0.000 020 830 729 321 671 205 134 999 154 509 660 585 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 585 7(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
0.000 020 830 729 321 671 205 134 999 154 509 660 585 7(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 585 7.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 585 7 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 171 4;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 171 4 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 342 8;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 342 8 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 685 6;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 685 6 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 371 2;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 371 2 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 138 742 4;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 138 742 4 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 277 484 8;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 277 484 8 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 554 969 6;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 554 969 6 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 109 939 2;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 109 939 2 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 219 878 4;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 219 878 4 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 439 756 8;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 439 756 8 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 879 513 6;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 879 513 6 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 759 027 2;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 759 027 2 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 518 054 4;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 518 054 4 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 036 108 8;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 036 108 8 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 558 072 217 6;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 558 072 217 6 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 116 144 435 2;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 116 144 435 2 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 232 288 870 4;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 232 288 870 4 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 464 577 740 8;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 464 577 740 8 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 929 155 481 6;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 929 155 481 6 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 858 310 963 2;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 858 310 963 2 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 716 621 926 4;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 716 621 926 4 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 433 243 852 8;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 433 243 852 8 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 974 866 487 705 6;
  • 24) 0.740 822 633 605 644 765 094 987 512 974 866 487 705 6 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 949 732 975 411 2;
  • 25) 0.481 645 267 211 289 530 189 975 025 949 732 975 411 2 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 899 465 950 822 4;
  • 26) 0.963 290 534 422 579 060 379 950 051 899 465 950 822 4 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 798 931 901 644 8;
  • 27) 0.926 581 068 845 158 120 759 900 103 798 931 901 644 8 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 597 863 803 289 6;
  • 28) 0.853 162 137 690 316 241 519 800 207 597 863 803 289 6 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 195 727 606 579 2;
  • 29) 0.706 324 275 380 632 483 039 600 415 195 727 606 579 2 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 391 455 213 158 4;
  • 30) 0.412 648 550 761 264 966 079 200 830 391 455 213 158 4 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 782 910 426 316 8;
  • 31) 0.825 297 101 522 529 932 158 401 660 782 910 426 316 8 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 565 820 852 633 6;
  • 32) 0.650 594 203 045 059 864 316 803 321 565 820 852 633 6 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 131 641 705 267 2;
  • 33) 0.301 188 406 090 119 728 633 606 643 131 641 705 267 2 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 263 283 410 534 4;
  • 34) 0.602 376 812 180 239 457 267 213 286 263 283 410 534 4 × 2 = 1 + 0.204 753 624 360 478 914 534 426 572 526 566 821 068 8;
