-0.000 806 264 623 585 362 514 063 654 156 856 105 006 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 006(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 006(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 006| = 0.000 806 264 623 585 362 514 063 654 156 856 105 006


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 006.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 006 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 012;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 012 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 024;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 024 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 048;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 048 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 096;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 096 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 360 192;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 360 192 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 720 384;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 720 384 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 440 768;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 440 768 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 881 536;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 881 536 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 763 072;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 763 072 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 526 144;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 526 144 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 052 288;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 052 288 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 104 576;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 104 576 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 209 152;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 209 152 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 418 304;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 418 304 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 848 836 608;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 848 836 608 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 697 673 216;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 697 673 216 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 395 346 432;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 395 346 432 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 790 692 864;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 790 692 864 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 581 385 728;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 581 385 728 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 162 771 456;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 162 771 456 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 325 542 912;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 325 542 912 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 651 085 824;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 651 085 824 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 302 171 648;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 302 171 648 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 754 604 343 296;
  • 25) 0.875 743 050 321 336 748 963 538 872 754 604 343 296 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 509 208 686 592;
  • 26) 0.751 486 100 642 673 497 927 077 745 509 208 686 592 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 018 417 373 184;
  • 27) 0.502 972 201 285 346 995 854 155 491 018 417 373 184 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 036 834 746 368;
  • 28) 0.005 944 402 570 693 991 708 310 982 036 834 746 368 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 073 669 492 736;
  • 29) 0.011 888 805 141 387 983 416 621 964 073 669 492 736 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 147 338 985 472;
  • 30) 0.023 777 610 282 775 966 833 243 928 147 338 985 472 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 294 677 970 944;
  • 31) 0.047 555 220 565 551 933 666 487 856 294 677 970 944 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 589 355 941 888;
  • 32) 0.095 110 441 131 103 867 332 975 712 589 355 941 888 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 178 711 883 776;
  • 33) 0.190 220 882 262 207 734 665 951 425 178 711 883 776 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 357 423 767 552;
  • 34) 0.380 441 764 524 415 469 331 902 850 357 423 767 552 × 2 = 0 + 0.760 883 529 048 830 938 663 805 700 714 847 535 104;
  • 35) 0.760 883 529 048 830 938 663 805 700 714 847 535 104 × 2 = 1 + 0.521 767 058 097 661 877 327 611 401 429 695 070 208;
  • 36) 0.521 767 058 097 661 877 327 611 401 429 695 070 208 × 2 = 1 + 0.043 534 116 195 323 754 655 222 802 859 390 140 416;
  • 37) 0.043 534 116 195 323 754 655 222 802 859 390 140 416 × 2 = 0 + 0.087 068 232 390 647 509 310 445 605 718 780 280 832;
  • 38) 0.087 068 232 390 647 509 310 445 605 718 780 280 832 × 2 = 0 + 0.174 136 464 781 295 018 620 891 211 437 560 561 664;
  • 39) 0.174 136 464 781 295 018 620 891 211 437 560 561 664 × 2 = 0 + 0.348 272 929 562 590 037 241 782 422 875 121 123 328;
  • 40) 0.348 272 929 562 590 037 241 782 422 875 121 123 328 × 2 = 0 + 0.696 545 859 125 180 074 483 564 845 750 242 246 656;
  • 41) 0.696 545 859 125 180 074 483 564 845 750 242 246 656 × 2 = 1 + 0.393 091 718 250 360 148 967 129 691 500 484 493 312;
  • 42) 0.393 091 718 250 360 148 967 129 691 500 484 493 312 × 2 = 0 + 0.786 183 436 500 720 297 934 259 383 000 968 986 624;
  • 43) 0.786 183 436 500 720 297 934 259 383 000 968 986 624 × 2 = 1 + 0.572 366 873 001 440 595 868 518 766 001 937 973 248;
  • 44) 0.572 366 873 001 440 595 868 518 766 001 937 973 248 × 2 = 1 + 0.144 733 746 002 881 191 737 037 532 003 875 946 496;
  • 45) 0.144 733 746 002 881 191 737 037 532 003 875 946 496 × 2 = 0 + 0.289 467 492 005 762 383 474 075 064 007 751 892 992;
  • 46) 0.289 467 492 005 762 383 474 075 064 007 751 892 992 × 2 = 0 + 0.578 934 984 011 524 766 948 150 128 015 503 785 984;
  • 47) 0.578 934 984 011 524 766 948 150 128 015 503 785 984 × 2 = 1 + 0.157 869 968 023 049 533 896 300 256 031 007 571 968;
  • 48) 0.157 869 968 023 049 533 896 300 256 031 007 571 968 × 2 = 0 + 0.315 739 936 046 099 067 792 600 512 062 015 143 936;
  • 49) 0.315 739 936 046 099 067 792 600 512 062 015 143 936 × 2 = 0 + 0.631 479 872 092 198 135 585 201 024 124 030 287 872;
  • 50) 0.631 479 872 092 198 135 585 201 024 124 030 287 872 × 2 = 1 + 0.262 959 744 184 396 271 170 402 048 248 060 575 744;
  • 51) 0.262 959 744 184 396 271 170 402 048 248 060 575 744 × 2 = 0 + 0.525 919 488 368 792 542 340 804 096 496 121 151 488;
  • 52) 0.525 919 488 368 792 542 340 804 096 496 121 151 488 × 2 = 1 + 0.051 838 976 737 585 084 681 608 192 992 242 302 976;
  • 53) 0.051 838 976 737 585 084 681 608 192 992 242 302 976 × 2 = 0 + 0.103 677 953 475 170 169 363 216 385 984 484 605 952;
  • 54) 0.103 677 953 475 170 169 363 216 385 984 484 605 952 × 2 = 0 + 0.207 355 906 950 340 338 726 432 771 968 969 211 904;
  • 55) 0.207 355 906 950 340 338 726 432 771 968 969 211 904 × 2 = 0 + 0.414 711 813 900 680 677 452 865 543 937 938 423 808;
  • 56) 0.414 711 813 900 680 677 452 865 543 937 938 423 808 × 2 = 0 + 0.829 423 627 801 361 354 905 731 087 875 876 847 616;
  • 57) 0.829 423 627 801 361 354 905 731 087 875 876 847 616 × 2 = 1 + 0.658 847 255 602 722 709 811 462 175 751 753 695 232;
  • 58) 0.658 847 255 602 722 709 811 462 175 751 753 695 232 × 2 = 1 + 0.317 694 511 205 445 419 622 924 351 503 507 390 464;
  • 59) 0.317 694 511 205 445 419 622 924 351 503 507 390 464 × 2 = 0 + 0.635 389 022 410 890 839 245 848 703 007 014 780 928;
  • 60) 0.635 389 022 410 890 839 245 848 703 007 014 780 928 × 2 = 1 + 0.270 778 044 821 781 678 491 697 406 014 029 561 856;
  • 61) 0.270 778 044 821 781 678 491 697 406 014 029 561 856 × 2 = 0 + 0.541 556 089 643 563 356 983 394 812 028 059 123 712;
  • 62) 0.541 556 089 643 563 356 983 394 812 028 059 123 712 × 2 = 1 + 0.083 112 179 287 126 713 966 789 624 056 118 247 424;
  • 63) 0.083 112 179 287 126 713 966 789 624 056 118 247 424 × 2 = 0 + 0.166 224 358 574 253 427 933 579 248 112 236 494 848;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 006(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 006(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 006(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 006 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