0.000 020 830 729 321 671 205 134 999 154 509 660 645 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 645 8(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
0.000 020 830 729 321 671 205 134 999 154 509 660 645 8(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 645 8.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 645 8 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 291 6;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 291 6 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 583 2;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 583 2 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 285 166 4;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 285 166 4 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 570 332 8;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 570 332 8 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 140 665 6;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 140 665 6 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 281 331 2;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 281 331 2 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 562 662 4;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 562 662 4 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 125 324 8;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 125 324 8 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 250 649 6;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 250 649 6 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 501 299 2;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 501 299 2 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 785 002 598 4;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 785 002 598 4 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 570 005 196 8;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 570 005 196 8 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 140 010 393 6;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 140 010 393 6 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 280 020 787 2;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 280 020 787 2 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 560 041 574 4;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 560 041 574 4 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 120 083 148 8;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 120 083 148 8 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 240 166 297 6;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 240 166 297 6 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 480 332 595 2;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 480 332 595 2 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 960 665 190 4;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 960 665 190 4 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 921 330 380 8;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 921 330 380 8 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 842 660 761 6;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 842 660 761 6 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 685 321 523 2;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 685 321 523 2 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 370 643 046 4;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 370 643 046 4 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 741 286 092 8;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 741 286 092 8 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 901 482 572 185 6;
  • 26) 0.963 290 534 422 579 060 379 950 051 901 482 572 185 6 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 802 965 144 371 2;
  • 27) 0.926 581 068 845 158 120 759 900 103 802 965 144 371 2 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 605 930 288 742 4;
  • 28) 0.853 162 137 690 316 241 519 800 207 605 930 288 742 4 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 211 860 577 484 8;
  • 29) 0.706 324 275 380 632 483 039 600 415 211 860 577 484 8 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 423 721 154 969 6;
  • 30) 0.412 648 550 761 264 966 079 200 830 423 721 154 969 6 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 847 442 309 939 2;
  • 31) 0.825 297 101 522 529 932 158 401 660 847 442 309 939 2 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 694 884 619 878 4;
  • 32) 0.650 594 203 045 059 864 316 803 321 694 884 619 878 4 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 389 769 239 756 8;
  • 33) 0.301 188 406 090 119 728 633 606 643 389 769 239 756 8 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 779 538 479 513 6;
  • 34) 0.602 376 812 180 239 457 267 213 286 779 538 479 513 6 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 559 076 959 027 2;
