0.000 020 830 729 321 671 205 134 999 154 509 660 618 025 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 618 025(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
0.000 020 830 729 321 671 205 134 999 154 509 660 618 025(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 618 025.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 618 025 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 236 05;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 236 05 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 472 1;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 472 1 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 944 2;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 944 2 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 888 4;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 888 4 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 776 8;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 776 8 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 553 6;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 553 6 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 107 2;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 107 2 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 214 4;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 214 4 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 428 8;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 428 8 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 472 857 6;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 472 857 6 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 945 715 2;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 945 715 2 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 891 430 4;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 891 430 4 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 782 860 8;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 782 860 8 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 565 721 6;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 565 721 6 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 131 443 2;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 131 443 2 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 262 886 4;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 262 886 4 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 525 772 8;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 525 772 8 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 473 051 545 6;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 473 051 545 6 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 946 103 091 2;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 946 103 091 2 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 892 206 182 4;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 892 206 182 4 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 784 412 364 8;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 784 412 364 8 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 568 824 729 6;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 568 824 729 6 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 137 649 459 2;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 137 649 459 2 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 275 298 918 4;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 275 298 918 4 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 550 597 836 8;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 550 597 836 8 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 101 195 673 6;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 101 195 673 6 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 202 391 347 2;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 202 391 347 2 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 404 782 694 4;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 404 782 694 4 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 809 565 388 8;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 809 565 388 8 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 817 619 130 777 6;
  • 31) 0.825 297 101 522 529 932 158 401 660 817 619 130 777 6 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 635 238 261 555 2;
  • 32) 0.650 594 203 045 059 864 316 803 321 635 238 261 555 2 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 270 476 523 110 4;
  • 33) 0.301 188 406 090 119 728 633 606 643 270 476 523 110 4 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 540 953 046 220 8;
  • 34) 0.602 376 812 180 239 457 267 213 286 540 953 046 220 8 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 081 906 092 441 6;
  • 35) 0.204 753 624 360 478 914 534 426 573 081 906 092 441 6 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 163 812 184 883 2;
  • 36) 0.409 507 248 720 957 829 068 853 146 163 812 184 883 2 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 327 624 369 766 4;
