-0.000 806 264 623 585 362 514 063 654 156 856 093 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 093 4(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
-0.000 806 264 623 585 362 514 063 654 156 856 093 4(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 093 4| = 0.000 806 264 623 585 362 514 063 654 156 856 093 4


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: 0.000 806 264 623 585 362 514 063 654 156 856 093 4.

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) 0.000 806 264 623 585 362 514 063 654 156 856 093 4 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 186 8;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 186 8 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 373 6;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 373 6 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 747 2;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 747 2 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 494 4;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 494 4 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 394 988 8;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 394 988 8 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 789 977 6;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 789 977 6 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 579 955 2;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 579 955 2 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 159 910 4;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 159 910 4 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 319 820 8;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 319 820 8 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 639 641 6;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 639 641 6 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 279 283 2;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 279 283 2 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 558 566 4;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 558 566 4 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 117 132 8;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 117 132 8 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 234 265 6;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 234 265 6 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 468 531 2;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 468 531 2 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 720 937 062 4;
  • 17) 0.839 358 371 290 317 721 675 638 823 720 937 062 4 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 441 874 124 8;
  • 18) 0.678 716 742 580 635 443 351 277 647 441 874 124 8 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 883 748 249 6;
  • 19) 0.357 433 485 161 270 886 702 555 294 883 748 249 6 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 767 496 499 2;
  • 20) 0.714 866 970 322 541 773 405 110 589 767 496 499 2 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 534 992 998 4;
  • 21) 0.429 733 940 645 083 546 810 221 179 534 992 998 4 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 069 985 996 8;
  • 22) 0.859 467 881 290 167 093 620 442 359 069 985 996 8 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 139 971 993 6;
  • 23) 0.718 935 762 580 334 187 240 884 718 139 971 993 6 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 279 943 987 2;
  • 24) 0.437 871 525 160 668 374 481 769 436 279 943 987 2 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 559 887 974 4;
  • 25) 0.875 743 050 321 336 748 963 538 872 559 887 974 4 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 119 775 948 8;
  • 26) 0.751 486 100 642 673 497 927 077 745 119 775 948 8 × 2 = 1 + 0.502 972 201 285 346 995 854 155 490 239 551 897 6;
  • 27) 0.502 972 201 285 346 995 854 155 490 239 551 897 6 × 2 = 1 + 0.005 944 402 570 693 991 708 310 980 479 103 795 2;
  • 28) 0.005 944 402 570 693 991 708 310 980 479 103 795 2 × 2 = 0 + 0.011 888 805 141 387 983 416 621 960 958 207 590 4;
  • 29) 0.011 888 805 141 387 983 416 621 960 958 207 590 4 × 2 = 0 + 0.023 777 610 282 775 966 833 243 921 916 415 180 8;
  • 30) 0.023 777 610 282 775 966 833 243 921 916 415 180 8 × 2 = 0 + 0.047 555 220 565 551 933 666 487 843 832 830 361 6;
  • 31) 0.047 555 220 565 551 933 666 487 843 832 830 361 6 × 2 = 0 + 0.095 110 441 131 103 867 332 975 687 665 660 723 2;
