1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 561 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 561(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
1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 561(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: 1.
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;
  • 1 ÷ 2 = 0 + 1;

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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 561.

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) 0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 561 × 2 = 1 + 0.236 067 977 499 789 696 409 173 668 731 276 235 440 618 359 611 122;
  • 2) 0.236 067 977 499 789 696 409 173 668 731 276 235 440 618 359 611 122 × 2 = 0 + 0.472 135 954 999 579 392 818 347 337 462 552 470 881 236 719 222 244;
  • 3) 0.472 135 954 999 579 392 818 347 337 462 552 470 881 236 719 222 244 × 2 = 0 + 0.944 271 909 999 158 785 636 694 674 925 104 941 762 473 438 444 488;
  • 4) 0.944 271 909 999 158 785 636 694 674 925 104 941 762 473 438 444 488 × 2 = 1 + 0.888 543 819 998 317 571 273 389 349 850 209 883 524 946 876 888 976;
  • 5) 0.888 543 819 998 317 571 273 389 349 850 209 883 524 946 876 888 976 × 2 = 1 + 0.777 087 639 996 635 142 546 778 699 700 419 767 049 893 753 777 952;
  • 6) 0.777 087 639 996 635 142 546 778 699 700 419 767 049 893 753 777 952 × 2 = 1 + 0.554 175 279 993 270 285 093 557 399 400 839 534 099 787 507 555 904;
  • 7) 0.554 175 279 993 270 285 093 557 399 400 839 534 099 787 507 555 904 × 2 = 1 + 0.108 350 559 986 540 570 187 114 798 801 679 068 199 575 015 111 808;
  • 8) 0.108 350 559 986 540 570 187 114 798 801 679 068 199 575 015 111 808 × 2 = 0 + 0.216 701 119 973 081 140 374 229 597 603 358 136 399 150 030 223 616;
  • 9) 0.216 701 119 973 081 140 374 229 597 603 358 136 399 150 030 223 616 × 2 = 0 + 0.433 402 239 946 162 280 748 459 195 206 716 272 798 300 060 447 232;
  • 10) 0.433 402 239 946 162 280 748 459 195 206 716 272 798 300 060 447 232 × 2 = 0 + 0.866 804 479 892 324 561 496 918 390 413 432 545 596 600 120 894 464;
  • 11) 0.866 804 479 892 324 561 496 918 390 413 432 545 596 600 120 894 464 × 2 = 1 + 0.733 608 959 784 649 122 993 836 780 826 865 091 193 200 241 788 928;
  • 12) 0.733 608 959 784 649 122 993 836 780 826 865 091 193 200 241 788 928 × 2 = 1 + 0.467 217 919 569 298 245 987 673 561 653 730 182 386 400 483 577 856;
  • 13) 0.467 217 919 569 298 245 987 673 561 653 730 182 386 400 483 577 856 × 2 = 0 + 0.934 435 839 138 596 491 975 347 123 307 460 364 772 800 967 155 712;
  • 14) 0.934 435 839 138 596 491 975 347 123 307 460 364 772 800 967 155 712 × 2 = 1 + 0.868 871 678 277 192 983 950 694 246 614 920 729 545 601 934 311 424;
  • 15) 0.868 871 678 277 192 983 950 694 246 614 920 729 545 601 934 311 424 × 2 = 1 + 0.737 743 356 554 385 967 901 388 493 229 841 459 091 203 868 622 848;
  • 16) 0.737 743 356 554 385 967 901 388 493 229 841 459 091 203 868 622 848 × 2 = 1 + 0.475 486 713 108 771 935 802 776 986 459 682 918 182 407 737 245 696;
  • 17) 0.475 486 713 108 771 935 802 776 986 459 682 918 182 407 737 245 696 × 2 = 0 + 0.950 973 426 217 543 871 605 553 972 919 365 836 364 815 474 491 392;
  • 18) 0.950 973 426 217 543 871 605 553 972 919 365 836 364 815 474 491 392 × 2 = 1 + 0.901 946 852 435 087 743 211 107 945 838 731 672 729 630 948 982 784;