  • 37) 0.819 014 497 441 915 658 137 706 292 327 624 369 766 4 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 655 248 739 532 8;
  • 38) 0.638 028 994 883 831 316 275 412 584 655 248 739 532 8 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 310 497 479 065 6;
  • 39) 0.276 057 989 767 662 632 550 825 169 310 497 479 065 6 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 620 994 958 131 2;
  • 40) 0.552 115 979 535 325 265 101 650 338 620 994 958 131 2 × 2 = 1 + 0.104 231 959 070 650 530 203 300 677 241 989 916 262 4;
  • 41) 0.104 231 959 070 650 530 203 300 677 241 989 916 262 4 × 2 = 0 + 0.208 463 918 141 301 060 406 601 354 483 979 832 524 8;
  • 42) 0.208 463 918 141 301 060 406 601 354 483 979 832 524 8 × 2 = 0 + 0.416 927 836 282 602 120 813 202 708 967 959 665 049 6;
  • 43) 0.416 927 836 282 602 120 813 202 708 967 959 665 049 6 × 2 = 0 + 0.833 855 672 565 204 241 626 405 417 935 919 330 099 2;
  • 44) 0.833 855 672 565 204 241 626 405 417 935 919 330 099 2 × 2 = 1 + 0.667 711 345 130 408 483 252 810 835 871 838 660 198 4;
  • 45) 0.667 711 345 130 408 483 252 810 835 871 838 660 198 4 × 2 = 1 + 0.335 422 690 260 816 966 505 621 671 743 677 320 396 8;
  • 46) 0.335 422 690 260 816 966 505 621 671 743 677 320 396 8 × 2 = 0 + 0.670 845 380 521 633 933 011 243 343 487 354 640 793 6;
  • 47) 0.670 845 380 521 633 933 011 243 343 487 354 640 793 6 × 2 = 1 + 0.341 690 761 043 267 866 022 486 686 974 709 281 587 2;
  • 48) 0.341 690 761 043 267 866 022 486 686 974 709 281 587 2 × 2 = 0 + 0.683 381 522 086 535 732 044 973 373 949 418 563 174 4;
  • 49) 0.683 381 522 086 535 732 044 973 373 949 418 563 174 4 × 2 = 1 + 0.366 763 044 173 071 464 089 946 747 898 837 126 348 8;
  • 50) 0.366 763 044 173 071 464 089 946 747 898 837 126 348 8 × 2 = 0 + 0.733 526 088 346 142 928 179 893 495 797 674 252 697 6;
  • 51) 0.733 526 088 346 142 928 179 893 495 797 674 252 697 6 × 2 = 1 + 0.467 052 176 692 285 856 359 786 991 595 348 505 395 2;
  • 52) 0.467 052 176 692 285 856 359 786 991 595 348 505 395 2 × 2 = 0 + 0.934 104 353 384 571 712 719 573 983 190 697 010 790 4;
  • 53) 0.934 104 353 384 571 712 719 573 983 190 697 010 790 4 × 2 = 1 + 0.868 208 706 769 143 425 439 147 966 381 394 021 580 8;
  • 54) 0.868 208 706 769 143 425 439 147 966 381 394 021 580 8 × 2 = 1 + 0.736 417 413 538 286 850 878 295 932 762 788 043 161 6;
  • 55) 0.736 417 413 538 286 850 878 295 932 762 788 043 161 6 × 2 = 1 + 0.472 834 827 076 573 701 756 591 865 525 576 086 323 2;
  • 56) 0.472 834 827 076 573 701 756 591 865 525 576 086 323 2 × 2 = 0 + 0.945 669 654 153 147 403 513 183 731 051 152 172 646 4;
  • 57) 0.945 669 654 153 147 403 513 183 731 051 152 172 646 4 × 2 = 1 + 0.891 339 308 306 294 807 026 367 462 102 304 345 292 8;
  • 58) 0.891 339 308 306 294 807 026 367 462 102 304 345 292 8 × 2 = 1 + 0.782 678 616 612 589 614 052 734 924 204 608 690 585 6;
  • 59) 0.782 678 616 612 589 614 052 734 924 204 608 690 585 6 × 2 = 1 + 0.565 357 233 225 179 228 105 469 848 409 217 381 171 2;
  • 60) 0.565 357 233 225 179 228 105 469 848 409 217 381 171 2 × 2 = 1 + 0.130 714 466 450 358 456 210 939 696 818 434 762 342 4;
  • 61) 0.130 714 466 450 358 456 210 939 696 818 434 762 342 4 × 2 = 0 + 0.261 428 932 900 716 912 421 879 393 636 869 524 684 8;
  • 62) 0.261 428 932 900 716 912 421 879 393 636 869 524 684 8 × 2 = 0 + 0.522 857 865 801 433 824 843 758 787 273 739 049 369 6;
  • 63) 0.522 857 865 801 433 824 843 758 787 273 739 049 369 6 × 2 = 1 + 0.045 715 731 602 867 649 687 517 574 547 478 098 739 2;
  • 64) 0.045 715 731 602 867 649 687 517 574 547 478 098 739 2 × 2 = 0 + 0.091 431 463 205 735 299 375 035 149 094 956 197 478 4;
  • 65) 0.091 431 463 205 735 299 375 035 149 094 956 197 478 4 × 2 = 0 + 0.182 862 926 411 470 598 750 070 298 189 912 394 956 8;
  • 66) 0.182 862 926 411 470 598 750 070 298 189 912 394 956 8 × 2 = 0 + 0.365 725 852 822 941 197 500 140 596 379 824 789 913 6;
  • 67) 0.365 725 852 822 941 197 500 140 596 379 824 789 913 6 × 2 = 0 + 0.731 451 705 645 882 395 000 281 192 759 649 579 827 2;
  • 68) 0.731 451 705 645 882 395 000 281 192 759 649 579 827 2 × 2 = 1 + 0.462 903 411 291 764 790 000 562 385 519 299 159 654 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:


0.000 020 830 729 321 671 205 134 999 154 509 660 618 025(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 618 025(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 618 025(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 0.000 020 830 729 321 671 205 134 999 154 509 660 618 025 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 0.000 020 830 729 321 671 205 134 999 154 509 660 618 041 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: My Manager Only Says “We” When Referring to “My Work”. Gemini Notebook YouTube Video of 11.13 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