0.000 020 830 729 321 671 205 134 999 154 509 660 632 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 632 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
0.000 020 830 729 321 671 205 134 999 154 509 660 632 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: 0.000 020 830 729 321 671 205 134 999 154 509 660 632 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) 0.000 020 830 729 321 671 205 134 999 154 509 660 632 8 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 265 6;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 265 6 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 531 2;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 531 2 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 285 062 4;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 285 062 4 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 570 124 8;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 570 124 8 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 140 249 6;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 140 249 6 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 280 499 2;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 280 499 2 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 560 998 4;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 560 998 4 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 121 996 8;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 121 996 8 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 243 993 6;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 243 993 6 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 487 987 2;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 487 987 2 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 975 974 4;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 975 974 4 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 951 948 8;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 951 948 8 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 903 897 6;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 903 897 6 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 807 795 2;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 807 795 2 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 615 590 4;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 615 590 4 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 119 231 180 8;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 119 231 180 8 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 238 462 361 6;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 238 462 361 6 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 476 924 723 2;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 476 924 723 2 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 953 849 446 4;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 953 849 446 4 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 907 698 892 8;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 907 698 892 8 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 815 397 785 6;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 815 397 785 6 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 630 795 571 2;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 630 795 571 2 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 261 591 142 4;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 261 591 142 4 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 523 182 284 8;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 523 182 284 8 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 901 046 364 569 6;
  • 26) 0.963 290 534 422 579 060 379 950 051 901 046 364 569 6 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 802 092 729 139 2;
  • 27) 0.926 581 068 845 158 120 759 900 103 802 092 729 139 2 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 604 185 458 278 4;
  • 28) 0.853 162 137 690 316 241 519 800 207 604 185 458 278 4 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 208 370 916 556 8;
  • 29) 0.706 324 275 380 632 483 039 600 415 208 370 916 556 8 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 416 741 833 113 6;
  • 30) 0.412 648 550 761 264 966 079 200 830 416 741 833 113 6 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 833 483 666 227 2;
  • 31) 0.825 297 101 522 529 932 158 401 660 833 483 666 227 2 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 666 967 332 454 4;
  • 32) 0.650 594 203 045 059 864 316 803 321 666 967 332 454 4 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 333 934 664 908 8;
  • 33) 0.301 188 406 090 119 728 633 606 643 333 934 664 908 8 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 667 869 329 817 6;
  • 34) 0.602 376 812 180 239 457 267 213 286 667 869 329 817 6 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 335 738 659 635 2;
