-0.000 806 264 623 585 362 514 063 654 156 856 294 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 294(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
-0.000 806 264 623 585 362 514 063 654 156 856 294(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 294| = 0.000 806 264 623 585 362 514 063 654 156 856 294


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: 0.000 806 264 623 585 362 514 063 654 156 856 294.

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) 0.000 806 264 623 585 362 514 063 654 156 856 294 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 588;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 588 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 425 176;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 425 176 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 850 352;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 850 352 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 700 704;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 700 704 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 401 408;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 401 408 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 802 816;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 802 816 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 605 632;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 605 632 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 211 264;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 211 264 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 422 528;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 422 528 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 845 056;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 845 056 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 690 112;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 690 112 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 483 380 224;
  • 13) 0.302 459 898 205 644 857 604 727 426 483 380 224 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 966 760 448;
  • 14) 0.604 919 796 411 289 715 209 454 852 966 760 448 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 933 520 896;
  • 15) 0.209 839 592 822 579 430 418 909 705 933 520 896 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 867 041 792;
  • 16) 0.419 679 185 645 158 860 837 819 411 867 041 792 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 734 083 584;
  • 17) 0.839 358 371 290 317 721 675 638 823 734 083 584 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 468 167 168;
  • 18) 0.678 716 742 580 635 443 351 277 647 468 167 168 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 936 334 336;
  • 19) 0.357 433 485 161 270 886 702 555 294 936 334 336 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 872 668 672;
  • 20) 0.714 866 970 322 541 773 405 110 589 872 668 672 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 745 337 344;
  • 21) 0.429 733 940 645 083 546 810 221 179 745 337 344 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 490 674 688;
  • 22) 0.859 467 881 290 167 093 620 442 359 490 674 688 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 981 349 376;
  • 23) 0.718 935 762 580 334 187 240 884 718 981 349 376 × 2 = 1 + 0.437 871 525 160 668 374 481 769 437 962 698 752;
  • 24) 0.437 871 525 160 668 374 481 769 437 962 698 752 × 2 = 0 + 0.875 743 050 321 336 748 963 538 875 925 397 504;
  • 25) 0.875 743 050 321 336 748 963 538 875 925 397 504 × 2 = 1 + 0.751 486 100 642 673 497 927 077 751 850 795 008;
  • 26) 0.751 486 100 642 673 497 927 077 751 850 795 008 × 2 = 1 + 0.502 972 201 285 346 995 854 155 503 701 590 016;
  • 27) 0.502 972 201 285 346 995 854 155 503 701 590 016 × 2 = 1 + 0.005 944 402 570 693 991 708 311 007 403 180 032;
  • 28) 0.005 944 402 570 693 991 708 311 007 403 180 032 × 2 = 0 + 0.011 888 805 141 387 983 416 622 014 806 360 064;
  • 29) 0.011 888 805 141 387 983 416 622 014 806 360 064 × 2 = 0 + 0.023 777 610 282 775 966 833 244 029 612 720 128;
  • 30) 0.023 777 610 282 775 966 833 244 029 612 720 128 × 2 = 0 + 0.047 555 220 565 551 933 666 488 059 225 440 256;
  • 31) 0.047 555 220 565 551 933 666 488 059 225 440 256 × 2 = 0 + 0.095 110 441 131 103 867 332 976 118 450 880 512;
