-0.000 806 264 623 585 362 514 063 654 156 856 104 981 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 104 981(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
-0.000 806 264 623 585 362 514 063 654 156 856 104 981(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 104 981| = 0.000 806 264 623 585 362 514 063 654 156 856 104 981


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: 0.000 806 264 623 585 362 514 063 654 156 856 104 981.

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) 0.000 806 264 623 585 362 514 063 654 156 856 104 981 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 209 962;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 209 962 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 419 924;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 419 924 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 839 848;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 839 848 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 679 696;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 679 696 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 359 392;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 359 392 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 718 784;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 718 784 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 437 568;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 437 568 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 875 136;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 875 136 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 750 272;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 750 272 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 500 544;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 500 544 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 001 088;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 001 088 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 002 176;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 002 176 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 004 352;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 004 352 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 008 704;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 008 704 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 848 017 408;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 848 017 408 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 696 034 816;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 696 034 816 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 392 069 632;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 392 069 632 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 784 139 264;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 784 139 264 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 568 278 528;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 568 278 528 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 136 557 056;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 136 557 056 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 273 114 112;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 273 114 112 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 546 228 224;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 546 228 224 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 092 456 448;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 092 456 448 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 754 184 912 896;
  • 25) 0.875 743 050 321 336 748 963 538 872 754 184 912 896 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 508 369 825 792;
  • 26) 0.751 486 100 642 673 497 927 077 745 508 369 825 792 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 016 739 651 584;
  • 27) 0.502 972 201 285 346 995 854 155 491 016 739 651 584 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 033 479 303 168;
  • 28) 0.005 944 402 570 693 991 708 310 982 033 479 303 168 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 066 958 606 336;
  • 29) 0.011 888 805 141 387 983 416 621 964 066 958 606 336 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 133 917 212 672;
  • 30) 0.023 777 610 282 775 966 833 243 928 133 917 212 672 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 267 834 425 344;
  • 31) 0.047 555 220 565 551 933 666 487 856 267 834 425 344 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 535 668 850 688;
