-0.000 806 264 623 585 362 514 063 654 156 856 102 98 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 102 98(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
-0.000 806 264 623 585 362 514 063 654 156 856 102 98(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. Start with the positive version of the number:

|-0.000 806 264 623 585 362 514 063 654 156 856 102 98| = 0.000 806 264 623 585 362 514 063 654 156 856 102 98


2. 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;

3. 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)


4. Convert to binary (base 2) the fractional part: 0.000 806 264 623 585 362 514 063 654 156 856 102 98.

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) 0.000 806 264 623 585 362 514 063 654 156 856 102 98 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 205 96;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 205 96 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 411 92;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 411 92 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 823 84;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 823 84 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 647 68;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 647 68 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 295 36;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 295 36 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 590 72;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 590 72 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 181 44;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 181 44 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 362 88;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 362 88 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 324 725 76;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 324 725 76 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 649 451 52;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 649 451 52 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 298 903 04;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 298 903 04 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 597 806 08;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 597 806 08 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 195 612 16;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 195 612 16 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 391 224 32;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 391 224 32 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 782 448 64;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 782 448 64 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 564 897 28;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 564 897 28 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 129 794 56;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 129 794 56 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 259 589 12;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 259 589 12 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 772 519 178 24;
  • 20) 0.714 866 970 322 541 773 405 110 589 772 519 178 24 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 545 038 356 48;
  • 21) 0.429 733 940 645 083 546 810 221 179 545 038 356 48 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 090 076 712 96;
  • 22) 0.859 467 881 290 167 093 620 442 359 090 076 712 96 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 180 153 425 92;
  • 23) 0.718 935 762 580 334 187 240 884 718 180 153 425 92 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 360 306 851 84;
  • 24) 0.437 871 525 160 668 374 481 769 436 360 306 851 84 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 720 613 703 68;
  • 25) 0.875 743 050 321 336 748 963 538 872 720 613 703 68 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 441 227 407 36;
  • 26) 0.751 486 100 642 673 497 927 077 745 441 227 407 36 × 2 = 1 + 0.502 972 201 285 346 995 854 155 490 882 454 814 72;
  • 27) 0.502 972 201 285 346 995 854 155 490 882 454 814 72 × 2 = 1 + 0.005 944 402 570 693 991 708 310 981 764 909 629 44;
  • 28) 0.005 944 402 570 693 991 708 310 981 764 909 629 44 × 2 = 0 + 0.011 888 805 141 387 983 416 621 963 529 819 258 88;
  • 29) 0.011 888 805 141 387 983 416 621 963 529 819 258 88 × 2 = 0 + 0.023 777 610 282 775 966 833 243 927 059 638 517 76;
  • 30) 0.023 777 610 282 775 966 833 243 927 059 638 517 76 × 2 = 0 + 0.047 555 220 565 551 933 666 487 854 119 277 035 52;
  • 31) 0.047 555 220 565 551 933 666 487 854 119 277 035 52 × 2 = 0 + 0.095 110 441 131 103 867 332 975 708 238 554 071 04;
  • 32) 0.095 110 441 131 103 867 332 975 708 238 554 071 04 × 2 = 0 + 0.190 220 882 262 207 734 665 951 416 477 108 142 08;
  • 33) 0.190 220 882 262 207 734 665 951 416 477 108 142 08 × 2 = 0 + 0.380 441 764 524 415 469 331 902 832 954 216 284 16;
  • 34) 0.380 441 764 524 415 469 331 902 832 954 216 284 16 × 2 = 0 + 0.760 883 529 048 830 938 663 805 665 908 432 568 32;
  • 35) 0.760 883 529 048 830 938 663 805 665 908 432 568 32 × 2 = 1 + 0.521 767 058 097 661 877 327 611 331 816 865 136 64;
