0 + 0.182 862 926 411 460 762 281 125 072 399 761 408;
  • 66) 0.182 862 926 411 460 762 281 125 072 399 761 408 × 2 = 0 + 0.365 725 852 822 921 524 562 250 144 799 522 816;
  • 67) 0.365 725 852 822 921 524 562 250 144 799 522 816 × 2 = 0 + 0.731 451 705 645 843 049 124 500 289 599 045 632;
  • 68) 0.731 451 705 645 843 049 124 500 289 599 045 632 × 2 = 1 + 0.462 903 411 291 686 098 249 000 579 198 091 264;

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:


0.000 020 830 729 321 671 205 134 999 154 509 394(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:

0.000 020 830 729 321 671 205 134 999 154 509 394(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:


0.000 020 830 729 321 671 205 134 999 154 509 394(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 0.000 020 830 729 321 671 205 134 999 154 509 394 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