-0.000 806 264 623 585 362 514 063 654 156 856 105 032 35 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 806 264 623 585 362 514 063 654 156 856 105 032 35(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
-0.000 806 264 623 585 362 514 063 654 156 856 105 032 35(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:

|-0.000 806 264 623 585 362 514 063 654 156 856 105 032 35| = 0.000 806 264 623 585 362 514 063 654 156 856 105 032 35


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: 0.000 806 264 623 585 362 514 063 654 156 856 105 032 35.

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) 0.000 806 264 623 585 362 514 063 654 156 856 105 032 35 × 2 = 0 + 0.001 612 529 247 170 725 028 127 308 313 712 210 064 7;
  • 2) 0.001 612 529 247 170 725 028 127 308 313 712 210 064 7 × 2 = 0 + 0.003 225 058 494 341 450 056 254 616 627 424 420 129 4;
  • 3) 0.003 225 058 494 341 450 056 254 616 627 424 420 129 4 × 2 = 0 + 0.006 450 116 988 682 900 112 509 233 254 848 840 258 8;
  • 4) 0.006 450 116 988 682 900 112 509 233 254 848 840 258 8 × 2 = 0 + 0.012 900 233 977 365 800 225 018 466 509 697 680 517 6;
  • 5) 0.012 900 233 977 365 800 225 018 466 509 697 680 517 6 × 2 = 0 + 0.025 800 467 954 731 600 450 036 933 019 395 361 035 2;
  • 6) 0.025 800 467 954 731 600 450 036 933 019 395 361 035 2 × 2 = 0 + 0.051 600 935 909 463 200 900 073 866 038 790 722 070 4;
  • 7) 0.051 600 935 909 463 200 900 073 866 038 790 722 070 4 × 2 = 0 + 0.103 201 871 818 926 401 800 147 732 077 581 444 140 8;
  • 8) 0.103 201 871 818 926 401 800 147 732 077 581 444 140 8 × 2 = 0 + 0.206 403 743 637 852 803 600 295 464 155 162 888 281 6;
  • 9) 0.206 403 743 637 852 803 600 295 464 155 162 888 281 6 × 2 = 0 + 0.412 807 487 275 705 607 200 590 928 310 325 776 563 2;
  • 10) 0.412 807 487 275 705 607 200 590 928 310 325 776 563 2 × 2 = 0 + 0.825 614 974 551 411 214 401 181 856 620 651 553 126 4;
  • 11) 0.825 614 974 551 411 214 401 181 856 620 651 553 126 4 × 2 = 1 + 0.651 229 949 102 822 428 802 363 713 241 303 106 252 8;
  • 12) 0.651 229 949 102 822 428 802 363 713 241 303 106 252 8 × 2 = 1 + 0.302 459 898 205 644 857 604 727 426 482 606 212 505 6;
  • 13) 0.302 459 898 205 644 857 604 727 426 482 606 212 505 6 × 2 = 0 + 0.604 919 796 411 289 715 209 454 852 965 212 425 011 2;
  • 14) 0.604 919 796 411 289 715 209 454 852 965 212 425 011 2 × 2 = 1 + 0.209 839 592 822 579 430 418 909 705 930 424 850 022 4;
  • 15) 0.209 839 592 822 579 430 418 909 705 930 424 850 022 4 × 2 = 0 + 0.419 679 185 645 158 860 837 819 411 860 849 700 044 8;
  • 16) 0.419 679 185 645 158 860 837 819 411 860 849 700 044 8 × 2 = 0 + 0.839 358 371 290 317 721 675 638 823 721 699 400 089 6;
  • 17) 0.839 358 371 290 317 721 675 638 823 721 699 400 089 6 × 2 = 1 + 0.678 716 742 580 635 443 351 277 647 443 398 800 179 2;
  • 18) 0.678 716 742 580 635 443 351 277 647 443 398 800 179 2 × 2 = 1 + 0.357 433 485 161 270 886 702 555 294 886 797 600 358 4;
  • 19) 0.357 433 485 161 270 886 702 555 294 886 797 600 358 4 × 2 = 0 + 0.714 866 970 322 541 773 405 110 589 773 595 200 716 8;
