0.000 020 830 729 321 671 205 134 999 154 504 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 504 9(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
0.000 020 830 729 321 671 205 134 999 154 504 9(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: 0.000 020 830 729 321 671 205 134 999 154 504 9.

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) 0.000 020 830 729 321 671 205 134 999 154 504 9 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 009 8;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 009 8 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 019 6;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 019 6 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 039 2;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 039 2 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 078 4;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 078 4 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 156 8;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 156 8 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 313 6;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 313 6 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 776 627 2;
  • 8) 0.002 666 333 353 173 914 257 279 891 776 627 2 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 553 254 4;
  • 9) 0.005 332 666 706 347 828 514 559 783 553 254 4 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 106 508 8;
  • 10) 0.010 665 333 412 695 657 029 119 567 106 508 8 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 213 017 6;
  • 11) 0.021 330 666 825 391 314 058 239 134 213 017 6 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 426 035 2;
  • 12) 0.042 661 333 650 782 628 116 478 268 426 035 2 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 852 070 4;
  • 13) 0.085 322 667 301 565 256 232 956 536 852 070 4 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 704 140 8;
  • 14) 0.170 645 334 603 130 512 465 913 073 704 140 8 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 408 281 6;
  • 15) 0.341 290 669 206 261 024 931 826 147 408 281 6 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 816 563 2;
  • 16) 0.682 581 338 412 522 049 863 652 294 816 563 2 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 633 126 4;
  • 17) 0.365 162 676 825 044 099 727 304 589 633 126 4 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 266 252 8;
  • 18) 0.730 325 353 650 088 199 454 609 179 266 252 8 × 2 = 1 + 0.460 650 707 300 176 398 909 218 358 532 505 6;
  • 19) 0.460 650 707 300 176 398 909 218 358 532 505 6 × 2 = 0 + 0.921 301 414 600 352 797 818 436 717 065 011 2;
  • 20) 0.921 301 414 600 352 797 818 436 717 065 011 2 × 2 = 1 + 0.842 602 829 200 705 595 636 873 434 130 022 4;
  • 21) 0.842 602 829 200 705 595 636 873 434 130 022 4 × 2 = 1 + 0.685 205 658 401 411 191 273 746 868 260 044 8;
  • 22) 0.685 205 658 401 411 191 273 746 868 260 044 8 × 2 = 1 + 0.370 411 316 802 822 382 547 493 736 520 089 6;
  • 23) 0.370 411 316 802 822 382 547 493 736 520 089 6 × 2 = 0 + 0.740 822 633 605 644 765 094 987 473 040 179 2;
  • 24) 0.740 822 633 605 644 765 094 987 473 040 179 2 × 2 = 1 + 0.481 645 267 211 289 530 189 974 946 080 358 4;
  • 25) 0.481 645 267 211 289 530 189 974 946 080 358 4 × 2 = 0 + 0.963 290 534 422 579 060 379 949 892 160 716 8;
  • 26) 0.963 290 534 422 579 060 379 949 892 160 716 8 × 2 = 1 + 0.926 581 068 845 158 120 759 899 784 321 433 6;
  • 27) 0.926 581 068 845 158 120 759 899 784 321 433 6 × 2 = 1 + 0.853 162 137 690 316 241 519 799 568 642 867 2;
  • 28) 0.853 162 137 690 316 241 519 799 568 642 867 2 × 2 = 1 + 0.706 324 275 380 632 483 039 599 137 285 734 4;
  • 29) 0.706 324 275 380 632 483 039 599 137 285 734 4 × 2 = 1 + 0.412 648 550 761 264 966 079 198 274 571 468 8;
  • 30) 0.412 648 550 761 264 966 079 198 274 571 468 8 × 2 = 0 + 0.825 297 101 522 529 932 158 396 549 142 937 6;
  • 31) 0.825 297 101 522 529 932 158 396 549 142 937 6 × 2 = 1 + 0.650 594 203 045 059 864 316 793 098 285 875 2;
  • 32) 0.650 594 203 045 059 864 316 793 098 285 875 2 × 2 = 1 + 0.301 188 406 090 119 728 633 586 196 571 750 4;
  • 33) 0.301 188 406 090 119 728 633 586 196 571 750 4 × 2 = 0 + 0.602 376 812 180 239 457 267 172 393 143 500 8;
  • 34) 0.602 376 812 180 239 457 267 172 393 143 500 8 × 2 = 1 + 0.204 753 624 360 478 914 534 344 786 287 001 6;
  • 35) 0.204 753 624 360 478 914 534 344 786 287 001 6 × 2 = 0 + 0.409 507 248 720 957 829 068 689 572 574 003 2;