  • 36) 0.521 767 058 097 661 877 327 611 331 816 865 136 64 × 2 = 1 + 0.043 534 116 195 323 754 655 222 663 633 730 273 28;
  • 37) 0.043 534 116 195 323 754 655 222 663 633 730 273 28 × 2 = 0 + 0.087 068 232 390 647 509 310 445 327 267 460 546 56;
  • 38) 0.087 068 232 390 647 509 310 445 327 267 460 546 56 × 2 = 0 + 0.174 136 464 781 295 018 620 890 654 534 921 093 12;
  • 39) 0.174 136 464 781 295 018 620 890 654 534 921 093 12 × 2 = 0 + 0.348 272 929 562 590 037 241 781 309 069 842 186 24;
  • 40) 0.348 272 929 562 590 037 241 781 309 069 842 186 24 × 2 = 0 + 0.696 545 859 125 180 074 483 562 618 139 684 372 48;
  • 41) 0.696 545 859 125 180 074 483 562 618 139 684 372 48 × 2 = 1 + 0.393 091 718 250 360 148 967 125 236 279 368 744 96;
  • 42) 0.393 091 718 250 360 148 967 125 236 279 368 744 96 × 2 = 0 + 0.786 183 436 500 720 297 934 250 472 558 737 489 92;
  • 43) 0.786 183 436 500 720 297 934 250 472 558 737 489 92 × 2 = 1 + 0.572 366 873 001 440 595 868 500 945 117 474 979 84;
  • 44) 0.572 366 873 001 440 595 868 500 945 117 474 979 84 × 2 = 1 + 0.144 733 746 002 881 191 737 001 890 234 949 959 68;
  • 45) 0.144 733 746 002 881 191 737 001 890 234 949 959 68 × 2 = 0 + 0.289 467 492 005 762 383 474 003 780 469 899 919 36;
  • 46) 0.289 467 492 005 762 383 474 003 780 469 899 919 36 × 2 = 0 + 0.578 934 984 011 524 766 948 007 560 939 799 838 72;
  • 47) 0.578 934 984 011 524 766 948 007 560 939 799 838 72 × 2 = 1 + 0.157 869 968 023 049 533 896 015 121 879 599 677 44;
  • 48) 0.157 869 968 023 049 533 896 015 121 879 599 677 44 × 2 = 0 + 0.315 739 936 046 099 067 792 030 243 759 199 354 88;
  • 49) 0.315 739 936 046 099 067 792 030 243 759 199 354 88 × 2 = 0 + 0.631 479 872 092 198 135 584 060 487 518 398 709 76;
  • 50) 0.631 479 872 092 198 135 584 060 487 518 398 709 76 × 2 = 1 + 0.262 959 744 184 396 271 168 120 975 036 797 419 52;
  • 51) 0.262 959 744 184 396 271 168 120 975 036 797 419 52 × 2 = 0 + 0.525 919 488 368 792 542 336 241 950 073 594 839 04;
  • 52) 0.525 919 488 368 792 542 336 241 950 073 594 839 04 × 2 = 1 + 0.051 838 976 737 585 084 672 483 900 147 189 678 08;
  • 53) 0.051 838 976 737 585 084 672 483 900 147 189 678 08 × 2 = 0 + 0.103 677 953 475 170 169 344 967 800 294 379 356 16;
  • 54) 0.103 677 953 475 170 169 344 967 800 294 379 356 16 × 2 = 0 + 0.207 355 906 950 340 338 689 935 600 588 758 712 32;
  • 55) 0.207 355 906 950 340 338 689 935 600 588 758 712 32 × 2 = 0 + 0.414 711 813 900 680 677 379 871 201 177 517 424 64;
  • 56) 0.414 711 813 900 680 677 379 871 201 177 517 424 64 × 2 = 0 + 0.829 423 627 801 361 354 759 742 402 355 034 849 28;
  • 57) 0.829 423 627 801 361 354 759 742 402 355 034 849 28 × 2 = 1 + 0.658 847 255 602 722 709 519 484 804 710 069 698 56;
  • 58) 0.658 847 255 602 722 709 519 484 804 710 069 698 56 × 2 = 1 + 0.317 694 511 205 445 419 038 969 609 420 139 397 12;
  • 59) 0.317 694 511 205 445 419 038 969 609 420 139 397 12 × 2 = 0 + 0.635 389 022 410 890 838 077 939 218 840 278 794 24;
  • 60) 0.635 389 022 410 890 838 077 939 218 840 278 794 24 × 2 = 1 + 0.270 778 044 821 781 676 155 878 437 680 557 588 48;
  • 61) 0.270 778 044 821 781 676 155 878 437 680 557 588 48 × 2 = 0 + 0.541 556 089 643 563 352 311 756 875 361 115 176 96;
  • 62) 0.541 556 089 643 563 352 311 756 875 361 115 176 96 × 2 = 1 + 0.083 112 179 287 126 704 623 513 750 722 230 353 92;
  • 63) 0.083 112 179 287 126 704 623 513 750 722 230 353 92 × 2 = 0 + 0.166 224 358 574 253 409 247 027 501 444 460 707 84;

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).


5. 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:


0.000 806 264 623 585 362 514 063 654 156 856 102 98(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2)

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 102 98(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 11 positions to the right, so that only one non zero digit remains to the left of it:


0.000 806 264 623 585 362 514 063 654 156 856 102 98(10) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2) =

0.0000 0000 0011 0100 1101 0110 1110 0000 0011 0000 1011 0010 0101 0000 1101 010(2) × 20 =

1.1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010(2) × 2-11


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): -11

Mantissa (not normalized):
1.1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-11 + 1 023)(10) =

1 012(10)


10. 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;
  • 1 012 ÷ 2 = 506 + 0;
  • 506 ÷ 2 = 253 + 0;
  • 253 ÷ 2 = 126 + 1;
  • 126 ÷ 2 = 63 + 0;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1012(10) =

011 1111 0100(2)


12. 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. 1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010 =

1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
011 1111 0100

Mantissa (52 bits) =
1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010


Decimal number -0.000 806 264 623 585 362 514 063 654 156 856 102 98 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0100 - 1010 0110 1011 0111 0000 0001 1000 0101 1001 0010 1000 0110 1010

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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