  • 19) 0.901 946 852 435 087 743 211 107 945 838 731 672 729 630 948 982 784 × 2 = 1 + 0.803 893 704 870 175 486 422 215 891 677 463 345 459 261 897 965 568;
  • 20) 0.803 893 704 870 175 486 422 215 891 677 463 345 459 261 897 965 568 × 2 = 1 + 0.607 787 409 740 350 972 844 431 783 354 926 690 918 523 795 931 136;
  • 21) 0.607 787 409 740 350 972 844 431 783 354 926 690 918 523 795 931 136 × 2 = 1 + 0.215 574 819 480 701 945 688 863 566 709 853 381 837 047 591 862 272;
  • 22) 0.215 574 819 480 701 945 688 863 566 709 853 381 837 047 591 862 272 × 2 = 0 + 0.431 149 638 961 403 891 377 727 133 419 706 763 674 095 183 724 544;
  • 23) 0.431 149 638 961 403 891 377 727 133 419 706 763 674 095 183 724 544 × 2 = 0 + 0.862 299 277 922 807 782 755 454 266 839 413 527 348 190 367 449 088;
  • 24) 0.862 299 277 922 807 782 755 454 266 839 413 527 348 190 367 449 088 × 2 = 1 + 0.724 598 555 845 615 565 510 908 533 678 827 054 696 380 734 898 176;
  • 25) 0.724 598 555 845 615 565 510 908 533 678 827 054 696 380 734 898 176 × 2 = 1 + 0.449 197 111 691 231 131 021 817 067 357 654 109 392 761 469 796 352;
  • 26) 0.449 197 111 691 231 131 021 817 067 357 654 109 392 761 469 796 352 × 2 = 0 + 0.898 394 223 382 462 262 043 634 134 715 308 218 785 522 939 592 704;
  • 27) 0.898 394 223 382 462 262 043 634 134 715 308 218 785 522 939 592 704 × 2 = 1 + 0.796 788 446 764 924 524 087 268 269 430 616 437 571 045 879 185 408;
  • 28) 0.796 788 446 764 924 524 087 268 269 430 616 437 571 045 879 185 408 × 2 = 1 + 0.593 576 893 529 849 048 174 536 538 861 232 875 142 091 758 370 816;
  • 29) 0.593 576 893 529 849 048 174 536 538 861 232 875 142 091 758 370 816 × 2 = 1 + 0.187 153 787 059 698 096 349 073 077 722 465 750 284 183 516 741 632;
  • 30) 0.187 153 787 059 698 096 349 073 077 722 465 750 284 183 516 741 632 × 2 = 0 + 0.374 307 574 119 396 192 698 146 155 444 931 500 568 367 033 483 264;
  • 31) 0.374 307 574 119 396 192 698 146 155 444 931 500 568 367 033 483 264 × 2 = 0 + 0.748 615 148 238 792 385 396 292 310 889 863 001 136 734 066 966 528;
  • 32) 0.748 615 148 238 792 385 396 292 310 889 863 001 136 734 066 966 528 × 2 = 1 + 0.497 230 296 477 584 770 792 584 621 779 726 002 273 468 133 933 056;
  • 33) 0.497 230 296 477 584 770 792 584 621 779 726 002 273 468 133 933 056 × 2 = 0 + 0.994 460 592 955 169 541 585 169 243 559 452 004 546 936 267 866 112;
  • 34) 0.994 460 592 955 169 541 585 169 243 559 452 004 546 936 267 866 112 × 2 = 1 + 0.988 921 185 910 339 083 170 338 487 118 904 009 093 872 535 732 224;
  • 35) 0.988 921 185 910 339 083 170 338 487 118 904 009 093 872 535 732 224 × 2 = 1 + 0.977 842 371 820 678 166 340 676 974 237 808 018 187 745 071 464 448;
  • 36) 0.977 842 371 820 678 166 340 676 974 237 808 018 187 745 071 464 448 × 2 = 1 + 0.955 684 743 641 356 332 681 353 948 475 616 036 375 490 142 928 896;
  • 37) 0.955 684 743 641 356 332 681 353 948 475 616 036 375 490 142 928 896 × 2 = 1 + 0.911 369 487 282 712 665 362 707 896 951 232 072 750 980 285 857 792;
  • 38) 0.911 369 487 282 712 665 362 707 896 951 232 072 750 980 285 857 792 × 2 = 1 + 0.822 738 974 565 425 330 725 415 793 902 464 145 501 960 571 715 584;
  • 39) 0.822 738 974 565 425 330 725 415 793 902 464 145 501 960 571 715 584 × 2 = 1 + 0.645 477 949 130 850 661 450 831 587 804 928 291 003 921 143 431 168;
  • 40) 0.645 477 949 130 850 661 450 831 587 804 928 291 003 921 143 431 168 × 2 = 1 + 0.290 955 898 261 701 322 901 663 175 609 856 582 007 842 286 862 336;