  • 35) 0.204 753 624 360 478 914 534 426 573 559 076 959 027 2 × 2 = 0 + 0.409 507 248 720 957 829 068 853 147 118 153 918 054 4;
  • 36) 0.409 507 248 720 957 829 068 853 147 118 153 918 054 4 × 2 = 0 + 0.819 014 497 441 915 658 137 706 294 236 307 836 108 8;
  • 37) 0.819 014 497 441 915 658 137 706 294 236 307 836 108 8 × 2 = 1 + 0.638 028 994 883 831 316 275 412 588 472 615 672 217 6;
  • 38) 0.638 028 994 883 831 316 275 412 588 472 615 672 217 6 × 2 = 1 + 0.276 057 989 767 662 632 550 825 176 945 231 344 435 2;
  • 39) 0.276 057 989 767 662 632 550 825 176 945 231 344 435 2 × 2 = 0 + 0.552 115 979 535 325 265 101 650 353 890 462 688 870 4;
  • 40) 0.552 115 979 535 325 265 101 650 353 890 462 688 870 4 × 2 = 1 + 0.104 231 959 070 650 530 203 300 707 780 925 377 740 8;
  • 41) 0.104 231 959 070 650 530 203 300 707 780 925 377 740 8 × 2 = 0 + 0.208 463 918 141 301 060 406 601 415 561 850 755 481 6;
  • 42) 0.208 463 918 141 301 060 406 601 415 561 850 755 481 6 × 2 = 0 + 0.416 927 836 282 602 120 813 202 831 123 701 510 963 2;
  • 43) 0.416 927 836 282 602 120 813 202 831 123 701 510 963 2 × 2 = 0 + 0.833 855 672 565 204 241 626 405 662 247 403 021 926 4;
  • 44) 0.833 855 672 565 204 241 626 405 662 247 403 021 926 4 × 2 = 1 + 0.667 711 345 130 408 483 252 811 324 494 806 043 852 8;
  • 45) 0.667 711 345 130 408 483 252 811 324 494 806 043 852 8 × 2 = 1 + 0.335 422 690 260 816 966 505 622 648 989 612 087 705 6;
  • 46) 0.335 422 690 260 816 966 505 622 648 989 612 087 705 6 × 2 = 0 + 0.670 845 380 521 633 933 011 245 297 979 224 175 411 2;
  • 47) 0.670 845 380 521 633 933 011 245 297 979 224 175 411 2 × 2 = 1 + 0.341 690 761 043 267 866 022 490 595 958 448 350 822 4;
  • 48) 0.341 690 761 043 267 866 022 490 595 958 448 350 822 4 × 2 = 0 + 0.683 381 522 086 535 732 044 981 191 916 896 701 644 8;
  • 49) 0.683 381 522 086 535 732 044 981 191 916 896 701 644 8 × 2 = 1 + 0.366 763 044 173 071 464 089 962 383 833 793 403 289 6;
  • 50) 0.366 763 044 173 071 464 089 962 383 833 793 403 289 6 × 2 = 0 + 0.733 526 088 346 142 928 179 924 767 667 586 806 579 2;
  • 51) 0.733 526 088 346 142 928 179 924 767 667 586 806 579 2 × 2 = 1 + 0.467 052 176 692 285 856 359 849 535 335 173 613 158 4;
  • 52) 0.467 052 176 692 285 856 359 849 535 335 173 613 158 4 × 2 = 0 + 0.934 104 353 384 571 712 719 699 070 670 347 226 316 8;
  • 53) 0.934 104 353 384 571 712 719 699 070 670 347 226 316 8 × 2 = 1 + 0.868 208 706 769 143 425 439 398 141 340 694 452 633 6;
  • 54) 0.868 208 706 769 143 425 439 398 141 340 694 452 633 6 × 2 = 1 + 0.736 417 413 538 286 850 878 796 282 681 388 905 267 2;
  • 55) 0.736 417 413 538 286 850 878 796 282 681 388 905 267 2 × 2 = 1 + 0.472 834 827 076 573 701 757 592 565 362 777 810 534 4;
  • 56) 0.472 834 827 076 573 701 757 592 565 362 777 810 534 4 × 2 = 0 + 0.945 669 654 153 147 403 515 185 130 725 555 621 068 8;
  • 57) 0.945 669 654 153 147 403 515 185 130 725 555 621 068 8 × 2 = 1 + 0.891 339 308 306 294 807 030 370 261 451 111 242 137 6;
  • 58) 0.891 339 308 306 294 807 030 370 261 451 111 242 137 6 × 2 = 1 + 0.782 678 616 612 589 614 060 740 522 902 222 484 275 2;
  • 59) 0.782 678 616 612 589 614 060 740 522 902 222 484 275 2 × 2 = 1 + 0.565 357 233 225 179 228 121 481 045 804 444 968 550 4;
  • 60) 0.565 357 233 225 179 228 121 481 045 804 444 968 550 4 × 2 = 1 + 0.130 714 466 450 358 456 242 962 091 608 889 937 100 8;
  • 61) 0.130 714 466 450 358 456 242 962 091 608 889 937 100 8 × 2 = 0 + 0.261 428 932 900 716 912 485 924 183 217 779 874 201 6;
  • 62) 0.261 428 932 900 716 912 485 924 183 217 779 874 201 6 × 2 = 0 + 0.522 857 865 801 433 824 971 848 366 435 559 748 403 2;
  • 63) 0.522 857 865 801 433 824 971 848 366 435 559 748 403 2 × 2 = 1 + 0.045 715 731 602 867 649 943 696 732 871 119 496 806 4;
  • 64) 0.045 715 731 602 867 649 943 696 732 871 119 496 806 4 × 2 = 0 + 0.091 431 463 205 735 299 887 393 465 742 238 993 612 8;
  • 65) 0.091 431 463 205 735 299 887 393 465 742 238 993 612 8 × 2 = 0 + 0.182 862 926 411 470 599 774 786 931 484 477 987 225 6;
  • 66) 0.182 862 926 411 470 599 774 786 931 484 477 987 225 6 × 2 = 0 + 0.365 725 852 822 941 199 549 573 862 968 955 974 451 2;
  • 67) 0.365 725 852 822 941 199 549 573 862 968 955 974 451 2 × 2 = 0 + 0.731 451 705 645 882 399 099 147 725 937 911 948 902 4;
  • 68) 0.731 451 705 645 882 399 099 147 725 937 911 948 902 4 × 2 = 1 + 0.462 903 411 291 764 798 198 295 451 875 823 897 804 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:


0.000 020 830 729 321 671 205 134 999 154 509 660 645 8(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 645 8(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 645 8(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 0.000 020 830 729 321 671 205 134 999 154 509 660 645 8 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