-0.000 806 264 623 585 362 514 063 654 156 856 100 59 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 100 59(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
-0.000 806 264 623 585 362 514 063 654 156 856 100 59(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 100 59| = 0.000 806 264 623 585 362 514 063 654 156 856 100 59


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: 0.000 806 264 623 585 362 514 063 654 156 856 100 59.

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) 0.000 806 264 623 585 362 514 063 654 156 856 100 59 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 201 18;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 201 18 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 402 36;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 402 36 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 804 72;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 804 72 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 609 44;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 609 44 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 218 88;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 218 88 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 437 76;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 437 76 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 580 875 52;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 580 875 52 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 161 751 04;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 161 751 04 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 323 502 08;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 323 502 08 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 647 004 16;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 647 004 16 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 294 008 32;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 294 008 32 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 588 016 64;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 588 016 64 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 176 033 28;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 176 033 28 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 352 066 56;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 352 066 56 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 704 133 12;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 704 133 12 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 408 266 24;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 408 266 24 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 442 816 532 48;
  • 18) 0.678 716 742 580 635 443 351 277 647 442 816 532 48 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 885 633 064 96;
  • 19) 0.357 433 485 161 270 886 702 555 294 885 633 064 96 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 771 266 129 92;
  • 20) 0.714 866 970 322 541 773 405 110 589 771 266 129 92 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 542 532 259 84;
  • 21) 0.429 733 940 645 083 546 810 221 179 542 532 259 84 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 085 064 519 68;
  • 22) 0.859 467 881 290 167 093 620 442 359 085 064 519 68 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 170 129 039 36;
  • 23) 0.718 935 762 580 334 187 240 884 718 170 129 039 36 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 340 258 078 72;
  • 24) 0.437 871 525 160 668 374 481 769 436 340 258 078 72 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 680 516 157 44;
  • 25) 0.875 743 050 321 336 748 963 538 872 680 516 157 44 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 361 032 314 88;
  • 26) 0.751 486 100 642 673 497 927 077 745 361 032 314 88 × 2 = 1 + 0.502 972 201 285 346 995 854 155 490 722 064 629 76;
  • 27) 0.502 972 201 285 346 995 854 155 490 722 064 629 76 × 2 = 1 + 0.005 944 402 570 693 991 708 310 981 444 129 259 52;
  • 28) 0.005 944 402 570 693 991 708 310 981 444 129 259 52 × 2 = 0 + 0.011 888 805 141 387 983 416 621 962 888 258 519 04;
  • 29) 0.011 888 805 141 387 983 416 621 962 888 258 519 04 × 2 = 0 + 0.023 777 610 282 775 966 833 243 925 776 517 038 08;
  • 30) 0.023 777 610 282 775 966 833 243 925 776 517 038 08 × 2 = 0 + 0.047 555 220 565 551 933 666 487 851 553 034 076 16;
  • 31) 0.047 555 220 565 551 933 666 487 851 553 034 076 16 × 2 = 0 + 0.095 110 441 131 103 867 332 975 703 106 068 152 32;