  • 41) 0.290 955 898 261 701 322 901 663 175 609 856 582 007 842 286 862 336 × 2 = 0 + 0.581 911 796 523 402 645 803 326 351 219 713 164 015 684 573 724 672;
  • 42) 0.581 911 796 523 402 645 803 326 351 219 713 164 015 684 573 724 672 × 2 = 1 + 0.163 823 593 046 805 291 606 652 702 439 426 328 031 369 147 449 344;
  • 43) 0.163 823 593 046 805 291 606 652 702 439 426 328 031 369 147 449 344 × 2 = 0 + 0.327 647 186 093 610 583 213 305 404 878 852 656 062 738 294 898 688;
  • 44) 0.327 647 186 093 610 583 213 305 404 878 852 656 062 738 294 898 688 × 2 = 0 + 0.655 294 372 187 221 166 426 610 809 757 705 312 125 476 589 797 376;
  • 45) 0.655 294 372 187 221 166 426 610 809 757 705 312 125 476 589 797 376 × 2 = 1 + 0.310 588 744 374 442 332 853 221 619 515 410 624 250 953 179 594 752;
  • 46) 0.310 588 744 374 442 332 853 221 619 515 410 624 250 953 179 594 752 × 2 = 0 + 0.621 177 488 748 884 665 706 443 239 030 821 248 501 906 359 189 504;
  • 47) 0.621 177 488 748 884 665 706 443 239 030 821 248 501 906 359 189 504 × 2 = 1 + 0.242 354 977 497 769 331 412 886 478 061 642 497 003 812 718 379 008;
  • 48) 0.242 354 977 497 769 331 412 886 478 061 642 497 003 812 718 379 008 × 2 = 0 + 0.484 709 954 995 538 662 825 772 956 123 284 994 007 625 436 758 016;
  • 49) 0.484 709 954 995 538 662 825 772 956 123 284 994 007 625 436 758 016 × 2 = 0 + 0.969 419 909 991 077 325 651 545 912 246 569 988 015 250 873 516 032;
  • 50) 0.969 419 909 991 077 325 651 545 912 246 569 988 015 250 873 516 032 × 2 = 1 + 0.938 839 819 982 154 651 303 091 824 493 139 976 030 501 747 032 064;
  • 51) 0.938 839 819 982 154 651 303 091 824 493 139 976 030 501 747 032 064 × 2 = 1 + 0.877 679 639 964 309 302 606 183 648 986 279 952 061 003 494 064 128;
  • 52) 0.877 679 639 964 309 302 606 183 648 986 279 952 061 003 494 064 128 × 2 = 1 + 0.755 359 279 928 618 605 212 367 297 972 559 904 122 006 988 128 256;
  • 53) 0.755 359 279 928 618 605 212 367 297 972 559 904 122 006 988 128 256 × 2 = 1 + 0.510 718 559 857 237 210 424 734 595 945 119 808 244 013 976 256 512;

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:


0.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 561(10) =

0.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(2)

5. Positive number before normalization:

1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 561(10) =

1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 561(10) =

1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(2) =

1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1(2) × 20


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): 0

Mantissa (not normalized):
1.1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 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) =

1023(10) =

011 1111 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111 1 =

1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111


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 1111 1111

Mantissa (52 bits) =
1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111


Decimal number 1.618 033 988 749 894 848 204 586 834 365 638 117 720 309 179 805 561 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 1001 1110 0011 0111 0111 1001 1011 1001 0111 1111 0100 1010 0111

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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