-0.000 806 264 623 585 362 514 063 654 156 856 104 814 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 104 814(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
-0.000 806 264 623 585 362 514 063 654 156 856 104 814(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 104 814| = 0.000 806 264 623 585 362 514 063 654 156 856 104 814


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: 0.000 806 264 623 585 362 514 063 654 156 856 104 814.

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) 0.000 806 264 623 585 362 514 063 654 156 856 104 814 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 209 628;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 209 628 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 419 256;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 419 256 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 838 512;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 838 512 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 677 024;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 677 024 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 354 048;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 354 048 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 708 096;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 708 096 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 416 192;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 416 192 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 832 384;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 832 384 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 664 768;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 664 768 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 329 536;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 329 536 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 302 659 072;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 302 659 072 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 605 318 144;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 605 318 144 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 210 636 288;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 210 636 288 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 421 272 576;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 421 272 576 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 842 545 152;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 842 545 152 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 685 090 304;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 685 090 304 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 370 180 608;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 370 180 608 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 740 361 216;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 740 361 216 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 480 722 432;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 480 722 432 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 546 961 444 864;
  • 21) 0.429 733 940 645 083 546 810 221 179 546 961 444 864 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 093 922 889 728;
  • 22) 0.859 467 881 290 167 093 620 442 359 093 922 889 728 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 187 845 779 456;
  • 23) 0.718 935 762 580 334 187 240 884 718 187 845 779 456 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 375 691 558 912;
  • 24) 0.437 871 525 160 668 374 481 769 436 375 691 558 912 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 751 383 117 824;
  • 25) 0.875 743 050 321 336 748 963 538 872 751 383 117 824 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 502 766 235 648;
  • 26) 0.751 486 100 642 673 497 927 077 745 502 766 235 648 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 005 532 471 296;
  • 27) 0.502 972 201 285 346 995 854 155 491 005 532 471 296 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 011 064 942 592;
  • 28) 0.005 944 402 570 693 991 708 310 982 011 064 942 592 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 022 129 885 184;
  • 29) 0.011 888 805 141 387 983 416 621 964 022 129 885 184 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 044 259 770 368;
  • 30) 0.023 777 610 282 775 966 833 243 928 044 259 770 368 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 088 519 540 736;
  • 31) 0.047 555 220 565 551 933 666 487 856 088 519 540 736 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 177 039 081 472;
