0.000 020 830 729 321 671 205 134 999 154 509 660 618 41 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 41(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 41(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 41.

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 41 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 236 82;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 236 82 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 473 64;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 473 64 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 947 28;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 947 28 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 894 56;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 894 56 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 789 12;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 789 12 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 578 24;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 578 24 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 559 156 48;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 559 156 48 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 118 312 96;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 118 312 96 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 236 625 92;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 236 625 92 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 473 251 84;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 473 251 84 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 946 503 68;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 946 503 68 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 893 007 36;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 893 007 36 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 786 014 72;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 786 014 72 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 572 029 44;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 572 029 44 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 144 058 88;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 144 058 88 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 288 117 76;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 288 117 76 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 576 235 52;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 576 235 52 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 473 152 471 04;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 473 152 471 04 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 946 304 942 08;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 946 304 942 08 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 892 609 884 16;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 892 609 884 16 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 785 219 768 32;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 785 219 768 32 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 570 439 536 64;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 570 439 536 64 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 140 879 073 28;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 140 879 073 28 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 281 758 146 56;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 281 758 146 56 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 563 516 293 12;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 563 516 293 12 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 127 032 586 24;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 127 032 586 24 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 254 065 172 48;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 254 065 172 48 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 508 130 344 96;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 508 130 344 96 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 409 016 260 689 92;
  • 30) 0.412 648 550 761 264 966 079 200 830 409 016 260 689 92 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 818 032 521 379 84;
  • 31) 0.825 297 101 522 529 932 158 401 660 818 032 521 379 84 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 636 065 042 759 68;
  • 32) 0.650 594 203 045 059 864 316 803 321 636 065 042 759 68 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 272 130 085 519 36;
  • 33) 0.301 188 406 090 119 728 633 606 643 272 130 085 519 36 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 544 260 171 038 72;
  • 34) 0.602 376 812 180 239 457 267 213 286 544 260 171 038 72 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 088 520 342 077 44;
