-0.000 000 000 000 176 557 58 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 176 557 58(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
-0.000 000 000 000 176 557 58(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:

|-0.000 000 000 000 176 557 58| = 0.000 000 000 000 176 557 58


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: 0.000 000 000 000 176 557 58.

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) 0.000 000 000 000 176 557 58 × 2 = 0 + 0.000 000 000 000 353 115 16;
  • 2) 0.000 000 000 000 353 115 16 × 2 = 0 + 0.000 000 000 000 706 230 32;
  • 3) 0.000 000 000 000 706 230 32 × 2 = 0 + 0.000 000 000 001 412 460 64;
  • 4) 0.000 000 000 001 412 460 64 × 2 = 0 + 0.000 000 000 002 824 921 28;
  • 5) 0.000 000 000 002 824 921 28 × 2 = 0 + 0.000 000 000 005 649 842 56;
  • 6) 0.000 000 000 005 649 842 56 × 2 = 0 + 0.000 000 000 011 299 685 12;
  • 7) 0.000 000 000 011 299 685 12 × 2 = 0 + 0.000 000 000 022 599 370 24;
  • 8) 0.000 000 000 022 599 370 24 × 2 = 0 + 0.000 000 000 045 198 740 48;
  • 9) 0.000 000 000 045 198 740 48 × 2 = 0 + 0.000 000 000 090 397 480 96;
  • 10) 0.000 000 000 090 397 480 96 × 2 = 0 + 0.000 000 000 180 794 961 92;
  • 11) 0.000 000 000 180 794 961 92 × 2 = 0 + 0.000 000 000 361 589 923 84;
  • 12) 0.000 000 000 361 589 923 84 × 2 = 0 + 0.000 000 000 723 179 847 68;
  • 13) 0.000 000 000 723 179 847 68 × 2 = 0 + 0.000 000 001 446 359 695 36;
  • 14) 0.000 000 001 446 359 695 36 × 2 = 0 + 0.000 000 002 892 719 390 72;
  • 15) 0.000 000 002 892 719 390 72 × 2 = 0 + 0.000 000 005 785 438 781 44;
  • 16) 0.000 000 005 785 438 781 44 × 2 = 0 + 0.000 000 011 570 877 562 88;
  • 17) 0.000 000 011 570 877 562 88 × 2 = 0 + 0.000 000 023 141 755 125 76;
  • 18) 0.000 000 023 141 755 125 76 × 2 = 0 + 0.000 000 046 283 510 251 52;
  • 19) 0.000 000 046 283 510 251 52 × 2 = 0 + 0.000 000 092 567 020 503 04;
  • 20) 0.000 000 092 567 020 503 04 × 2 = 0 + 0.000 000 185 134 041 006 08;
  • 21) 0.000 000 185 134 041 006 08 × 2 = 0 + 0.000 000 370 268 082 012 16;
  • 22) 0.000 000 370 268 082 012 16 × 2 = 0 + 0.000 000 740 536 164 024 32;
  • 23) 0.000 000 740 536 164 024 32 × 2 = 0 + 0.000 001 481 072 328 048 64;
  • 24) 0.000 001 481 072 328 048 64 × 2 = 0 + 0.000 002 962 144 656 097 28;
  • 25) 0.000 002 962 144 656 097 28 × 2 = 0 + 0.000 005 924 289 312 194 56;
  • 26) 0.000 005 924 289 312 194 56 × 2 = 0 + 0.000 011 848 578 624 389 12;
  • 27) 0.000 011 848 578 624 389 12 × 2 = 0 + 0.000 023 697 157 248 778 24;
  • 28) 0.000 023 697 157 248 778 24 × 2 = 0 + 0.000 047 394 314 497 556 48;
  • 29) 0.000 047 394 314 497 556 48 × 2 = 0 + 0.000 094 788 628 995 112 96;
  • 30) 0.000 094 788 628 995 112 96 × 2 = 0 + 0.000 189 577 257 990 225 92;
  • 31) 0.000 189 577 257 990 225 92 × 2 = 0 + 0.000 379 154 515 980 451 84;
  • 32) 0.000 379 154 515 980 451 84 × 2 = 0 + 0.000 758 309 031 960 903 68;
  • 33) 0.000 758 309 031 960 903 68 × 2 = 0 + 0.001 516 618 063 921 807 36;
  • 34) 0.001 516 618 063 921 807 36 × 2 = 0 + 0.003 033 236 127 843 614 72;
  • 35) 0.003 033 236 127 843 614 72 × 2 = 0 + 0.006 066 472 255 687 229 44;
  • 36) 0.006 066 472 255 687 229 44 × 2 = 0 + 0.012 132 944 511 374 458 88;
  • 37) 0.012 132 944 511 374 458 88 × 2 = 0 + 0.024 265 889 022 748 917 76;
  • 38) 0.024 265 889 022 748 917 76 × 2 = 0 + 0.048 531 778 045 497 835 52;
  • 39) 0.048 531 778 045 497 835 52 × 2 = 0 + 0.097 063 556 090 995 671 04;
  • 40) 0.097 063 556 090 995 671 04 × 2 = 0 + 0.194 127 112 181 991 342 08;
  • 41) 0.194 127 112 181 991 342 08 × 2 = 0 + 0.388 254 224 363 982 684 16;
  • 42) 0.388 254 224 363 982 684 16 × 2 = 0 + 0.776 508 448 727 965 368 32;
  • 43) 0.776 508 448 727 965 368 32 × 2 = 1 + 0.553 016 897 455 930 736 64;
  • 44) 0.553 016 897 455 930 736 64 × 2 = 1 + 0.106 033 794 911 861 473 28;
  • 45) 0.106 033 794 911 861 473 28 × 2 = 0 + 0.212 067 589 823 722 946 56;