  • 20) 0.714 866 970 322 541 773 405 110 589 773 595 200 716 8 × 2 = 1 + 0.429 733 940 645 083 546 810 221 179 547 190 401 433 6;
  • 21) 0.429 733 940 645 083 546 810 221 179 547 190 401 433 6 × 2 = 0 + 0.859 467 881 290 167 093 620 442 359 094 380 802 867 2;
  • 22) 0.859 467 881 290 167 093 620 442 359 094 380 802 867 2 × 2 = 1 + 0.718 935 762 580 334 187 240 884 718 188 761 605 734 4;
  • 23) 0.718 935 762 580 334 187 240 884 718 188 761 605 734 4 × 2 = 1 + 0.437 871 525 160 668 374 481 769 436 377 523 211 468 8;
  • 24) 0.437 871 525 160 668 374 481 769 436 377 523 211 468 8 × 2 = 0 + 0.875 743 050 321 336 748 963 538 872 755 046 422 937 6;
  • 25) 0.875 743 050 321 336 748 963 538 872 755 046 422 937 6 × 2 = 1 + 0.751 486 100 642 673 497 927 077 745 510 092 845 875 2;
  • 26) 0.751 486 100 642 673 497 927 077 745 510 092 845 875 2 × 2 = 1 + 0.502 972 201 285 346 995 854 155 491 020 185 691 750 4;
  • 27) 0.502 972 201 285 346 995 854 155 491 020 185 691 750 4 × 2 = 1 + 0.005 944 402 570 693 991 708 310 982 040 371 383 500 8;
  • 28) 0.005 944 402 570 693 991 708 310 982 040 371 383 500 8 × 2 = 0 + 0.011 888 805 141 387 983 416 621 964 080 742 767 001 6;
  • 29) 0.011 888 805 141 387 983 416 621 964 080 742 767 001 6 × 2 = 0 + 0.023 777 610 282 775 966 833 243 928 161 485 534 003 2;
  • 30) 0.023 777 610 282 775 966 833 243 928 161 485 534 003 2 × 2 = 0 + 0.047 555 220 565 551 933 666 487 856 322 971 068 006 4;
  • 31) 0.047 555 220 565 551 933 666 487 856 322 971 068 006 4 × 2 = 0 + 0.095 110 441 131 103 867 332 975 712 645 942 136 012 8;
  • 32) 0.095 110 441 131 103 867 332 975 712 645 942 136 012 8 × 2 = 0 + 0.190 220 882 262 207 734 665 951 425 291 884 272 025 6;
  • 33) 0.190 220 882 262 207 734 665 951 425 291 884 272 025 6 × 2 = 0 + 0.380 441 764 524 415 469 331 902 850 583 768 544 051 2;
  • 34) 0.380 441 764 524 415 469 331 902 850 583 768 544 051 2 × 2 = 0 + 0.760 883 529 048 830 938 663 805 701 167 537 088 102 4;
  • 35) 0.760 883 529 048 830 938 663 805 701 167 537 088 102 4 × 2 = 1 + 0.521 767 058 097 661 877 327 611 402 335 074 176 204 8;
  • 36) 0.521 767 058 097 661 877 327 611 402 335 074 176 204 8 × 2 = 1 + 0.043 534 116 195 323 754 655 222 804 670 148 352 409 6;
  • 37) 0.043 534 116 195 323 754 655 222 804 670 148 352 409 6 × 2 = 0 + 0.087 068 232 390 647 509 310 445 609 340 296 704 819 2;
  • 38) 0.087 068 232 390 647 509 310 445 609 340 296 704 819 2 × 2 = 0 + 0.174 136 464 781 295 018 620 891 218 680 593 409 638 4;
  • 39) 0.174 136 464 781 295 018 620 891 218 680 593 409 638 4 × 2 = 0 + 0.348 272 929 562 590 037 241 782 437 361 186 819 276 8;
  • 40) 0.348 272 929 562 590 037 241 782 437 361 186 819 276 8 × 2 = 0 + 0.696 545 859 125 180 074 483 564 874 722 373 638 553 6;
  • 41) 0.696 545 859 125 180 074 483 564 874 722 373 638 553 6 × 2 = 1 + 0.393 091 718 250 360 148 967 129 749 444 747 277 107 2;
  • 42) 0.393 091 718 250 360 148 967 129 749 444 747 277 107 2 × 2 = 0 + 0.786 183 436 500 720 297 934 259 498 889 494 554 214 4;
  • 43) 0.786 183 436 500 720 297 934 259 498 889 494 554 214 4 × 2 = 1 + 0.572 366 873 001 440 595 868 518 997 778 989 108 428 8;