  • 35) 0.204 753 624 360 478 914 534 426 572 526 566 821 068 8 × 2 = 0 + 0.409 507 248 720 957 829 068 853 145 053 133 642 137 6;
  • 36) 0.409 507 248 720 957 829 068 853 145 053 133 642 137 6 × 2 = 0 + 0.819 014 497 441 915 658 137 706 290 106 267 284 275 2;
  • 37) 0.819 014 497 441 915 658 137 706 290 106 267 284 275 2 × 2 = 1 + 0.638 028 994 883 831 316 275 412 580 212 534 568 550 4;
  • 38) 0.638 028 994 883 831 316 275 412 580 212 534 568 550 4 × 2 = 1 + 0.276 057 989 767 662 632 550 825 160 425 069 137 100 8;
  • 39) 0.276 057 989 767 662 632 550 825 160 425 069 137 100 8 × 2 = 0 + 0.552 115 979 535 325 265 101 650 320 850 138 274 201 6;
  • 40) 0.552 115 979 535 325 265 101 650 320 850 138 274 201 6 × 2 = 1 + 0.104 231 959 070 650 530 203 300 641 700 276 548 403 2;
  • 41) 0.104 231 959 070 650 530 203 300 641 700 276 548 403 2 × 2 = 0 + 0.208 463 918 141 301 060 406 601 283 400 553 096 806 4;
  • 42) 0.208 463 918 141 301 060 406 601 283 400 553 096 806 4 × 2 = 0 + 0.416 927 836 282 602 120 813 202 566 801 106 193 612 8;
  • 43) 0.416 927 836 282 602 120 813 202 566 801 106 193 612 8 × 2 = 0 + 0.833 855 672 565 204 241 626 405 133 602 212 387 225 6;
  • 44) 0.833 855 672 565 204 241 626 405 133 602 212 387 225 6 × 2 = 1 + 0.667 711 345 130 408 483 252 810 267 204 424 774 451 2;
  • 45) 0.667 711 345 130 408 483 252 810 267 204 424 774 451 2 × 2 = 1 + 0.335 422 690 260 816 966 505 620 534 408 849 548 902 4;
  • 46) 0.335 422 690 260 816 966 505 620 534 408 849 548 902 4 × 2 = 0 + 0.670 845 380 521 633 933 011 241 068 817 699 097 804 8;
  • 47) 0.670 845 380 521 633 933 011 241 068 817 699 097 804 8 × 2 = 1 + 0.341 690 761 043 267 866 022 482 137 635 398 195 609 6;
  • 48) 0.341 690 761 043 267 866 022 482 137 635 398 195 609 6 × 2 = 0 + 0.683 381 522 086 535 732 044 964 275 270 796 391 219 2;
  • 49) 0.683 381 522 086 535 732 044 964 275 270 796 391 219 2 × 2 = 1 + 0.366 763 044 173 071 464 089 928 550 541 592 782 438 4;
  • 50) 0.366 763 044 173 071 464 089 928 550 541 592 782 438 4 × 2 = 0 + 0.733 526 088 346 142 928 179 857 101 083 185 564 876 8;
  • 51) 0.733 526 088 346 142 928 179 857 101 083 185 564 876 8 × 2 = 1 + 0.467 052 176 692 285 856 359 714 202 166 371 129 753 6;
  • 52) 0.467 052 176 692 285 856 359 714 202 166 371 129 753 6 × 2 = 0 + 0.934 104 353 384 571 712 719 428 404 332 742 259 507 2;
  • 53) 0.934 104 353 384 571 712 719 428 404 332 742 259 507 2 × 2 = 1 + 0.868 208 706 769 143 425 438 856 808 665 484 519 014 4;
  • 54) 0.868 208 706 769 143 425 438 856 808 665 484 519 014 4 × 2 = 1 + 0.736 417 413 538 286 850 877 713 617 330 969 038 028 8;
  • 55) 0.736 417 413 538 286 850 877 713 617 330 969 038 028 8 × 2 = 1 + 0.472 834 827 076 573 701 755 427 234 661 938 076 057 6;
  • 56) 0.472 834 827 076 573 701 755 427 234 661 938 076 057 6 × 2 = 0 + 0.945 669 654 153 147 403 510 854 469 323 876 152 115 2;
  • 57) 0.945 669 654 153 147 403 510 854 469 323 876 152 115 2 × 2 = 1 + 0.891 339 308 306 294 807 021 708 938 647 752 304 230 4;
  • 58) 0.891 339 308 306 294 807 021 708 938 647 752 304 230 4 × 2 = 1 + 0.782 678 616 612 589 614 043 417 877 295 504 608 460 8;
  • 59) 0.782 678 616 612 589 614 043 417 877 295 504 608 460 8 × 2 = 1 + 0.565 357 233 225 179 228 086 835 754 591 009 216 921 6;
  • 60) 0.565 357 233 225 179 228 086 835 754 591 009 216 921 6 × 2 = 1 + 0.130 714 466 450 358 456 173 671 509 182 018 433 843 2;
  • 61) 0.130 714 466 450 358 456 173 671 509 182 018 433 843 2 × 2 = 0 + 0.261 428 932 900 716 912 347 343 018 364 036 867 686 4;
  • 62) 0.261 428 932 900 716 912 347 343 018 364 036 867 686 4 × 2 = 0 + 0.522 857 865 801 433 824 694 686 036 728 073 735 372 8;
  • 63) 0.522 857 865 801 433 824 694 686 036 728 073 735 372 8 × 2 = 1 + 0.045 715 731 602 867 649 389 372 073 456 147 470 745 6;
  • 64) 0.045 715 731 602 867 649 389 372 073 456 147 470 745 6 × 2 = 0 + 0.091 431 463 205 735 298 778 744 146 912 294 941 491 2;
  • 65) 0.091 431 463 205 735 298 778 744 146 912 294 941 491 2 × 2 = 0 + 0.182 862 926 411 470 597 557 488 293 824 589 882 982 4;
  • 66) 0.182 862 926 411 470 597 557 488 293 824 589 882 982 4 × 2 = 0 + 0.365 725 852 822 941 195 114 976 587 649 179 765 964 8;
  • 67) 0.365 725 852 822 941 195 114 976 587 649 179 765 964 8 × 2 = 0 + 0.731 451 705 645 882 390 229 953 175 298 359 531 929 6;
  • 68) 0.731 451 705 645 882 390 229 953 175 298 359 531 929 6 × 2 = 1 + 0.462 903 411 291 764 780 459 906 350 596 719 063 859 2;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 585 7(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 585 7(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 585 7(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 0.000 020 830 729 321 671 205 134 999 154 509 660 585 7 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