  • 32) 0.095 110 441 131 103 867 332 975 712 177 039 081 472 × 2 = 0 + 0.190 220 882 262 207 734 665 951 424 354 078 162 944;
  • 33) 0.190 220 882 262 207 734 665 951 424 354 078 162 944 × 2 = 0 + 0.380 441 764 524 415 469 331 902 848 708 156 325 888;
  • 34) 0.380 441 764 524 415 469 331 902 848 708 156 325 888 × 2 = 0 + 0.760 883 529 048 830 938 663 805 697 416 312 651 776;
  • 35) 0.760 883 529 048 830 938 663 805 697 416 312 651 776 × 2 = 1 + 0.521 767 058 097 661 877 327 611 394 832 625 303 552;
  • 36) 0.521 767 058 097 661 877 327 611 394 832 625 303 552 × 2 = 1 + 0.043 534 116 195 323 754 655 222 789 665 250 607 104;
  • 37) 0.043 534 116 195 323 754 655 222 789 665 250 607 104 × 2 = 0 + 0.087 068 232 390 647 509 310 445 579 330 501 214 208;
  • 38) 0.087 068 232 390 647 509 310 445 579 330 501 214 208 × 2 = 0 + 0.174 136 464 781 295 018 620 891 158 661 002 428 416;
  • 39) 0.174 136 464 781 295 018 620 891 158 661 002 428 416 × 2 = 0 + 0.348 272 929 562 590 037 241 782 317 322 004 856 832;
  • 40) 0.348 272 929 562 590 037 241 782 317 322 004 856 832 × 2 = 0 + 0.696 545 859 125 180 074 483 564 634 644 009 713 664;
  • 41) 0.696 545 859 125 180 074 483 564 634 644 009 713 664 × 2 = 1 + 0.393 091 718 250 360 148 967 129 269 288 019 427 328;
  • 42) 0.393 091 718 250 360 148 967 129 269 288 019 427 328 × 2 = 0 + 0.786 183 436 500 720 297 934 258 538 576 038 854 656;
  • 43) 0.786 183 436 500 720 297 934 258 538 576 038 854 656 × 2 = 1 + 0.572 366 873 001 440 595 868 517 077 152 077 709 312;
  • 44) 0.572 366 873 001 440 595 868 517 077 152 077 709 312 × 2 = 1 + 0.144 733 746 002 881 191 737 034 154 304 155 418 624;
  • 45) 0.144 733 746 002 881 191 737 034 154 304 155 418 624 × 2 = 0 + 0.289 467 492 005 762 383 474 068 308 608 310 837 248;
  • 46) 0.289 467 492 005 762 383 474 068 308 608 310 837 248 × 2 = 0 + 0.578 934 984 011 524 766 948 136 617 216 621 674 496;
  • 47) 0.578 934 984 011 524 766 948 136 617 216 621 674 496 × 2 = 1 + 0.157 869 968 023 049 533 896 273 234 433 243 348 992;
  • 48) 0.157 869 968 023 049 533 896 273 234 433 243 348 992 × 2 = 0 + 0.315 739 936 046 099 067 792 546 468 866 486 697 984;
  • 49) 0.315 739 936 046 099 067 792 546 468 866 486 697 984 × 2 = 0 + 0.631 479 872 092 198 135 585 092 937 732 973 395 968;
  • 50) 0.631 479 872 092 198 135 585 092 937 732 973 395 968 × 2 = 1 + 0.262 959 744 184 396 271 170 185 875 465 946 791 936;
  • 51) 0.262 959 744 184 396 271 170 185 875 465 946 791 936 × 2 = 0 + 0.525 919 488 368 792 542 340 371 750 931 893 583 872;
  • 52) 0.525 919 488 368 792 542 340 371 750 931 893 583 872 × 2 = 1 + 0.051 838 976 737 585 084 680 743 501 863 787 167 744;
  • 53) 0.051 838 976 737 585 084 680 743 501 863 787 167 744 × 2 = 0 + 0.103 677 953 475 170 169 361 487 003 727 574 335 488;
  • 54) 0.103 677 953 475 170 169 361 487 003 727 574 335 488 × 2 = 0 + 0.207 355 906 950 340 338 722 974 007 455 148 670 976;
  • 55) 0.207 355 906 950 340 338 722 974 007 455 148 670 976 × 2 = 0 + 0.414 711 813 900 680 677 445 948 014 910 297 341 952;
  • 56) 0.414 711 813 900 680 677 445 948 014 910 297 341 952 × 2 = 0 + 0.829 423 627 801 361 354 891 896 029 820 594 683 904;
  • 57) 0.829 423 627 801 361 354 891 896 029 820 594 683 904 × 2 = 1 + 0.658 847 255 602 722 709 783 792 059 641 189 367 808;
  • 58) 0.658 847 255 602 722 709 783 792 059 641 189 367 808 × 2 = 1 + 0.317 694 511 205 445 419 567 584 119 282 378 735 616;
  • 59) 0.317 694 511 205 445 419 567 584 119 282 378 735 616 × 2 = 0 + 0.635 389 022 410 890 839 135 168 238 564 757 471 232;
  • 60) 0.635 389 022 410 890 839 135 168 238 564 757 471 232 × 2 = 1 + 0.270 778 044 821 781 678 270 336 477 129 514 942 464;
  • 61) 0.270 778 044 821 781 678 270 336 477 129 514 942 464 × 2 = 0 + 0.541 556 089 643 563 356 540 672 954 259 029 884 928;
  • 62) 0.541 556 089 643 563 356 540 672 954 259 029 884 928 × 2 = 1 + 0.083 112 179 287 126 713 081 345 908 518 059 769 856;
  • 63) 0.083 112 179 287 126 713 081 345 908 518 059 769 856 × 2 = 0 + 0.166 224 358 574 253 426 162 691 817 036 119 539 712;

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:


0.000 806 264 623 585 362 514 063 654 156 856 104 814(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 104 814(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:


0.000 806 264 623 585 362 514 063 654 156 856 104 814(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 -0.000 806 264 623 585 362 514 063 654 156 856 104 814 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