  • 32) 0.095 110 441 131 103 867 332 975 687 665 660 723 2 × 2 = 0 + 0.190 220 882 262 207 734 665 951 375 331 321 446 4;
  • 33) 0.190 220 882 262 207 734 665 951 375 331 321 446 4 × 2 = 0 + 0.380 441 764 524 415 469 331 902 750 662 642 892 8;
  • 34) 0.380 441 764 524 415 469 331 902 750 662 642 892 8 × 2 = 0 + 0.760 883 529 048 830 938 663 805 501 325 285 785 6;
  • 35) 0.760 883 529 048 830 938 663 805 501 325 285 785 6 × 2 = 1 + 0.521 767 058 097 661 877 327 611 002 650 571 571 2;
  • 36) 0.521 767 058 097 661 877 327 611 002 650 571 571 2 × 2 = 1 + 0.043 534 116 195 323 754 655 222 005 301 143 142 4;
  • 37) 0.043 534 116 195 323 754 655 222 005 301 143 142 4 × 2 = 0 + 0.087 068 232 390 647 509 310 444 010 602 286 284 8;
  • 38) 0.087 068 232 390 647 509 310 444 010 602 286 284 8 × 2 = 0 + 0.174 136 464 781 295 018 620 888 021 204 572 569 6;
  • 39) 0.174 136 464 781 295 018 620 888 021 204 572 569 6 × 2 = 0 + 0.348 272 929 562 590 037 241 776 042 409 145 139 2;
  • 40) 0.348 272 929 562 590 037 241 776 042 409 145 139 2 × 2 = 0 + 0.696 545 859 125 180 074 483 552 084 818 290 278 4;
  • 41) 0.696 545 859 125 180 074 483 552 084 818 290 278 4 × 2 = 1 + 0.393 091 718 250 360 148 967 104 169 636 580 556 8;
  • 42) 0.393 091 718 250 360 148 967 104 169 636 580 556 8 × 2 = 0 + 0.786 183 436 500 720 297 934 208 339 273 161 113 6;
  • 43) 0.786 183 436 500 720 297 934 208 339 273 161 113 6 × 2 = 1 + 0.572 366 873 001 440 595 868 416 678 546 322 227 2;
  • 44) 0.572 366 873 001 440 595 868 416 678 546 322 227 2 × 2 = 1 + 0.144 733 746 002 881 191 736 833 357 092 644 454 4;
  • 45) 0.144 733 746 002 881 191 736 833 357 092 644 454 4 × 2 = 0 + 0.289 467 492 005 762 383 473 666 714 185 288 908 8;
  • 46) 0.289 467 492 005 762 383 473 666 714 185 288 908 8 × 2 = 0 + 0.578 934 984 011 524 766 947 333 428 370 577 817 6;
  • 47) 0.578 934 984 011 524 766 947 333 428 370 577 817 6 × 2 = 1 + 0.157 869 968 023 049 533 894 666 856 741 155 635 2;
  • 48) 0.157 869 968 023 049 533 894 666 856 741 155 635 2 × 2 = 0 + 0.315 739 936 046 099 067 789 333 713 482 311 270 4;
  • 49) 0.315 739 936 046 099 067 789 333 713 482 311 270 4 × 2 = 0 + 0.631 479 872 092 198 135 578 667 426 964 622 540 8;
  • 50) 0.631 479 872 092 198 135 578 667 426 964 622 540 8 × 2 = 1 + 0.262 959 744 184 396 271 157 334 853 929 245 081 6;
  • 51) 0.262 959 744 184 396 271 157 334 853 929 245 081 6 × 2 = 0 + 0.525 919 488 368 792 542 314 669 707 858 490 163 2;
  • 52) 0.525 919 488 368 792 542 314 669 707 858 490 163 2 × 2 = 1 + 0.051 838 976 737 585 084 629 339 415 716 980 326 4;
  • 53) 0.051 838 976 737 585 084 629 339 415 716 980 326 4 × 2 = 0 + 0.103 677 953 475 170 169 258 678 831 433 960 652 8;
  • 54) 0.103 677 953 475 170 169 258 678 831 433 960 652 8 × 2 = 0 + 0.207 355 906 950 340 338 517 357 662 867 921 305 6;
  • 55) 0.207 355 906 950 340 338 517 357 662 867 921 305 6 × 2 = 0 + 0.414 711 813 900 680 677 034 715 325 735 842 611 2;
  • 56) 0.414 711 813 900 680 677 034 715 325 735 842 611 2 × 2 = 0 + 0.829 423 627 801 361 354 069 430 651 471 685 222 4;
  • 57) 0.829 423 627 801 361 354 069 430 651 471 685 222 4 × 2 = 1 + 0.658 847 255 602 722 708 138 861 302 943 370 444 8;
  • 58) 0.658 847 255 602 722 708 138 861 302 943 370 444 8 × 2 = 1 + 0.317 694 511 205 445 416 277 722 605 886 740 889 6;
  • 59) 0.317 694 511 205 445 416 277 722 605 886 740 889 6 × 2 = 0 + 0.635 389 022 410 890 832 555 445 211 773 481 779 2;
  • 60) 0.635 389 022 410 890 832 555 445 211 773 481 779 2 × 2 = 1 + 0.270 778 044 821 781 665 110 890 423 546 963 558 4;
  • 61) 0.270 778 044 821 781 665 110 890 423 546 963 558 4 × 2 = 0 + 0.541 556 089 643 563 330 221 780 847 093 927 116 8;
  • 62) 0.541 556 089 643 563 330 221 780 847 093 927 116 8 × 2 = 1 + 0.083 112 179 287 126 660 443 561 694 187 854 233 6;
  • 63) 0.083 112 179 287 126 660 443 561 694 187 854 233 6 × 2 = 0 + 0.166 224 358 574 253 320 887 123 388 375 708 467 2;

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:


0.000 806 264 623 585 362 514 063 654 156 856 093 4(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 093 4(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:


0.000 806 264 623 585 362 514 063 654 156 856 093 4(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 -0.000 806 264 623 585 362 514 063 654 156 856 093 4 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