  • 35) 0.204 753 624 360 478 914 534 426 573 088 520 342 077 44 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 177 040 684 154 88;
  • 36) 0.409 507 248 720 957 829 068 853 146 177 040 684 154 88 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 354 081 368 309 76;
  • 37) 0.819 014 497 441 915 658 137 706 292 354 081 368 309 76 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 708 162 736 619 52;
  • 38) 0.638 028 994 883 831 316 275 412 584 708 162 736 619 52 × 2 = 1 + 0.276 057 989 767 662 632 550 825 169 416 325 473 239 04;
  • 39) 0.276 057 989 767 662 632 550 825 169 416 325 473 239 04 × 2 = 0 + 0.552 115 979 535 325 265 101 650 338 832 650 946 478 08;
  • 40) 0.552 115 979 535 325 265 101 650 338 832 650 946 478 08 × 2 = 1 + 0.104 231 959 070 650 530 203 300 677 665 301 892 956 16;
  • 41) 0.104 231 959 070 650 530 203 300 677 665 301 892 956 16 × 2 = 0 + 0.208 463 918 141 301 060 406 601 355 330 603 785 912 32;
  • 42) 0.208 463 918 141 301 060 406 601 355 330 603 785 912 32 × 2 = 0 + 0.416 927 836 282 602 120 813 202 710 661 207 571 824 64;
  • 43) 0.416 927 836 282 602 120 813 202 710 661 207 571 824 64 × 2 = 0 + 0.833 855 672 565 204 241 626 405 421 322 415 143 649 28;
  • 44) 0.833 855 672 565 204 241 626 405 421 322 415 143 649 28 × 2 = 1 + 0.667 711 345 130 408 483 252 810 842 644 830 287 298 56;
  • 45) 0.667 711 345 130 408 483 252 810 842 644 830 287 298 56 × 2 = 1 + 0.335 422 690 260 816 966 505 621 685 289 660 574 597 12;
  • 46) 0.335 422 690 260 816 966 505 621 685 289 660 574 597 12 × 2 = 0 + 0.670 845 380 521 633 933 011 243 370 579 321 149 194 24;
  • 47) 0.670 845 380 521 633 933 011 243 370 579 321 149 194 24 × 2 = 1 + 0.341 690 761 043 267 866 022 486 741 158 642 298 388 48;
  • 48) 0.341 690 761 043 267 866 022 486 741 158 642 298 388 48 × 2 = 0 + 0.683 381 522 086 535 732 044 973 482 317 284 596 776 96;
  • 49) 0.683 381 522 086 535 732 044 973 482 317 284 596 776 96 × 2 = 1 + 0.366 763 044 173 071 464 089 946 964 634 569 193 553 92;
  • 50) 0.366 763 044 173 071 464 089 946 964 634 569 193 553 92 × 2 = 0 + 0.733 526 088 346 142 928 179 893 929 269 138 387 107 84;
  • 51) 0.733 526 088 346 142 928 179 893 929 269 138 387 107 84 × 2 = 1 + 0.467 052 176 692 285 856 359 787 858 538 276 774 215 68;
  • 52) 0.467 052 176 692 285 856 359 787 858 538 276 774 215 68 × 2 = 0 + 0.934 104 353 384 571 712 719 575 717 076 553 548 431 36;
  • 53) 0.934 104 353 384 571 712 719 575 717 076 553 548 431 36 × 2 = 1 + 0.868 208 706 769 143 425 439 151 434 153 107 096 862 72;
  • 54) 0.868 208 706 769 143 425 439 151 434 153 107 096 862 72 × 2 = 1 + 0.736 417 413 538 286 850 878 302 868 306 214 193 725 44;
  • 55) 0.736 417 413 538 286 850 878 302 868 306 214 193 725 44 × 2 = 1 + 0.472 834 827 076 573 701 756 605 736 612 428 387 450 88;
  • 56) 0.472 834 827 076 573 701 756 605 736 612 428 387 450 88 × 2 = 0 + 0.945 669 654 153 147 403 513 211 473 224 856 774 901 76;
  • 57) 0.945 669 654 153 147 403 513 211 473 224 856 774 901 76 × 2 = 1 + 0.891 339 308 306 294 807 026 422 946 449 713 549 803 52;
  • 58) 0.891 339 308 306 294 807 026 422 946 449 713 549 803 52 × 2 = 1 + 0.782 678 616 612 589 614 052 845 892 899 427 099 607 04;
  • 59) 0.782 678 616 612 589 614 052 845 892 899 427 099 607 04 × 2 = 1 + 0.565 357 233 225 179 228 105 691 785 798 854 199 214 08;
  • 60) 0.565 357 233 225 179 228 105 691 785 798 854 199 214 08 × 2 = 1 + 0.130 714 466 450 358 456 211 383 571 597 708 398 428 16;
  • 61) 0.130 714 466 450 358 456 211 383 571 597 708 398 428 16 × 2 = 0 + 0.261 428 932 900 716 912 422 767 143 195 416 796 856 32;
  • 62) 0.261 428 932 900 716 912 422 767 143 195 416 796 856 32 × 2 = 0 + 0.522 857 865 801 433 824 845 534 286 390 833 593 712 64;
  • 63) 0.522 857 865 801 433 824 845 534 286 390 833 593 712 64 × 2 = 1 + 0.045 715 731 602 867 649 691 068 572 781 667 187 425 28;
  • 64) 0.045 715 731 602 867 649 691 068 572 781 667 187 425 28 × 2 = 0 + 0.091 431 463 205 735 299 382 137 145 563 334 374 850 56;
  • 65) 0.091 431 463 205 735 299 382 137 145 563 334 374 850 56 × 2 = 0 + 0.182 862 926 411 470 598 764 274 291 126 668 749 701 12;
  • 66) 0.182 862 926 411 470 598 764 274 291 126 668 749 701 12 × 2 = 0 + 0.365 725 852 822 941 197 528 548 582 253 337 499 402 24;
  • 67) 0.365 725 852 822 941 197 528 548 582 253 337 499 402 24 × 2 = 0 + 0.731 451 705 645 882 395 057 097 164 506 674 998 804 48;
  • 68) 0.731 451 705 645 882 395 057 097 164 506 674 998 804 48 × 2 = 1 + 0.462 903 411 291 764 790 114 194 329 013 349 997 608 96;

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 41(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 41(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 41(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 41 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