  • 35) 0.204 753 624 360 478 914 534 426 573 335 738 659 635 2 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 671 477 319 270 4;
  • 36) 0.409 507 248 720 957 829 068 853 146 671 477 319 270 4 × 2 = 0 + 0.819 014 497 441 915 658 137 706 293 342 954 638 540 8;
  • 37) 0.819 014 497 441 915 658 137 706 293 342 954 638 540 8 × 2 = 1 + 0.638 028 994 883 831 316 275 412 586 685 909 277 081 6;
  • 38) 0.638 028 994 883 831 316 275 412 586 685 909 277 081 6 × 2 = 1 + 0.276 057 989 767 662 632 550 825 173 371 818 554 163 2;
  • 39) 0.276 057 989 767 662 632 550 825 173 371 818 554 163 2 × 2 = 0 + 0.552 115 979 535 325 265 101 650 346 743 637 108 326 4;
  • 40) 0.552 115 979 535 325 265 101 650 346 743 637 108 326 4 × 2 = 1 + 0.104 231 959 070 650 530 203 300 693 487 274 216 652 8;
  • 41) 0.104 231 959 070 650 530 203 300 693 487 274 216 652 8 × 2 = 0 + 0.208 463 918 141 301 060 406 601 386 974 548 433 305 6;
  • 42) 0.208 463 918 141 301 060 406 601 386 974 548 433 305 6 × 2 = 0 + 0.416 927 836 282 602 120 813 202 773 949 096 866 611 2;
  • 43) 0.416 927 836 282 602 120 813 202 773 949 096 866 611 2 × 2 = 0 + 0.833 855 672 565 204 241 626 405 547 898 193 733 222 4;
  • 44) 0.833 855 672 565 204 241 626 405 547 898 193 733 222 4 × 2 = 1 + 0.667 711 345 130 408 483 252 811 095 796 387 466 444 8;
  • 45) 0.667 711 345 130 408 483 252 811 095 796 387 466 444 8 × 2 = 1 + 0.335 422 690 260 816 966 505 622 191 592 774 932 889 6;
  • 46) 0.335 422 690 260 816 966 505 622 191 592 774 932 889 6 × 2 = 0 + 0.670 845 380 521 633 933 011 244 383 185 549 865 779 2;
  • 47) 0.670 845 380 521 633 933 011 244 383 185 549 865 779 2 × 2 = 1 + 0.341 690 761 043 267 866 022 488 766 371 099 731 558 4;
  • 48) 0.341 690 761 043 267 866 022 488 766 371 099 731 558 4 × 2 = 0 + 0.683 381 522 086 535 732 044 977 532 742 199 463 116 8;
  • 49) 0.683 381 522 086 535 732 044 977 532 742 199 463 116 8 × 2 = 1 + 0.366 763 044 173 071 464 089 955 065 484 398 926 233 6;
  • 50) 0.366 763 044 173 071 464 089 955 065 484 398 926 233 6 × 2 = 0 + 0.733 526 088 346 142 928 179 910 130 968 797 852 467 2;
  • 51) 0.733 526 088 346 142 928 179 910 130 968 797 852 467 2 × 2 = 1 + 0.467 052 176 692 285 856 359 820 261 937 595 704 934 4;
  • 52) 0.467 052 176 692 285 856 359 820 261 937 595 704 934 4 × 2 = 0 + 0.934 104 353 384 571 712 719 640 523 875 191 409 868 8;
  • 53) 0.934 104 353 384 571 712 719 640 523 875 191 409 868 8 × 2 = 1 + 0.868 208 706 769 143 425 439 281 047 750 382 819 737 6;
  • 54) 0.868 208 706 769 143 425 439 281 047 750 382 819 737 6 × 2 = 1 + 0.736 417 413 538 286 850 878 562 095 500 765 639 475 2;
  • 55) 0.736 417 413 538 286 850 878 562 095 500 765 639 475 2 × 2 = 1 + 0.472 834 827 076 573 701 757 124 191 001 531 278 950 4;
  • 56) 0.472 834 827 076 573 701 757 124 191 001 531 278 950 4 × 2 = 0 + 0.945 669 654 153 147 403 514 248 382 003 062 557 900 8;
  • 57) 0.945 669 654 153 147 403 514 248 382 003 062 557 900 8 × 2 = 1 + 0.891 339 308 306 294 807 028 496 764 006 125 115 801 6;
  • 58) 0.891 339 308 306 294 807 028 496 764 006 125 115 801 6 × 2 = 1 + 0.782 678 616 612 589 614 056 993 528 012 250 231 603 2;
  • 59) 0.782 678 616 612 589 614 056 993 528 012 250 231 603 2 × 2 = 1 + 0.565 357 233 225 179 228 113 987 056 024 500 463 206 4;
  • 60) 0.565 357 233 225 179 228 113 987 056 024 500 463 206 4 × 2 = 1 + 0.130 714 466 450 358 456 227 974 112 049 000 926 412 8;
  • 61) 0.130 714 466 450 358 456 227 974 112 049 000 926 412 8 × 2 = 0 + 0.261 428 932 900 716 912 455 948 224 098 001 852 825 6;
  • 62) 0.261 428 932 900 716 912 455 948 224 098 001 852 825 6 × 2 = 0 + 0.522 857 865 801 433 824 911 896 448 196 003 705 651 2;
  • 63) 0.522 857 865 801 433 824 911 896 448 196 003 705 651 2 × 2 = 1 + 0.045 715 731 602 867 649 823 792 896 392 007 411 302 4;
  • 64) 0.045 715 731 602 867 649 823 792 896 392 007 411 302 4 × 2 = 0 + 0.091 431 463 205 735 299 647 585 792 784 014 822 604 8;
  • 65) 0.091 431 463 205 735 299 647 585 792 784 014 822 604 8 × 2 = 0 + 0.182 862 926 411 470 599 295 171 585 568 029 645 209 6;
  • 66) 0.182 862 926 411 470 599 295 171 585 568 029 645 209 6 × 2 = 0 + 0.365 725 852 822 941 198 590 343 171 136 059 290 419 2;
  • 67) 0.365 725 852 822 941 198 590 343 171 136 059 290 419 2 × 2 = 0 + 0.731 451 705 645 882 397 180 686 342 272 118 580 838 4;
  • 68) 0.731 451 705 645 882 397 180 686 342 272 118 580 838 4 × 2 = 1 + 0.462 903 411 291 764 794 361 372 684 544 237 161 676 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:


0.000 020 830 729 321 671 205 134 999 154 509 660 632 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:

0.000 020 830 729 321 671 205 134 999 154 509 660 632 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:


0.000 020 830 729 321 671 205 134 999 154 509 660 632 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 0.000 020 830 729 321 671 205 134 999 154 509 660 632 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