  • 36) 0.409 507 248 720 957 829 068 689 572 574 003 2 × 2 = 0 + 0.819 014 497 441 915 658 137 379 145 148 006 4;
  • 37) 0.819 014 497 441 915 658 137 379 145 148 006 4 × 2 = 1 + 0.638 028 994 883 831 316 274 758 290 296 012 8;
  • 38) 0.638 028 994 883 831 316 274 758 290 296 012 8 × 2 = 1 + 0.276 057 989 767 662 632 549 516 580 592 025 6;
  • 39) 0.276 057 989 767 662 632 549 516 580 592 025 6 × 2 = 0 + 0.552 115 979 535 325 265 099 033 161 184 051 2;
  • 40) 0.552 115 979 535 325 265 099 033 161 184 051 2 × 2 = 1 + 0.104 231 959 070 650 530 198 066 322 368 102 4;
  • 41) 0.104 231 959 070 650 530 198 066 322 368 102 4 × 2 = 0 + 0.208 463 918 141 301 060 396 132 644 736 204 8;
  • 42) 0.208 463 918 141 301 060 396 132 644 736 204 8 × 2 = 0 + 0.416 927 836 282 602 120 792 265 289 472 409 6;
  • 43) 0.416 927 836 282 602 120 792 265 289 472 409 6 × 2 = 0 + 0.833 855 672 565 204 241 584 530 578 944 819 2;
  • 44) 0.833 855 672 565 204 241 584 530 578 944 819 2 × 2 = 1 + 0.667 711 345 130 408 483 169 061 157 889 638 4;
  • 45) 0.667 711 345 130 408 483 169 061 157 889 638 4 × 2 = 1 + 0.335 422 690 260 816 966 338 122 315 779 276 8;
  • 46) 0.335 422 690 260 816 966 338 122 315 779 276 8 × 2 = 0 + 0.670 845 380 521 633 932 676 244 631 558 553 6;
  • 47) 0.670 845 380 521 633 932 676 244 631 558 553 6 × 2 = 1 + 0.341 690 761 043 267 865 352 489 263 117 107 2;
  • 48) 0.341 690 761 043 267 865 352 489 263 117 107 2 × 2 = 0 + 0.683 381 522 086 535 730 704 978 526 234 214 4;
  • 49) 0.683 381 522 086 535 730 704 978 526 234 214 4 × 2 = 1 + 0.366 763 044 173 071 461 409 957 052 468 428 8;
  • 50) 0.366 763 044 173 071 461 409 957 052 468 428 8 × 2 = 0 + 0.733 526 088 346 142 922 819 914 104 936 857 6;
  • 51) 0.733 526 088 346 142 922 819 914 104 936 857 6 × 2 = 1 + 0.467 052 176 692 285 845 639 828 209 873 715 2;
  • 52) 0.467 052 176 692 285 845 639 828 209 873 715 2 × 2 = 0 + 0.934 104 353 384 571 691 279 656 419 747 430 4;
  • 53) 0.934 104 353 384 571 691 279 656 419 747 430 4 × 2 = 1 + 0.868 208 706 769 143 382 559 312 839 494 860 8;
  • 54) 0.868 208 706 769 143 382 559 312 839 494 860 8 × 2 = 1 + 0.736 417 413 538 286 765 118 625 678 989 721 6;
  • 55) 0.736 417 413 538 286 765 118 625 678 989 721 6 × 2 = 1 + 0.472 834 827 076 573 530 237 251 357 979 443 2;
  • 56) 0.472 834 827 076 573 530 237 251 357 979 443 2 × 2 = 0 + 0.945 669 654 153 147 060 474 502 715 958 886 4;
  • 57) 0.945 669 654 153 147 060 474 502 715 958 886 4 × 2 = 1 + 0.891 339 308 306 294 120 949 005 431 917 772 8;
  • 58) 0.891 339 308 306 294 120 949 005 431 917 772 8 × 2 = 1 + 0.782 678 616 612 588 241 898 010 863 835 545 6;
  • 59) 0.782 678 616 612 588 241 898 010 863 835 545 6 × 2 = 1 + 0.565 357 233 225 176 483 796 021 727 671 091 2;
  • 60) 0.565 357 233 225 176 483 796 021 727 671 091 2 × 2 = 1 + 0.130 714 466 450 352 967 592 043 455 342 182 4;
  • 61) 0.130 714 466 450 352 967 592 043 455 342 182 4 × 2 = 0 + 0.261 428 932 900 705 935 184 086 910 684 364 8;
  • 62) 0.261 428 932 900 705 935 184 086 910 684 364 8 × 2 = 0 + 0.522 857 865 801 411 870 368 173 821 368 729 6;
  • 63) 0.522 857 865 801 411 870 368 173 821 368 729 6 × 2 = 1 + 0.045 715 731 602 823 740 736 347 642 737 459 2;
  • 64) 0.045 715 731 602 823 740 736 347 642 737 459 2 × 2 = 0 + 0.091 431 463 205 647 481 472 695 285 474 918 4;
  • 65) 0.091 431 463 205 647 481 472 695 285 474 918 4 × 2 = 0 + 0.182 862 926 411 294 962 945 390 570 949 836 8;
  • 66) 0.182 862 926 411 294 962 945 390 570 949 836 8 × 2 = 0 + 0.365 725 852 822 589 925 890 781 141 899 673 6;
  • 67) 0.365 725 852 822 589 925 890 781 141 899 673 6 × 2 = 0 + 0.731 451 705 645 179 851 781 562 283 799 347 2;
  • 68) 0.731 451 705 645 179 851 781 562 283 799 347 2 × 2 = 1 + 0.462 903 411 290 359 703 563 124 567 598 694 4;

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:


0.000 020 830 729 321 671 205 134 999 154 504 9(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:

0.000 020 830 729 321 671 205 134 999 154 504 9(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:


0.000 020 830 729 321 671 205 134 999 154 504 9(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 0.000 020 830 729 321 671 205 134 999 154 504 9 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 0.000 020 830 729 321 671 205 134 999 154 502 8 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