  • 32) 0.095 110 441 131 103 867 332 975 703 106 068 152 32 × 2 = 0 + 0.190 220 882 262 207 734 665 951 406 212 136 304 64;
  • 33) 0.190 220 882 262 207 734 665 951 406 212 136 304 64 × 2 = 0 + 0.380 441 764 524 415 469 331 902 812 424 272 609 28;
  • 34) 0.380 441 764 524 415 469 331 902 812 424 272 609 28 × 2 = 0 + 0.760 883 529 048 830 938 663 805 624 848 545 218 56;
  • 35) 0.760 883 529 048 830 938 663 805 624 848 545 218 56 × 2 = 1 + 0.521 767 058 097 661 877 327 611 249 697 090 437 12;
  • 36) 0.521 767 058 097 661 877 327 611 249 697 090 437 12 × 2 = 1 + 0.043 534 116 195 323 754 655 222 499 394 180 874 24;
  • 37) 0.043 534 116 195 323 754 655 222 499 394 180 874 24 × 2 = 0 + 0.087 068 232 390 647 509 310 444 998 788 361 748 48;
  • 38) 0.087 068 232 390 647 509 310 444 998 788 361 748 48 × 2 = 0 + 0.174 136 464 781 295 018 620 889 997 576 723 496 96;
  • 39) 0.174 136 464 781 295 018 620 889 997 576 723 496 96 × 2 = 0 + 0.348 272 929 562 590 037 241 779 995 153 446 993 92;
  • 40) 0.348 272 929 562 590 037 241 779 995 153 446 993 92 × 2 = 0 + 0.696 545 859 125 180 074 483 559 990 306 893 987 84;
  • 41) 0.696 545 859 125 180 074 483 559 990 306 893 987 84 × 2 = 1 + 0.393 091 718 250 360 148 967 119 980 613 787 975 68;
  • 42) 0.393 091 718 250 360 148 967 119 980 613 787 975 68 × 2 = 0 + 0.786 183 436 500 720 297 934 239 961 227 575 951 36;
  • 43) 0.786 183 436 500 720 297 934 239 961 227 575 951 36 × 2 = 1 + 0.572 366 873 001 440 595 868 479 922 455 151 902 72;
  • 44) 0.572 366 873 001 440 595 868 479 922 455 151 902 72 × 2 = 1 + 0.144 733 746 002 881 191 736 959 844 910 303 805 44;
  • 45) 0.144 733 746 002 881 191 736 959 844 910 303 805 44 × 2 = 0 + 0.289 467 492 005 762 383 473 919 689 820 607 610 88;
  • 46) 0.289 467 492 005 762 383 473 919 689 820 607 610 88 × 2 = 0 + 0.578 934 984 011 524 766 947 839 379 641 215 221 76;
  • 47) 0.578 934 984 011 524 766 947 839 379 641 215 221 76 × 2 = 1 + 0.157 869 968 023 049 533 895 678 759 282 430 443 52;
  • 48) 0.157 869 968 023 049 533 895 678 759 282 430 443 52 × 2 = 0 + 0.315 739 936 046 099 067 791 357 518 564 860 887 04;
  • 49) 0.315 739 936 046 099 067 791 357 518 564 860 887 04 × 2 = 0 + 0.631 479 872 092 198 135 582 715 037 129 721 774 08;
  • 50) 0.631 479 872 092 198 135 582 715 037 129 721 774 08 × 2 = 1 + 0.262 959 744 184 396 271 165 430 074 259 443 548 16;
  • 51) 0.262 959 744 184 396 271 165 430 074 259 443 548 16 × 2 = 0 + 0.525 919 488 368 792 542 330 860 148 518 887 096 32;
  • 52) 0.525 919 488 368 792 542 330 860 148 518 887 096 32 × 2 = 1 + 0.051 838 976 737 585 084 661 720 297 037 774 192 64;
  • 53) 0.051 838 976 737 585 084 661 720 297 037 774 192 64 × 2 = 0 + 0.103 677 953 475 170 169 323 440 594 075 548 385 28;
  • 54) 0.103 677 953 475 170 169 323 440 594 075 548 385 28 × 2 = 0 + 0.207 355 906 950 340 338 646 881 188 151 096 770 56;
  • 55) 0.207 355 906 950 340 338 646 881 188 151 096 770 56 × 2 = 0 + 0.414 711 813 900 680 677 293 762 376 302 193 541 12;
  • 56) 0.414 711 813 900 680 677 293 762 376 302 193 541 12 × 2 = 0 + 0.829 423 627 801 361 354 587 524 752 604 387 082 24;
  • 57) 0.829 423 627 801 361 354 587 524 752 604 387 082 24 × 2 = 1 + 0.658 847 255 602 722 709 175 049 505 208 774 164 48;
  • 58) 0.658 847 255 602 722 709 175 049 505 208 774 164 48 × 2 = 1 + 0.317 694 511 205 445 418 350 099 010 417 548 328 96;
  • 59) 0.317 694 511 205 445 418 350 099 010 417 548 328 96 × 2 = 0 + 0.635 389 022 410 890 836 700 198 020 835 096 657 92;
  • 60) 0.635 389 022 410 890 836 700 198 020 835 096 657 92 × 2 = 1 + 0.270 778 044 821 781 673 400 396 041 670 193 315 84;
  • 61) 0.270 778 044 821 781 673 400 396 041 670 193 315 84 × 2 = 0 + 0.541 556 089 643 563 346 800 792 083 340 386 631 68;
  • 62) 0.541 556 089 643 563 346 800 792 083 340 386 631 68 × 2 = 1 + 0.083 112 179 287 126 693 601 584 166 680 773 263 36;
  • 63) 0.083 112 179 287 126 693 601 584 166 680 773 263 36 × 2 = 0 + 0.166 224 358 574 253 387 203 168 333 361 546 526 72;

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:


0.000 806 264 623 585 362 514 063 654 156 856 100 59(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 100 59(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:


0.000 806 264 623 585 362 514 063 654 156 856 100 59(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 -0.000 806 264 623 585 362 514 063 654 156 856 100 59 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