0.000 020 830 729 321 671 205 134 999 154 758 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 758(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
0.000 020 830 729 321 671 205 134 999 154 758(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: 0.000 020 830 729 321 671 205 134 999 154 758.

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) 0.000 020 830 729 321 671 205 134 999 154 758 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 516;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 516 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 619 032;
  • 3) 0.000 083 322 917 286 684 820 539 996 619 032 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 238 064;
  • 4) 0.000 166 645 834 573 369 641 079 993 238 064 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 476 128;
  • 5) 0.000 333 291 669 146 739 282 159 986 476 128 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 952 256;
  • 6) 0.000 666 583 338 293 478 564 319 972 952 256 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 904 512;
  • 7) 0.001 333 166 676 586 957 128 639 945 904 512 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 809 024;
  • 8) 0.002 666 333 353 173 914 257 279 891 809 024 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 618 048;
  • 9) 0.005 332 666 706 347 828 514 559 783 618 048 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 236 096;
  • 10) 0.010 665 333 412 695 657 029 119 567 236 096 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 472 192;
  • 11) 0.021 330 666 825 391 314 058 239 134 472 192 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 944 384;
  • 12) 0.042 661 333 650 782 628 116 478 268 944 384 × 2 = 0 + 0.085 322 667 301 565 256 232 956 537 888 768;
  • 13) 0.085 322 667 301 565 256 232 956 537 888 768 × 2 = 0 + 0.170 645 334 603 130 512 465 913 075 777 536;
  • 14) 0.170 645 334 603 130 512 465 913 075 777 536 × 2 = 0 + 0.341 290 669 206 261 024 931 826 151 555 072;
  • 15) 0.341 290 669 206 261 024 931 826 151 555 072 × 2 = 0 + 0.682 581 338 412 522 049 863 652 303 110 144;
  • 16) 0.682 581 338 412 522 049 863 652 303 110 144 × 2 = 1 + 0.365 162 676 825 044 099 727 304 606 220 288;
  • 17) 0.365 162 676 825 044 099 727 304 606 220 288 × 2 = 0 + 0.730 325 353 650 088 199 454 609 212 440 576;
  • 18) 0.730 325 353 650 088 199 454 609 212 440 576 × 2 = 1 + 0.460 650 707 300 176 398 909 218 424 881 152;
  • 19) 0.460 650 707 300 176 398 909 218 424 881 152 × 2 = 0 + 0.921 301 414 600 352 797 818 436 849 762 304;
  • 20) 0.921 301 414 600 352 797 818 436 849 762 304 × 2 = 1 + 0.842 602 829 200 705 595 636 873 699 524 608;
  • 21) 0.842 602 829 200 705 595 636 873 699 524 608 × 2 = 1 + 0.685 205 658 401 411 191 273 747 399 049 216;
  • 22) 0.685 205 658 401 411 191 273 747 399 049 216 × 2 = 1 + 0.370 411 316 802 822 382 547 494 798 098 432;
  • 23) 0.370 411 316 802 822 382 547 494 798 098 432 × 2 = 0 + 0.740 822 633 605 644 765 094 989 596 196 864;
  • 24) 0.740 822 633 605 644 765 094 989 596 196 864 × 2 = 1 + 0.481 645 267 211 289 530 189 979 192 393 728;
  • 25) 0.481 645 267 211 289 530 189 979 192 393 728 × 2 = 0 + 0.963 290 534 422 579 060 379 958 384 787 456;
  • 26) 0.963 290 534 422 579 060 379 958 384 787 456 × 2 = 1 + 0.926 581 068 845 158 120 759 916 769 574 912;
  • 27) 0.926 581 068 845 158 120 759 916 769 574 912 × 2 = 1 + 0.853 162 137 690 316 241 519 833 539 149 824;
  • 28) 0.853 162 137 690 316 241 519 833 539 149 824 × 2 = 1 + 0.706 324 275 380 632 483 039 667 078 299 648;
  • 29) 0.706 324 275 380 632 483 039 667 078 299 648 × 2 = 1 + 0.412 648 550 761 264 966 079 334 156 599 296;
  • 30) 0.412 648 550 761 264 966 079 334 156 599 296 × 2 = 0 + 0.825 297 101 522 529 932 158 668 313 198 592;
  • 31) 0.825 297 101 522 529 932 158 668 313 198 592 × 2 = 1 + 0.650 594 203 045 059 864 317 336 626 397 184;
  • 32) 0.650 594 203 045 059 864 317 336 626 397 184 × 2 = 1 + 0.301 188 406 090 119 728 634 673 252 794 368;
  • 33) 0.301 188 406 090 119 728 634 673 252 794 368 × 2 = 0 + 0.602 376 812 180 239 457 269 346 505 588 736;
  • 34) 0.602 376 812 180 239 457 269 346 505 588 736 × 2 = 1 + 0.204 753 624 360 478 914 538 693 011 177 472;
  • 35) 0.204 753 624 360 478 914 538 693 011 177 472 × 2 = 0 + 0.409 507 248 720 957 829 077 386 022 354 944;
  • 36) 0.409 507 248 720 957 829 077 386 022 354 944 × 2 = 0 + 0.819 014 497 441 915 658 154 772 044 709 888;
