0.000 020 830 729 321 671 205 134 999 154 509 660 626 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 626 6(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
0.000 020 830 729 321 671 205 134 999 154 509 660 626 6(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 626 6.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 626 6 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 253 2;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 253 2 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 506 4;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 506 4 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 285 012 8;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 285 012 8 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 570 025 6;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 570 025 6 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 140 051 2;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 140 051 2 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 280 102 4;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 280 102 4 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 560 204 8;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 560 204 8 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 120 409 6;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 120 409 6 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 240 819 2;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 240 819 2 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 481 638 4;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 481 638 4 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 963 276 8;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 963 276 8 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 926 553 6;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 926 553 6 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 853 107 2;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 853 107 2 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 706 214 4;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 706 214 4 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 412 428 8;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 412 428 8 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 824 857 6;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 824 857 6 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 237 649 715 2;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 237 649 715 2 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 475 299 430 4;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 475 299 430 4 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 950 598 860 8;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 950 598 860 8 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 901 197 721 6;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 901 197 721 6 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 802 395 443 2;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 802 395 443 2 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 604 790 886 4;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 604 790 886 4 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 209 581 772 8;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 209 581 772 8 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 419 163 545 6;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 419 163 545 6 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 838 327 091 2;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 838 327 091 2 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 676 654 182 4;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 676 654 182 4 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 603 353 308 364 8;
  • 28) 0.853 162 137 690 316 241 519 800 207 603 353 308 364 8 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 206 706 616 729 6;
  • 29) 0.706 324 275 380 632 483 039 600 415 206 706 616 729 6 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 413 413 233 459 2;
  • 30) 0.412 648 550 761 264 966 079 200 830 413 413 233 459 2 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 826 826 466 918 4;
  • 31) 0.825 297 101 522 529 932 158 401 660 826 826 466 918 4 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 653 652 933 836 8;
  • 32) 0.650 594 203 045 059 864 316 803 321 653 652 933 836 8 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 307 305 867 673 6;
  • 33) 0.301 188 406 090 119 728 633 606 643 307 305 867 673 6 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 614 611 735 347 2;
  • 34) 0.602 376 812 180 239 457 267 213 286 614 611 735 347 2 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 229 223 470 694 4;
  • 35) 0.204 753 624 360 478 914 534 426 573 229 223 470 694 4 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 458 446 941 388 8;
  • 36) 0.409 507 248 720 957 829 068 853 146 458 446 941 388 8 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 916 893 882 777 6;
  • 37) 0.819 014 497 441 915 658 137 706 292 916 893 882 777 6 × 2 = 1 + 0.638 028 994 883 831 316 275 412 585 833 787 765 555 2;
  • 38) 0.638 028 994 883 831 316 275 412 585 833 787 765 555 2 × 2 = 1 + 0.276 057 989 767 662 632 550 825 171 667 575 531 110 4;
  • 39) 0.276 057 989 767 662 632 550 825 171 667 575 531 110 4 × 2 = 0 + 0.552 115 979 535 325 265 101 650 343 335 151 062 220 8;
  • 40) 0.552 115 979 535 325 265 101 650 343 335 151 062 220 8 × 2 = 1 + 0.104 231 959 070 650 530 203 300 686 670 302 124 441 6;
  • 41) 0.104 231 959 070 650 530 203 300 686 670 302 124 441 6 × 2 = 0 + 0.208 463 918 141 301 060 406 601 373 340 604 248 883 2;
  • 42) 0.208 463 918 141 301 060 406 601 373 340 604 248 883 2 × 2 = 0 + 0.416 927 836 282 602 120 813 202 746 681 208 497 766 4;
  • 43) 0.416 927 836 282 602 120 813 202 746 681 208 497 766 4 × 2 = 0 + 0.833 855 672 565 204 241 626 405 493 362 416 995 532 8;
  • 44) 0.833 855 672 565 204 241 626 405 493 362 416 995 532 8 × 2 = 1 + 0.667 711 345 130 408 483 252 810 986 724 833 991 065 6;
  • 45) 0.667 711 345 130 408 483 252 810 986 724 833 991 065 6 × 2 = 1 + 0.335 422 690 260 816 966 505 621 973 449 667 982 131 2;
  • 46) 0.335 422 690 260 816 966 505 621 973 449 667 982 131 2 × 2 = 0 + 0.670 845 380 521 633 933 011 243 946 899 335 964 262 4;
  • 47) 0.670 845 380 521 633 933 011 243 946 899 335 964 262 4 × 2 = 1 + 0.341 690 761 043 267 866 022 487 893 798 671 928 524 8;
  • 48) 0.341 690 761 043 267 866 022 487 893 798 671 928 524 8 × 2 = 0 + 0.683 381 522 086 535 732 044 975 787 597 343 857 049 6;
  • 49) 0.683 381 522 086 535 732 044 975 787 597 343 857 049 6 × 2 = 1 + 0.366 763 044 173 071 464 089 951 575 194 687 714 099 2;
  • 50) 0.366 763 044 173 071 464 089 951 575 194 687 714 099 2 × 2 = 0 + 0.733 526 088 346 142 928 179 903 150 389 375 428 198 4;
  • 51) 0.733 526 088 346 142 928 179 903 150 389 375 428 198 4 × 2 = 1 + 0.467 052 176 692 285 856 359 806 300 778 750 856 396 8;
  • 52) 0.467 052 176 692 285 856 359 806 300 778 750 856 396 8 × 2 = 0 + 0.934 104 353 384 571 712 719 612 601 557 501 712 793 6;
  • 53) 0.934 104 353 384 571 712 719 612 601 557 501 712 793 6 × 2 = 1 + 0.868 208 706 769 143 425 439 225 203 115 003 425 587 2;
  • 54) 0.868 208 706 769 143 425 439 225 203 115 003 425 587 2 × 2 = 1 + 0.736 417 413 538 286 850 878 450 406 230 006 851 174 4;
  • 55) 0.736 417 413 538 286 850 878 450 406 230 006 851 174 4 × 2 = 1 + 0.472 834 827 076 573 701 756 900 812 460 013 702 348 8;
  • 56) 0.472 834 827 076 573 701 756 900 812 460 013 702 348 8 × 2 = 0 + 0.945 669 654 153 147 403 513 801 624 920 027 404 697 6;
  • 57) 0.945 669 654 153 147 403 513 801 624 920 027 404 697 6 × 2 = 1 + 0.891 339 308 306 294 807 027 603 249 840 054 809 395 2;
  • 58) 0.891 339 308 306 294 807 027 603 249 840 054 809 395 2 × 2 = 1 + 0.782 678 616 612 589 614 055 206 499 680 109 618 790 4;
  • 59) 0.782 678 616 612 589 614 055 206 499 680 109 618 790 4 × 2 = 1 + 0.565 357 233 225 179 228 110 412 999 360 219 237 580 8;
  • 60) 0.565 357 233 225 179 228 110 412 999 360 219 237 580 8 × 2 = 1 + 0.130 714 466 450 358 456 220 825 998 720 438 475 161 6;
  • 61) 0.130 714 466 450 358 456 220 825 998 720 438 475 161 6 × 2 = 0 + 0.261 428 932 900 716 912 441 651 997 440 876 950 323 2;
  • 62) 0.261 428 932 900 716 912 441 651 997 440 876 950 323 2 × 2 = 0 + 0.522 857 865 801 433 824 883 303 994 881 753 900 646 4;
  • 63) 0.522 857 865 801 433 824 883 303 994 881 753 900 646 4 × 2 = 1 + 0.045 715 731 602 867 649 766 607 989 763 507 801 292 8;
  • 64) 0.045 715 731 602 867 649 766 607 989 763 507 801 292 8 × 2 = 0 + 0.091 431 463 205 735 299 533 215 979 527 015 602 585 6;
  • 65) 0.091 431 463 205 735 299 533 215 979 527 015 602 585 6 × 2 = 0 + 0.182 862 926 411 470 599 066 431 959 054 031 205 171 2;
  • 66) 0.182 862 926 411 470 599 066 431 959 054 031 205 171 2 × 2 = 0 + 0.365 725 852 822 941 198 132 863 918 108 062 410 342 4;
  • 67) 0.365 725 852 822 941 198 132 863 918 108 062 410 342 4 × 2 = 0 + 0.731 451 705 645 882 396 265 727 836 216 124 820 684 8;
  • 68) 0.731 451 705 645 882 396 265 727 836 216 124 820 684 8 × 2 = 1 + 0.462 903 411 291 764 792 531 455 672 432 249 641 369 6;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 626 6(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 626 6(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 626 6(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 0.000 020 830 729 321 671 205 134 999 154 509 660 626 6 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