0.000 020 830 729 321 671 205 134 999 154 509 660 605 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 605 3(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
0.000 020 830 729 321 671 205 134 999 154 509 660 605 3(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 605 3.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 605 3 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 210 6;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 210 6 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 421 2;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 421 2 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 842 4;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 842 4 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 684 8;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 684 8 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 369 6;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 369 6 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 278 739 2;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 278 739 2 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 557 478 4;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 557 478 4 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 114 956 8;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 114 956 8 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 229 913 6;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 229 913 6 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 459 827 2;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 459 827 2 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 919 654 4;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 919 654 4 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 839 308 8;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 839 308 8 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 678 617 6;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 678 617 6 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 357 235 2;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 357 235 2 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 558 714 470 4;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 558 714 470 4 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 117 428 940 8;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 117 428 940 8 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 234 857 881 6;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 234 857 881 6 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 469 715 763 2;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 469 715 763 2 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 939 431 526 4;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 939 431 526 4 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 878 863 052 8;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 878 863 052 8 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 757 726 105 6;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 757 726 105 6 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 515 452 211 2;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 515 452 211 2 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 030 904 422 4;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 030 904 422 4 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 061 808 844 8;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 061 808 844 8 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 123 617 689 6;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 123 617 689 6 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 800 247 235 379 2;
  • 27) 0.926 581 068 845 158 120 759 900 103 800 247 235 379 2 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 600 494 470 758 4;
  • 28) 0.853 162 137 690 316 241 519 800 207 600 494 470 758 4 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 200 988 941 516 8;
  • 29) 0.706 324 275 380 632 483 039 600 415 200 988 941 516 8 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 401 977 883 033 6;
  • 30) 0.412 648 550 761 264 966 079 200 830 401 977 883 033 6 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 803 955 766 067 2;
  • 31) 0.825 297 101 522 529 932 158 401 660 803 955 766 067 2 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 607 911 532 134 4;
  • 32) 0.650 594 203 045 059 864 316 803 321 607 911 532 134 4 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 215 823 064 268 8;
  • 33) 0.301 188 406 090 119 728 633 606 643 215 823 064 268 8 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 431 646 128 537 6;
  • 34) 0.602 376 812 180 239 457 267 213 286 431 646 128 537 6 × 2 = 1 + 0.204 753 624 360 478 914 534 426 572 863 292 257 075 2;