  • 32) 0.095 110 441 131 103 867 332 975 712 535 668 850 688 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 071 337 701 376;
  • 33) 0.190 220 882 262 207 734 665 951 425 071 337 701 376 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 142 675 402 752;
  • 34) 0.380 441 764 524 415 469 331 902 850 142 675 402 752 × 2 = 0 + 0.760 883 529 048 830 938 663 805 700 285 350 805 504;
  • 35) 0.760 883 529 048 830 938 663 805 700 285 350 805 504 × 2 = 1 + 0.521 767 058 097 661 877 327 611 400 570 701 611 008;
  • 36) 0.521 767 058 097 661 877 327 611 400 570 701 611 008 × 2 = 1 + 0.043 534 116 195 323 754 655 222 801 141 403 222 016;
  • 37) 0.043 534 116 195 323 754 655 222 801 141 403 222 016 × 2 = 0 + 0.087 068 232 390 647 509 310 445 602 282 806 444 032;
  • 38) 0.087 068 232 390 647 509 310 445 602 282 806 444 032 × 2 = 0 + 0.174 136 464 781 295 018 620 891 204 565 612 888 064;
  • 39) 0.174 136 464 781 295 018 620 891 204 565 612 888 064 × 2 = 0 + 0.348 272 929 562 590 037 241 782 409 131 225 776 128;
  • 40) 0.348 272 929 562 590 037 241 782 409 131 225 776 128 × 2 = 0 + 0.696 545 859 125 180 074 483 564 818 262 451 552 256;
  • 41) 0.696 545 859 125 180 074 483 564 818 262 451 552 256 × 2 = 1 + 0.393 091 718 250 360 148 967 129 636 524 903 104 512;
  • 42) 0.393 091 718 250 360 148 967 129 636 524 903 104 512 × 2 = 0 + 0.786 183 436 500 720 297 934 259 273 049 806 209 024;
  • 43) 0.786 183 436 500 720 297 934 259 273 049 806 209 024 × 2 = 1 + 0.572 366 873 001 440 595 868 518 546 099 612 418 048;
  • 44) 0.572 366 873 001 440 595 868 518 546 099 612 418 048 × 2 = 1 + 0.144 733 746 002 881 191 737 037 092 199 224 836 096;
  • 45) 0.144 733 746 002 881 191 737 037 092 199 224 836 096 × 2 = 0 + 0.289 467 492 005 762 383 474 074 184 398 449 672 192;
  • 46) 0.289 467 492 005 762 383 474 074 184 398 449 672 192 × 2 = 0 + 0.578 934 984 011 524 766 948 148 368 796 899 344 384;
  • 47) 0.578 934 984 011 524 766 948 148 368 796 899 344 384 × 2 = 1 + 0.157 869 968 023 049 533 896 296 737 593 798 688 768;
  • 48) 0.157 869 968 023 049 533 896 296 737 593 798 688 768 × 2 = 0 + 0.315 739 936 046 099 067 792 593 475 187 597 377 536;
  • 49) 0.315 739 936 046 099 067 792 593 475 187 597 377 536 × 2 = 0 + 0.631 479 872 092 198 135 585 186 950 375 194 755 072;
  • 50) 0.631 479 872 092 198 135 585 186 950 375 194 755 072 × 2 = 1 + 0.262 959 744 184 396 271 170 373 900 750 389 510 144;
  • 51) 0.262 959 744 184 396 271 170 373 900 750 389 510 144 × 2 = 0 + 0.525 919 488 368 792 542 340 747 801 500 779 020 288;
  • 52) 0.525 919 488 368 792 542 340 747 801 500 779 020 288 × 2 = 1 + 0.051 838 976 737 585 084 681 495 603 001 558 040 576;
  • 53) 0.051 838 976 737 585 084 681 495 603 001 558 040 576 × 2 = 0 + 0.103 677 953 475 170 169 362 991 206 003 116 081 152;
  • 54) 0.103 677 953 475 170 169 362 991 206 003 116 081 152 × 2 = 0 + 0.207 355 906 950 340 338 725 982 412 006 232 162 304;
  • 55) 0.207 355 906 950 340 338 725 982 412 006 232 162 304 × 2 = 0 + 0.414 711 813 900 680 677 451 964 824 012 464 324 608;
  • 56) 0.414 711 813 900 680 677 451 964 824 012 464 324 608 × 2 = 0 + 0.829 423 627 801 361 354 903 929 648 024 928 649 216;
  • 57) 0.829 423 627 801 361 354 903 929 648 024 928 649 216 × 2 = 1 + 0.658 847 255 602 722 709 807 859 296 049 857 298 432;
  • 58) 0.658 847 255 602 722 709 807 859 296 049 857 298 432 × 2 = 1 + 0.317 694 511 205 445 419 615 718 592 099 714 596 864;
  • 59) 0.317 694 511 205 445 419 615 718 592 099 714 596 864 × 2 = 0 + 0.635 389 022 410 890 839 231 437 184 199 429 193 728;
  • 60) 0.635 389 022 410 890 839 231 437 184 199 429 193 728 × 2 = 1 + 0.270 778 044 821 781 678 462 874 368 398 858 387 456;
  • 61) 0.270 778 044 821 781 678 462 874 368 398 858 387 456 × 2 = 0 + 0.541 556 089 643 563 356 925 748 736 797 716 774 912;
  • 62) 0.541 556 089 643 563 356 925 748 736 797 716 774 912 × 2 = 1 + 0.083 112 179 287 126 713 851 497 473 595 433 549 824;
  • 63) 0.083 112 179 287 126 713 851 497 473 595 433 549 824 × 2 = 0 + 0.166 224 358 574 253 427 702 994 947 190 867 099 648;

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:


0.000 806 264 623 585 362 514 063 654 156 856 104 981(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 104 981(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:


0.000 806 264 623 585 362 514 063 654 156 856 104 981(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 -0.000 806 264 623 585 362 514 063 654 156 856 104 981 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