  • 46) 0.212 067 589 823 722 946 56 × 2 = 0 + 0.424 135 179 647 445 893 12;
  • 47) 0.424 135 179 647 445 893 12 × 2 = 0 + 0.848 270 359 294 891 786 24;
  • 48) 0.848 270 359 294 891 786 24 × 2 = 1 + 0.696 540 718 589 783 572 48;
  • 49) 0.696 540 718 589 783 572 48 × 2 = 1 + 0.393 081 437 179 567 144 96;
  • 50) 0.393 081 437 179 567 144 96 × 2 = 0 + 0.786 162 874 359 134 289 92;
  • 51) 0.786 162 874 359 134 289 92 × 2 = 1 + 0.572 325 748 718 268 579 84;
  • 52) 0.572 325 748 718 268 579 84 × 2 = 1 + 0.144 651 497 436 537 159 68;
  • 53) 0.144 651 497 436 537 159 68 × 2 = 0 + 0.289 302 994 873 074 319 36;
  • 54) 0.289 302 994 873 074 319 36 × 2 = 0 + 0.578 605 989 746 148 638 72;
  • 55) 0.578 605 989 746 148 638 72 × 2 = 1 + 0.157 211 979 492 297 277 44;
  • 56) 0.157 211 979 492 297 277 44 × 2 = 0 + 0.314 423 958 984 594 554 88;
  • 57) 0.314 423 958 984 594 554 88 × 2 = 0 + 0.628 847 917 969 189 109 76;
  • 58) 0.628 847 917 969 189 109 76 × 2 = 1 + 0.257 695 835 938 378 219 52;
  • 59) 0.257 695 835 938 378 219 52 × 2 = 0 + 0.515 391 671 876 756 439 04;
  • 60) 0.515 391 671 876 756 439 04 × 2 = 1 + 0.030 783 343 753 512 878 08;
  • 61) 0.030 783 343 753 512 878 08 × 2 = 0 + 0.061 566 687 507 025 756 16;
  • 62) 0.061 566 687 507 025 756 16 × 2 = 0 + 0.123 133 375 014 051 512 32;
  • 63) 0.123 133 375 014 051 512 32 × 2 = 0 + 0.246 266 750 028 103 024 64;
  • 64) 0.246 266 750 028 103 024 64 × 2 = 0 + 0.492 533 500 056 206 049 28;
  • 65) 0.492 533 500 056 206 049 28 × 2 = 0 + 0.985 067 000 112 412 098 56;
  • 66) 0.985 067 000 112 412 098 56 × 2 = 1 + 0.970 134 000 224 824 197 12;
  • 67) 0.970 134 000 224 824 197 12 × 2 = 1 + 0.940 268 000 449 648 394 24;
  • 68) 0.940 268 000 449 648 394 24 × 2 = 1 + 0.880 536 000 899 296 788 48;
  • 69) 0.880 536 000 899 296 788 48 × 2 = 1 + 0.761 072 001 798 593 576 96;
  • 70) 0.761 072 001 798 593 576 96 × 2 = 1 + 0.522 144 003 597 187 153 92;
  • 71) 0.522 144 003 597 187 153 92 × 2 = 1 + 0.044 288 007 194 374 307 84;
  • 72) 0.044 288 007 194 374 307 84 × 2 = 0 + 0.088 576 014 388 748 615 68;
  • 73) 0.088 576 014 388 748 615 68 × 2 = 0 + 0.177 152 028 777 497 231 36;
  • 74) 0.177 152 028 777 497 231 36 × 2 = 0 + 0.354 304 057 554 994 462 72;
  • 75) 0.354 304 057 554 994 462 72 × 2 = 0 + 0.708 608 115 109 988 925 44;
  • 76) 0.708 608 115 109 988 925 44 × 2 = 1 + 0.417 216 230 219 977 850 88;
  • 77) 0.417 216 230 219 977 850 88 × 2 = 0 + 0.834 432 460 439 955 701 76;
  • 78) 0.834 432 460 439 955 701 76 × 2 = 1 + 0.668 864 920 879 911 403 52;
  • 79) 0.668 864 920 879 911 403 52 × 2 = 1 + 0.337 729 841 759 822 807 04;
  • 80) 0.337 729 841 759 822 807 04 × 2 = 0 + 0.675 459 683 519 645 614 08;
  • 81) 0.675 459 683 519 645 614 08 × 2 = 1 + 0.350 919 367 039 291 228 16;
  • 82) 0.350 919 367 039 291 228 16 × 2 = 0 + 0.701 838 734 078 582 456 32;
  • 83) 0.701 838 734 078 582 456 32 × 2 = 1 + 0.403 677 468 157 164 912 64;
  • 84) 0.403 677 468 157 164 912 64 × 2 = 0 + 0.807 354 936 314 329 825 28;
  • 85) 0.807 354 936 314 329 825 28 × 2 = 1 + 0.614 709 872 628 659 650 56;
  • 86) 0.614 709 872 628 659 650 56 × 2 = 1 + 0.229 419 745 257 319 301 12;
  • 87) 0.229 419 745 257 319 301 12 × 2 = 0 + 0.458 839 490 514 638 602 24;
  • 88) 0.458 839 490 514 638 602 24 × 2 = 0 + 0.917 678 981 029 277 204 48;
  • 89) 0.917 678 981 029 277 204 48 × 2 = 1 + 0.835 357 962 058 554 408 96;
  • 90) 0.835 357 962 058 554 408 96 × 2 = 1 + 0.670 715 924 117 108 817 92;
  • 91) 0.670 715 924 117 108 817 92 × 2 = 1 + 0.341 431 848 234 217 635 84;
  • 92) 0.341 431 848 234 217 635 84 × 2 = 0 + 0.682 863 696 468 435 271 68;
  • 93) 0.682 863 696 468 435 271 68 × 2 = 1 + 0.365 727 392 936 870 543 36;
  • 94) 0.365 727 392 936 870 543 36 × 2 = 0 + 0.731 454 785 873 741 086 72;
  • 95) 0.731 454 785 873 741 086 72 × 2 = 1 + 0.462 909 571 747 482 173 44;