  • 35) 0.204 753 624 360 478 914 534 426 572 863 292 257 075 2 × 2 = 0 + 0.409 507 248 720 957 829 068 853 145 726 584 514 150 4;
  • 36) 0.409 507 248 720 957 829 068 853 145 726 584 514 150 4 × 2 = 0 + 0.819 014 497 441 915 658 137 706 291 453 169 028 300 8;
  • 37) 0.819 014 497 441 915 658 137 706 291 453 169 028 300 8 × 2 = 1 + 0.638 028 994 883 831 316 275 412 582 906 338 056 601 6;
  • 38) 0.638 028 994 883 831 316 275 412 582 906 338 056 601 6 × 2 = 1 + 0.276 057 989 767 662 632 550 825 165 812 676 113 203 2;
  • 39) 0.276 057 989 767 662 632 550 825 165 812 676 113 203 2 × 2 = 0 + 0.552 115 979 535 325 265 101 650 331 625 352 226 406 4;
  • 40) 0.552 115 979 535 325 265 101 650 331 625 352 226 406 4 × 2 = 1 + 0.104 231 959 070 650 530 203 300 663 250 704 452 812 8;
  • 41) 0.104 231 959 070 650 530 203 300 663 250 704 452 812 8 × 2 = 0 + 0.208 463 918 141 301 060 406 601 326 501 408 905 625 6;
  • 42) 0.208 463 918 141 301 060 406 601 326 501 408 905 625 6 × 2 = 0 + 0.416 927 836 282 602 120 813 202 653 002 817 811 251 2;
  • 43) 0.416 927 836 282 602 120 813 202 653 002 817 811 251 2 × 2 = 0 + 0.833 855 672 565 204 241 626 405 306 005 635 622 502 4;
  • 44) 0.833 855 672 565 204 241 626 405 306 005 635 622 502 4 × 2 = 1 + 0.667 711 345 130 408 483 252 810 612 011 271 245 004 8;
  • 45) 0.667 711 345 130 408 483 252 810 612 011 271 245 004 8 × 2 = 1 + 0.335 422 690 260 816 966 505 621 224 022 542 490 009 6;
  • 46) 0.335 422 690 260 816 966 505 621 224 022 542 490 009 6 × 2 = 0 + 0.670 845 380 521 633 933 011 242 448 045 084 980 019 2;
  • 47) 0.670 845 380 521 633 933 011 242 448 045 084 980 019 2 × 2 = 1 + 0.341 690 761 043 267 866 022 484 896 090 169 960 038 4;
  • 48) 0.341 690 761 043 267 866 022 484 896 090 169 960 038 4 × 2 = 0 + 0.683 381 522 086 535 732 044 969 792 180 339 920 076 8;
  • 49) 0.683 381 522 086 535 732 044 969 792 180 339 920 076 8 × 2 = 1 + 0.366 763 044 173 071 464 089 939 584 360 679 840 153 6;
  • 50) 0.366 763 044 173 071 464 089 939 584 360 679 840 153 6 × 2 = 0 + 0.733 526 088 346 142 928 179 879 168 721 359 680 307 2;
  • 51) 0.733 526 088 346 142 928 179 879 168 721 359 680 307 2 × 2 = 1 + 0.467 052 176 692 285 856 359 758 337 442 719 360 614 4;
  • 52) 0.467 052 176 692 285 856 359 758 337 442 719 360 614 4 × 2 = 0 + 0.934 104 353 384 571 712 719 516 674 885 438 721 228 8;
  • 53) 0.934 104 353 384 571 712 719 516 674 885 438 721 228 8 × 2 = 1 + 0.868 208 706 769 143 425 439 033 349 770 877 442 457 6;
  • 54) 0.868 208 706 769 143 425 439 033 349 770 877 442 457 6 × 2 = 1 + 0.736 417 413 538 286 850 878 066 699 541 754 884 915 2;
  • 55) 0.736 417 413 538 286 850 878 066 699 541 754 884 915 2 × 2 = 1 + 0.472 834 827 076 573 701 756 133 399 083 509 769 830 4;
  • 56) 0.472 834 827 076 573 701 756 133 399 083 509 769 830 4 × 2 = 0 + 0.945 669 654 153 147 403 512 266 798 167 019 539 660 8;
  • 57) 0.945 669 654 153 147 403 512 266 798 167 019 539 660 8 × 2 = 1 + 0.891 339 308 306 294 807 024 533 596 334 039 079 321 6;
  • 58) 0.891 339 308 306 294 807 024 533 596 334 039 079 321 6 × 2 = 1 + 0.782 678 616 612 589 614 049 067 192 668 078 158 643 2;
  • 59) 0.782 678 616 612 589 614 049 067 192 668 078 158 643 2 × 2 = 1 + 0.565 357 233 225 179 228 098 134 385 336 156 317 286 4;
  • 60) 0.565 357 233 225 179 228 098 134 385 336 156 317 286 4 × 2 = 1 + 0.130 714 466 450 358 456 196 268 770 672 312 634 572 8;
  • 61) 0.130 714 466 450 358 456 196 268 770 672 312 634 572 8 × 2 = 0 + 0.261 428 932 900 716 912 392 537 541 344 625 269 145 6;
  • 62) 0.261 428 932 900 716 912 392 537 541 344 625 269 145 6 × 2 = 0 + 0.522 857 865 801 433 824 785 075 082 689 250 538 291 2;
  • 63) 0.522 857 865 801 433 824 785 075 082 689 250 538 291 2 × 2 = 1 + 0.045 715 731 602 867 649 570 150 165 378 501 076 582 4;
  • 64) 0.045 715 731 602 867 649 570 150 165 378 501 076 582 4 × 2 = 0 + 0.091 431 463 205 735 299 140 300 330 757 002 153 164 8;
  • 65) 0.091 431 463 205 735 299 140 300 330 757 002 153 164 8 × 2 = 0 + 0.182 862 926 411 470 598 280 600 661 514 004 306 329 6;
  • 66) 0.182 862 926 411 470 598 280 600 661 514 004 306 329 6 × 2 = 0 + 0.365 725 852 822 941 196 561 201 323 028 008 612 659 2;
  • 67) 0.365 725 852 822 941 196 561 201 323 028 008 612 659 2 × 2 = 0 + 0.731 451 705 645 882 393 122 402 646 056 017 225 318 4;
  • 68) 0.731 451 705 645 882 393 122 402 646 056 017 225 318 4 × 2 = 1 + 0.462 903 411 291 764 786 244 805 292 112 034 450 636 8;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 605 3(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 605 3(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 605 3(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 0.000 020 830 729 321 671 205 134 999 154 509 660 605 3 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