  • 32) 0.095 110 441 131 103 867 332 976 118 450 880 512 × 2 = 0 + 0.190 220 882 262 207 734 665 952 236 901 761 024;
  • 33) 0.190 220 882 262 207 734 665 952 236 901 761 024 × 2 = 0 + 0.380 441 764 524 415 469 331 904 473 803 522 048;
  • 34) 0.380 441 764 524 415 469 331 904 473 803 522 048 × 2 = 0 + 0.760 883 529 048 830 938 663 808 947 607 044 096;
  • 35) 0.760 883 529 048 830 938 663 808 947 607 044 096 × 2 = 1 + 0.521 767 058 097 661 877 327 617 895 214 088 192;
  • 36) 0.521 767 058 097 661 877 327 617 895 214 088 192 × 2 = 1 + 0.043 534 116 195 323 754 655 235 790 428 176 384;
  • 37) 0.043 534 116 195 323 754 655 235 790 428 176 384 × 2 = 0 + 0.087 068 232 390 647 509 310 471 580 856 352 768;
  • 38) 0.087 068 232 390 647 509 310 471 580 856 352 768 × 2 = 0 + 0.174 136 464 781 295 018 620 943 161 712 705 536;
  • 39) 0.174 136 464 781 295 018 620 943 161 712 705 536 × 2 = 0 + 0.348 272 929 562 590 037 241 886 323 425 411 072;
  • 40) 0.348 272 929 562 590 037 241 886 323 425 411 072 × 2 = 0 + 0.696 545 859 125 180 074 483 772 646 850 822 144;
  • 41) 0.696 545 859 125 180 074 483 772 646 850 822 144 × 2 = 1 + 0.393 091 718 250 360 148 967 545 293 701 644 288;
  • 42) 0.393 091 718 250 360 148 967 545 293 701 644 288 × 2 = 0 + 0.786 183 436 500 720 297 935 090 587 403 288 576;
  • 43) 0.786 183 436 500 720 297 935 090 587 403 288 576 × 2 = 1 + 0.572 366 873 001 440 595 870 181 174 806 577 152;
  • 44) 0.572 366 873 001 440 595 870 181 174 806 577 152 × 2 = 1 + 0.144 733 746 002 881 191 740 362 349 613 154 304;
  • 45) 0.144 733 746 002 881 191 740 362 349 613 154 304 × 2 = 0 + 0.289 467 492 005 762 383 480 724 699 226 308 608;
  • 46) 0.289 467 492 005 762 383 480 724 699 226 308 608 × 2 = 0 + 0.578 934 984 011 524 766 961 449 398 452 617 216;
  • 47) 0.578 934 984 011 524 766 961 449 398 452 617 216 × 2 = 1 + 0.157 869 968 023 049 533 922 898 796 905 234 432;
  • 48) 0.157 869 968 023 049 533 922 898 796 905 234 432 × 2 = 0 + 0.315 739 936 046 099 067 845 797 593 810 468 864;
  • 49) 0.315 739 936 046 099 067 845 797 593 810 468 864 × 2 = 0 + 0.631 479 872 092 198 135 691 595 187 620 937 728;
  • 50) 0.631 479 872 092 198 135 691 595 187 620 937 728 × 2 = 1 + 0.262 959 744 184 396 271 383 190 375 241 875 456;
  • 51) 0.262 959 744 184 396 271 383 190 375 241 875 456 × 2 = 0 + 0.525 919 488 368 792 542 766 380 750 483 750 912;
  • 52) 0.525 919 488 368 792 542 766 380 750 483 750 912 × 2 = 1 + 0.051 838 976 737 585 085 532 761 500 967 501 824;
  • 53) 0.051 838 976 737 585 085 532 761 500 967 501 824 × 2 = 0 + 0.103 677 953 475 170 171 065 523 001 935 003 648;
  • 54) 0.103 677 953 475 170 171 065 523 001 935 003 648 × 2 = 0 + 0.207 355 906 950 340 342 131 046 003 870 007 296;
  • 55) 0.207 355 906 950 340 342 131 046 003 870 007 296 × 2 = 0 + 0.414 711 813 900 680 684 262 092 007 740 014 592;
  • 56) 0.414 711 813 900 680 684 262 092 007 740 014 592 × 2 = 0 + 0.829 423 627 801 361 368 524 184 015 480 029 184;
  • 57) 0.829 423 627 801 361 368 524 184 015 480 029 184 × 2 = 1 + 0.658 847 255 602 722 737 048 368 030 960 058 368;
  • 58) 0.658 847 255 602 722 737 048 368 030 960 058 368 × 2 = 1 + 0.317 694 511 205 445 474 096 736 061 920 116 736;
  • 59) 0.317 694 511 205 445 474 096 736 061 920 116 736 × 2 = 0 + 0.635 389 022 410 890 948 193 472 123 840 233 472;
  • 60) 0.635 389 022 410 890 948 193 472 123 840 233 472 × 2 = 1 + 0.270 778 044 821 781 896 386 944 247 680 466 944;
  • 61) 0.270 778 044 821 781 896 386 944 247 680 466 944 × 2 = 0 + 0.541 556 089 643 563 792 773 888 495 360 933 888;
  • 62) 0.541 556 089 643 563 792 773 888 495 360 933 888 × 2 = 1 + 0.083 112 179 287 127 585 547 776 990 721 867 776;
  • 63) 0.083 112 179 287 127 585 547 776 990 721 867 776 × 2 = 0 + 0.166 224 358 574 255 171 095 553 981 443 735 552;

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:


0.000 806 264 623 585 362 514 063 654 156 856 294(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 294(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:


0.000 806 264 623 585 362 514 063 654 156 856 294(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 -0.000 806 264 623 585 362 514 063 654 156 856 294 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