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:


0.000 000 000 000 176 557 58(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0001 1011 0010 0101 0000 0111 1110 0001 0110 1010 1100 1110 101(2)

6. Positive number before normalization:

0.000 000 000 000 176 557 58(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0001 1011 0010 0101 0000 0111 1110 0001 0110 1010 1100 1110 101(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 43 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 176 557 58(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0001 1011 0010 0101 0000 0111 1110 0001 0110 1010 1100 1110 101(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0001 1011 0010 0101 0000 0111 1110 0001 0110 1010 1100 1110 101(2) × 20 =

1.1000 1101 1001 0010 1000 0011 1111 0000 1011 0101 0110 0111 0101(2) × 2-43


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): -43

Mantissa (not normalized):
1.1000 1101 1001 0010 1000 0011 1111 0000 1011 0101 0110 0111 0101


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-43 + 2(11-1) - 1 =

(-43 + 1 023)(10) =

980(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;
  • 980 ÷ 2 = 490 + 0;
  • 490 ÷ 2 = 245 + 0;
  • 245 ÷ 2 = 122 + 1;
  • 122 ÷ 2 = 61 + 0;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

980(10) =

011 1101 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. 1000 1101 1001 0010 1000 0011 1111 0000 1011 0101 0110 0111 0101 =

1000 1101 1001 0010 1000 0011 1111 0000 1011 0101 0110 0111 0101


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 1101 0100

Mantissa (52 bits) =
1000 1101 1001 0010 1000 0011 1111 0000 1011 0101 0110 0111 0101


Decimal number -0.000 000 000 000 176 557 58 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1101 0100 - 1000 1101 1001 0010 1000 0011 1111 0000 1011 0101 0110 0111 0101

Share

» