0.000 020 830 729 321 671 205 134 999 154 509 660 616 878 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 616 878(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
0.000 020 830 729 321 671 205 134 999 154 509 660 616 878(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 616 878.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 616 878 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 233 756;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 233 756 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 467 512;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 467 512 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 935 024;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 935 024 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 870 048;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 870 048 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 740 096;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 740 096 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 480 192;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 480 192 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 558 960 384;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 558 960 384 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 117 920 768;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 117 920 768 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 235 841 536;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 235 841 536 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 471 683 072;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 471 683 072 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 943 366 144;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 943 366 144 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 886 732 288;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 886 732 288 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 773 464 576;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 773 464 576 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 546 929 152;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 546 929 152 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 093 858 304;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 093 858 304 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 187 716 608;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 187 716 608 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 375 433 216;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 375 433 216 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 750 866 432;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 750 866 432 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 501 732 864;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 501 732 864 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 891 003 465 728;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 891 003 465 728 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 782 006 931 456;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 782 006 931 456 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 564 013 862 912;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 564 013 862 912 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 128 027 725 824;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 128 027 725 824 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 256 055 451 648;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 256 055 451 648 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 512 110 903 296;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 512 110 903 296 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 024 221 806 592;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 024 221 806 592 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 048 443 613 184;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 048 443 613 184 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 096 887 226 368;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 096 887 226 368 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 193 774 452 736;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 193 774 452 736 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 816 387 548 905 472;
  • 31) 0.825 297 101 522 529 932 158 401 660 816 387 548 905 472 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 632 775 097 810 944;
  • 32) 0.650 594 203 045 059 864 316 803 321 632 775 097 810 944 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 265 550 195 621 888;
  • 33) 0.301 188 406 090 119 728 633 606 643 265 550 195 621 888 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 531 100 391 243 776;
  • 34) 0.602 376 812 180 239 457 267 213 286 531 100 391 243 776 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 062 200 782 487 552;
  • 35) 0.204 753 624 360 478 914 534 426 573 062 200 782 487 552 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 124 401 564 975 104;
  • 36) 0.409 507 248 720 957 829 068 853 146 124 401 564 975 104 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 248 803 129 950 208;
  • 37) 0.819 014 497 441 915 658 137 706 292 248 803 129 950 208 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 497 606 259 900 416;
  • 38) 0.638 028 994 883 831 316 275 412 584 497 606 259 900 416 × 2 = 1 + 0.276 057 989 767 662 632 550 825 168 995 212 519 800 832;
  • 39) 0.276 057 989 767 662 632 550 825 168 995 212 519 800 832 × 2 = 0 + 0.552 115 979 535 325 265 101 650 337 990 425 039 601 664;
  • 40) 0.552 115 979 535 325 265 101 650 337 990 425 039 601 664 × 2 = 1 + 0.104 231 959 070 650 530 203 300 675 980 850 079 203 328;
  • 41) 0.104 231 959 070 650 530 203 300 675 980 850 079 203 328 × 2 = 0 + 0.208 463 918 141 301 060 406 601 351 961 700 158 406 656;
  • 42) 0.208 463 918 141 301 060 406 601 351 961 700 158 406 656 × 2 = 0 + 0.416 927 836 282 602 120 813 202 703 923 400 316 813 312;
  • 43) 0.416 927 836 282 602 120 813 202 703 923 400 316 813 312 × 2 = 0 + 0.833 855 672 565 204 241 626 405 407 846 800 633 626 624;
  • 44) 0.833 855 672 565 204 241 626 405 407 846 800 633 626 624 × 2 = 1 + 0.667 711 345 130 408 483 252 810 815 693 601 267 253 248;
  • 45) 0.667 711 345 130 408 483 252 810 815 693 601 267 253 248 × 2 = 1 + 0.335 422 690 260 816 966 505 621 631 387 202 534 506 496;
  • 46) 0.335 422 690 260 816 966 505 621 631 387 202 534 506 496 × 2 = 0 + 0.670 845 380 521 633 933 011 243 262 774 405 069 012 992;
  • 47) 0.670 845 380 521 633 933 011 243 262 774 405 069 012 992 × 2 = 1 + 0.341 690 761 043 267 866 022 486 525 548 810 138 025 984;
  • 48) 0.341 690 761 043 267 866 022 486 525 548 810 138 025 984 × 2 = 0 + 0.683 381 522 086 535 732 044 973 051 097 620 276 051 968;
  • 49) 0.683 381 522 086 535 732 044 973 051 097 620 276 051 968 × 2 = 1 + 0.366 763 044 173 071 464 089 946 102 195 240 552 103 936;
  • 50) 0.366 763 044 173 071 464 089 946 102 195 240 552 103 936 × 2 = 0 + 0.733 526 088 346 142 928 179 892 204 390 481 104 207 872;
  • 51) 0.733 526 088 346 142 928 179 892 204 390 481 104 207 872 × 2 = 1 + 0.467 052 176 692 285 856 359 784 408 780 962 208 415 744;
  • 52) 0.467 052 176 692 285 856 359 784 408 780 962 208 415 744 × 2 = 0 + 0.934 104 353 384 571 712 719 568 817 561 924 416 831 488;
  • 53) 0.934 104 353 384 571 712 719 568 817 561 924 416 831 488 × 2 = 1 + 0.868 208 706 769 143 425 439 137 635 123 848 833 662 976;
  • 54) 0.868 208 706 769 143 425 439 137 635 123 848 833 662 976 × 2 = 1 + 0.736 417 413 538 286 850 878 275 270 247 697 667 325 952;
  • 55) 0.736 417 413 538 286 850 878 275 270 247 697 667 325 952 × 2 = 1 + 0.472 834 827 076 573 701 756 550 540 495 395 334 651 904;
  • 56) 0.472 834 827 076 573 701 756 550 540 495 395 334 651 904 × 2 = 0 + 0.945 669 654 153 147 403 513 101 080 990 790 669 303 808;
  • 57) 0.945 669 654 153 147 403 513 101 080 990 790 669 303 808 × 2 = 1 + 0.891 339 308 306 294 807 026 202 161 981 581 338 607 616;
  • 58) 0.891 339 308 306 294 807 026 202 161 981 581 338 607 616 × 2 = 1 + 0.782 678 616 612 589 614 052 404 323 963 162 677 215 232;
  • 59) 0.782 678 616 612 589 614 052 404 323 963 162 677 215 232 × 2 = 1 + 0.565 357 233 225 179 228 104 808 647 926 325 354 430 464;
  • 60) 0.565 357 233 225 179 228 104 808 647 926 325 354 430 464 × 2 = 1 + 0.130 714 466 450 358 456 209 617 295 852 650 708 860 928;
  • 61) 0.130 714 466 450 358 456 209 617 295 852 650 708 860 928 × 2 = 0 + 0.261 428 932 900 716 912 419 234 591 705 301 417 721 856;
  • 62) 0.261 428 932 900 716 912 419 234 591 705 301 417 721 856 × 2 = 0 + 0.522 857 865 801 433 824 838 469 183 410 602 835 443 712;
  • 63) 0.522 857 865 801 433 824 838 469 183 410 602 835 443 712 × 2 = 1 + 0.045 715 731 602 867 649 676 938 366 821 205 670 887 424;
  • 64) 0.045 715 731 602 867 649 676 938 366 821 205 670 887 424 × 2 = 0 + 0.091 431 463 205 735 299 353 876 733 642 411 341 774 848;
  • 65) 0.091 431 463 205 735 299 353 876 733 642 411 341 774 848 × 2 = 0 + 0.182 862 926 411 470 598 707 753 467 284 822 683 549 696;
  • 66) 0.182 862 926 411 470 598 707 753 467 284 822 683 549 696 × 2 = 0 + 0.365 725 852 822 941 197 415 506 934 569 645 367 099 392;
  • 67) 0.365 725 852 822 941 197 415 506 934 569 645 367 099 392 × 2 = 0 + 0.731 451 705 645 882 394 831 013 869 139 290 734 198 784;
  • 68) 0.731 451 705 645 882 394 831 013 869 139 290 734 198 784 × 2 = 1 + 0.462 903 411 291 764 789 662 027 738 278 581 468 397 568;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 616 878(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 616 878(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 616 878(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 0.000 020 830 729 321 671 205 134 999 154 509 660 616 878 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