  • 37) 0.819 014 497 441 915 658 154 772 044 709 888 × 2 = 1 + 0.638 028 994 883 831 316 309 544 089 419 776;
  • 38) 0.638 028 994 883 831 316 309 544 089 419 776 × 2 = 1 + 0.276 057 989 767 662 632 619 088 178 839 552;
  • 39) 0.276 057 989 767 662 632 619 088 178 839 552 × 2 = 0 + 0.552 115 979 535 325 265 238 176 357 679 104;
  • 40) 0.552 115 979 535 325 265 238 176 357 679 104 × 2 = 1 + 0.104 231 959 070 650 530 476 352 715 358 208;
  • 41) 0.104 231 959 070 650 530 476 352 715 358 208 × 2 = 0 + 0.208 463 918 141 301 060 952 705 430 716 416;
  • 42) 0.208 463 918 141 301 060 952 705 430 716 416 × 2 = 0 + 0.416 927 836 282 602 121 905 410 861 432 832;
  • 43) 0.416 927 836 282 602 121 905 410 861 432 832 × 2 = 0 + 0.833 855 672 565 204 243 810 821 722 865 664;
  • 44) 0.833 855 672 565 204 243 810 821 722 865 664 × 2 = 1 + 0.667 711 345 130 408 487 621 643 445 731 328;
  • 45) 0.667 711 345 130 408 487 621 643 445 731 328 × 2 = 1 + 0.335 422 690 260 816 975 243 286 891 462 656;
  • 46) 0.335 422 690 260 816 975 243 286 891 462 656 × 2 = 0 + 0.670 845 380 521 633 950 486 573 782 925 312;
  • 47) 0.670 845 380 521 633 950 486 573 782 925 312 × 2 = 1 + 0.341 690 761 043 267 900 973 147 565 850 624;
  • 48) 0.341 690 761 043 267 900 973 147 565 850 624 × 2 = 0 + 0.683 381 522 086 535 801 946 295 131 701 248;
  • 49) 0.683 381 522 086 535 801 946 295 131 701 248 × 2 = 1 + 0.366 763 044 173 071 603 892 590 263 402 496;
  • 50) 0.366 763 044 173 071 603 892 590 263 402 496 × 2 = 0 + 0.733 526 088 346 143 207 785 180 526 804 992;
  • 51) 0.733 526 088 346 143 207 785 180 526 804 992 × 2 = 1 + 0.467 052 176 692 286 415 570 361 053 609 984;
  • 52) 0.467 052 176 692 286 415 570 361 053 609 984 × 2 = 0 + 0.934 104 353 384 572 831 140 722 107 219 968;
  • 53) 0.934 104 353 384 572 831 140 722 107 219 968 × 2 = 1 + 0.868 208 706 769 145 662 281 444 214 439 936;
  • 54) 0.868 208 706 769 145 662 281 444 214 439 936 × 2 = 1 + 0.736 417 413 538 291 324 562 888 428 879 872;
  • 55) 0.736 417 413 538 291 324 562 888 428 879 872 × 2 = 1 + 0.472 834 827 076 582 649 125 776 857 759 744;
  • 56) 0.472 834 827 076 582 649 125 776 857 759 744 × 2 = 0 + 0.945 669 654 153 165 298 251 553 715 519 488;
  • 57) 0.945 669 654 153 165 298 251 553 715 519 488 × 2 = 1 + 0.891 339 308 306 330 596 503 107 431 038 976;
  • 58) 0.891 339 308 306 330 596 503 107 431 038 976 × 2 = 1 + 0.782 678 616 612 661 193 006 214 862 077 952;
  • 59) 0.782 678 616 612 661 193 006 214 862 077 952 × 2 = 1 + 0.565 357 233 225 322 386 012 429 724 155 904;
  • 60) 0.565 357 233 225 322 386 012 429 724 155 904 × 2 = 1 + 0.130 714 466 450 644 772 024 859 448 311 808;
  • 61) 0.130 714 466 450 644 772 024 859 448 311 808 × 2 = 0 + 0.261 428 932 901 289 544 049 718 896 623 616;
  • 62) 0.261 428 932 901 289 544 049 718 896 623 616 × 2 = 0 + 0.522 857 865 802 579 088 099 437 793 247 232;
  • 63) 0.522 857 865 802 579 088 099 437 793 247 232 × 2 = 1 + 0.045 715 731 605 158 176 198 875 586 494 464;
  • 64) 0.045 715 731 605 158 176 198 875 586 494 464 × 2 = 0 + 0.091 431 463 210 316 352 397 751 172 988 928;
  • 65) 0.091 431 463 210 316 352 397 751 172 988 928 × 2 = 0 + 0.182 862 926 420 632 704 795 502 345 977 856;
  • 66) 0.182 862 926 420 632 704 795 502 345 977 856 × 2 = 0 + 0.365 725 852 841 265 409 591 004 691 955 712;
  • 67) 0.365 725 852 841 265 409 591 004 691 955 712 × 2 = 0 + 0.731 451 705 682 530 819 182 009 383 911 424;
  • 68) 0.731 451 705 682 530 819 182 009 383 911 424 × 2 = 1 + 0.462 903 411 365 061 638 364 018 767 822 848;

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:


0.000 020 830 729 321 671 205 134 999 154 758(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:

0.000 020 830 729 321 671 205 134 999 154 758(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:


0.000 020 830 729 321 671 205 134 999 154 758(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 0.000 020 830 729 321 671 205 134 999 154 758 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

Share

» Convert decimal 0.000 020 830 729 321 671 205 134 999 154 668 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 08, 2026 [August]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: August - by our visitors

» Improve your social skills: Why Do Colleagues From Other Teams Have Better Results? Gemini Notebook YouTube Video of 11.31 minutes

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