0.000 020 830 729 321 671 205 134 999 154 507 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 507 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
0.000 020 830 729 321 671 205 134 999 154 507 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: 0.000 020 830 729 321 671 205 134 999 154 507 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) 0.000 020 830 729 321 671 205 134 999 154 507 6 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 015 2;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 015 2 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 030 4;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 030 4 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 060 8;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 060 8 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 121 6;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 121 6 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 243 2;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 243 2 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 486 4;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 486 4 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 776 972 8;
  • 8) 0.002 666 333 353 173 914 257 279 891 776 972 8 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 553 945 6;
  • 9) 0.005 332 666 706 347 828 514 559 783 553 945 6 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 107 891 2;
  • 10) 0.010 665 333 412 695 657 029 119 567 107 891 2 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 215 782 4;
  • 11) 0.021 330 666 825 391 314 058 239 134 215 782 4 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 431 564 8;
  • 12) 0.042 661 333 650 782 628 116 478 268 431 564 8 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 863 129 6;
  • 13) 0.085 322 667 301 565 256 232 956 536 863 129 6 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 726 259 2;
  • 14) 0.170 645 334 603 130 512 465 913 073 726 259 2 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 452 518 4;
  • 15) 0.341 290 669 206 261 024 931 826 147 452 518 4 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 905 036 8;
  • 16) 0.682 581 338 412 522 049 863 652 294 905 036 8 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 810 073 6;
  • 17) 0.365 162 676 825 044 099 727 304 589 810 073 6 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 620 147 2;
  • 18) 0.730 325 353 650 088 199 454 609 179 620 147 2 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 240 294 4;
  • 19) 0.460 650 707 300 176 398 909 218 359 240 294 4 × 2 = 0 + 0.921 301 414 600 352 797 818 436 718 480 588 8;
  • 20) 0.921 301 414 600 352 797 818 436 718 480 588 8 × 2 = 1 + 0.842 602 829 200 705 595 636 873 436 961 177 6;
  • 21) 0.842 602 829 200 705 595 636 873 436 961 177 6 × 2 = 1 + 0.685 205 658 401 411 191 273 746 873 922 355 2;
  • 22) 0.685 205 658 401 411 191 273 746 873 922 355 2 × 2 = 1 + 0.370 411 316 802 822 382 547 493 747 844 710 4;
  • 23) 0.370 411 316 802 822 382 547 493 747 844 710 4 × 2 = 0 + 0.740 822 633 605 644 765 094 987 495 689 420 8;
  • 24) 0.740 822 633 605 644 765 094 987 495 689 420 8 × 2 = 1 + 0.481 645 267 211 289 530 189 974 991 378 841 6;
  • 25) 0.481 645 267 211 289 530 189 974 991 378 841 6 × 2 = 0 + 0.963 290 534 422 579 060 379 949 982 757 683 2;
  • 26) 0.963 290 534 422 579 060 379 949 982 757 683 2 × 2 = 1 + 0.926 581 068 845 158 120 759 899 965 515 366 4;
  • 27) 0.926 581 068 845 158 120 759 899 965 515 366 4 × 2 = 1 + 0.853 162 137 690 316 241 519 799 931 030 732 8;
  • 28) 0.853 162 137 690 316 241 519 799 931 030 732 8 × 2 = 1 + 0.706 324 275 380 632 483 039 599 862 061 465 6;
  • 29) 0.706 324 275 380 632 483 039 599 862 061 465 6 × 2 = 1 + 0.412 648 550 761 264 966 079 199 724 122 931 2;
  • 30) 0.412 648 550 761 264 966 079 199 724 122 931 2 × 2 = 0 + 0.825 297 101 522 529 932 158 399 448 245 862 4;
  • 31) 0.825 297 101 522 529 932 158 399 448 245 862 4 × 2 = 1 + 0.650 594 203 045 059 864 316 798 896 491 724 8;
  • 32) 0.650 594 203 045 059 864 316 798 896 491 724 8 × 2 = 1 + 0.301 188 406 090 119 728 633 597 792 983 449 6;
  • 33) 0.301 188 406 090 119 728 633 597 792 983 449 6 × 2 = 0 + 0.602 376 812 180 239 457 267 195 585 966 899 2;
  • 34) 0.602 376 812 180 239 457 267 195 585 966 899 2 × 2 = 1 + 0.204 753 624 360 478 914 534 391 171 933 798 4;
  • 35) 0.204 753 624 360 478 914 534 391 171 933 798 4 × 2 = 0 + 0.409 507 248 720 957 829 068 782 343 867 596 8;
  • 36) 0.409 507 248 720 957 829 068 782 343 867 596 8 × 2 = 0 + 0.819 014 497 441 915 658 137 564 687 735 193 6;