  • 44) 0.572 366 873 001 440 595 868 518 997 778 989 108 428 8 × 2 = 1 + 0.144 733 746 002 881 191 737 037 995 557 978 216 857 6;
  • 45) 0.144 733 746 002 881 191 737 037 995 557 978 216 857 6 × 2 = 0 + 0.289 467 492 005 762 383 474 075 991 115 956 433 715 2;
  • 46) 0.289 467 492 005 762 383 474 075 991 115 956 433 715 2 × 2 = 0 + 0.578 934 984 011 524 766 948 151 982 231 912 867 430 4;
  • 47) 0.578 934 984 011 524 766 948 151 982 231 912 867 430 4 × 2 = 1 + 0.157 869 968 023 049 533 896 303 964 463 825 734 860 8;
  • 48) 0.157 869 968 023 049 533 896 303 964 463 825 734 860 8 × 2 = 0 + 0.315 739 936 046 099 067 792 607 928 927 651 469 721 6;
  • 49) 0.315 739 936 046 099 067 792 607 928 927 651 469 721 6 × 2 = 0 + 0.631 479 872 092 198 135 585 215 857 855 302 939 443 2;
  • 50) 0.631 479 872 092 198 135 585 215 857 855 302 939 443 2 × 2 = 1 + 0.262 959 744 184 396 271 170 431 715 710 605 878 886 4;
  • 51) 0.262 959 744 184 396 271 170 431 715 710 605 878 886 4 × 2 = 0 + 0.525 919 488 368 792 542 340 863 431 421 211 757 772 8;
  • 52) 0.525 919 488 368 792 542 340 863 431 421 211 757 772 8 × 2 = 1 + 0.051 838 976 737 585 084 681 726 862 842 423 515 545 6;
  • 53) 0.051 838 976 737 585 084 681 726 862 842 423 515 545 6 × 2 = 0 + 0.103 677 953 475 170 169 363 453 725 684 847 031 091 2;
  • 54) 0.103 677 953 475 170 169 363 453 725 684 847 031 091 2 × 2 = 0 + 0.207 355 906 950 340 338 726 907 451 369 694 062 182 4;
  • 55) 0.207 355 906 950 340 338 726 907 451 369 694 062 182 4 × 2 = 0 + 0.414 711 813 900 680 677 453 814 902 739 388 124 364 8;
  • 56) 0.414 711 813 900 680 677 453 814 902 739 388 124 364 8 × 2 = 0 + 0.829 423 627 801 361 354 907 629 805 478 776 248 729 6;
  • 57) 0.829 423 627 801 361 354 907 629 805 478 776 248 729 6 × 2 = 1 + 0.658 847 255 602 722 709 815 259 610 957 552 497 459 2;
  • 58) 0.658 847 255 602 722 709 815 259 610 957 552 497 459 2 × 2 = 1 + 0.317 694 511 205 445 419 630 519 221 915 104 994 918 4;
  • 59) 0.317 694 511 205 445 419 630 519 221 915 104 994 918 4 × 2 = 0 + 0.635 389 022 410 890 839 261 038 443 830 209 989 836 8;
  • 60) 0.635 389 022 410 890 839 261 038 443 830 209 989 836 8 × 2 = 1 + 0.270 778 044 821 781 678 522 076 887 660 419 979 673 6;
  • 61) 0.270 778 044 821 781 678 522 076 887 660 419 979 673 6 × 2 = 0 + 0.541 556 089 643 563 357 044 153 775 320 839 959 347 2;
  • 62) 0.541 556 089 643 563 357 044 153 775 320 839 959 347 2 × 2 = 1 + 0.083 112 179 287 126 714 088 307 550 641 679 918 694 4;
  • 63) 0.083 112 179 287 126 714 088 307 550 641 679 918 694 4 × 2 = 0 + 0.166 224 358 574 253 428 176 615 101 283 359 837 388 8;

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:


0.000 806 264 623 585 362 514 063 654 156 856 105 032 35(10) =

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

6. Positive number before normalization:

0.000 806 264 623 585 362 514 063 654 156 856 105 032 35(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:


0.000 806 264 623 585 362 514 063 654 156 856 105 032 35(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 -0.000 806 264 623 585 362 514 063 654 156 856 105 032 35 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