  • 37) 0.819 014 497 441 915 658 137 564 687 735 193 6 × 2 = 1 + 0.638 028 994 883 831 316 275 129 375 470 387 2;
  • 38) 0.638 028 994 883 831 316 275 129 375 470 387 2 × 2 = 1 + 0.276 057 989 767 662 632 550 258 750 940 774 4;
  • 39) 0.276 057 989 767 662 632 550 258 750 940 774 4 × 2 = 0 + 0.552 115 979 535 325 265 100 517 501 881 548 8;
  • 40) 0.552 115 979 535 325 265 100 517 501 881 548 8 × 2 = 1 + 0.104 231 959 070 650 530 201 035 003 763 097 6;
  • 41) 0.104 231 959 070 650 530 201 035 003 763 097 6 × 2 = 0 + 0.208 463 918 141 301 060 402 070 007 526 195 2;
  • 42) 0.208 463 918 141 301 060 402 070 007 526 195 2 × 2 = 0 + 0.416 927 836 282 602 120 804 140 015 052 390 4;
  • 43) 0.416 927 836 282 602 120 804 140 015 052 390 4 × 2 = 0 + 0.833 855 672 565 204 241 608 280 030 104 780 8;
  • 44) 0.833 855 672 565 204 241 608 280 030 104 780 8 × 2 = 1 + 0.667 711 345 130 408 483 216 560 060 209 561 6;
  • 45) 0.667 711 345 130 408 483 216 560 060 209 561 6 × 2 = 1 + 0.335 422 690 260 816 966 433 120 120 419 123 2;
  • 46) 0.335 422 690 260 816 966 433 120 120 419 123 2 × 2 = 0 + 0.670 845 380 521 633 932 866 240 240 838 246 4;
  • 47) 0.670 845 380 521 633 932 866 240 240 838 246 4 × 2 = 1 + 0.341 690 761 043 267 865 732 480 481 676 492 8;
  • 48) 0.341 690 761 043 267 865 732 480 481 676 492 8 × 2 = 0 + 0.683 381 522 086 535 731 464 960 963 352 985 6;
  • 49) 0.683 381 522 086 535 731 464 960 963 352 985 6 × 2 = 1 + 0.366 763 044 173 071 462 929 921 926 705 971 2;
  • 50) 0.366 763 044 173 071 462 929 921 926 705 971 2 × 2 = 0 + 0.733 526 088 346 142 925 859 843 853 411 942 4;
  • 51) 0.733 526 088 346 142 925 859 843 853 411 942 4 × 2 = 1 + 0.467 052 176 692 285 851 719 687 706 823 884 8;
  • 52) 0.467 052 176 692 285 851 719 687 706 823 884 8 × 2 = 0 + 0.934 104 353 384 571 703 439 375 413 647 769 6;
  • 53) 0.934 104 353 384 571 703 439 375 413 647 769 6 × 2 = 1 + 0.868 208 706 769 143 406 878 750 827 295 539 2;
  • 54) 0.868 208 706 769 143 406 878 750 827 295 539 2 × 2 = 1 + 0.736 417 413 538 286 813 757 501 654 591 078 4;
  • 55) 0.736 417 413 538 286 813 757 501 654 591 078 4 × 2 = 1 + 0.472 834 827 076 573 627 515 003 309 182 156 8;
  • 56) 0.472 834 827 076 573 627 515 003 309 182 156 8 × 2 = 0 + 0.945 669 654 153 147 255 030 006 618 364 313 6;
  • 57) 0.945 669 654 153 147 255 030 006 618 364 313 6 × 2 = 1 + 0.891 339 308 306 294 510 060 013 236 728 627 2;
  • 58) 0.891 339 308 306 294 510 060 013 236 728 627 2 × 2 = 1 + 0.782 678 616 612 589 020 120 026 473 457 254 4;
  • 59) 0.782 678 616 612 589 020 120 026 473 457 254 4 × 2 = 1 + 0.565 357 233 225 178 040 240 052 946 914 508 8;
  • 60) 0.565 357 233 225 178 040 240 052 946 914 508 8 × 2 = 1 + 0.130 714 466 450 356 080 480 105 893 829 017 6;
  • 61) 0.130 714 466 450 356 080 480 105 893 829 017 6 × 2 = 0 + 0.261 428 932 900 712 160 960 211 787 658 035 2;
  • 62) 0.261 428 932 900 712 160 960 211 787 658 035 2 × 2 = 0 + 0.522 857 865 801 424 321 920 423 575 316 070 4;
  • 63) 0.522 857 865 801 424 321 920 423 575 316 070 4 × 2 = 1 + 0.045 715 731 602 848 643 840 847 150 632 140 8;
  • 64) 0.045 715 731 602 848 643 840 847 150 632 140 8 × 2 = 0 + 0.091 431 463 205 697 287 681 694 301 264 281 6;
  • 65) 0.091 431 463 205 697 287 681 694 301 264 281 6 × 2 = 0 + 0.182 862 926 411 394 575 363 388 602 528 563 2;
  • 66) 0.182 862 926 411 394 575 363 388 602 528 563 2 × 2 = 0 + 0.365 725 852 822 789 150 726 777 205 057 126 4;
  • 67) 0.365 725 852 822 789 150 726 777 205 057 126 4 × 2 = 0 + 0.731 451 705 645 578 301 453 554 410 114 252 8;
  • 68) 0.731 451 705 645 578 301 453 554 410 114 252 8 × 2 = 1 + 0.462 903 411 291 156 602 907 108 820 228 505 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:


0.000 020 830 729 321 671 205 134 999 154 507 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:

0.000 020 830 729 321 671 205 134 999 154 507 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:


0.000 020 830 729 321 671 205 134 999 154 507 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 0.000 020 830 729 321 671 205 134 999 154 507 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

Share

» Convert decimal 0.000 020 830 729 321 671 205 134 999 154 505 9 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 08, 2026 [August]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: August - by our visitors

» Improve your social skills: Your Greatest Competitor, Your Most Powerful Ally. Gemini Notebook YouTube Video of 7.54 minutes

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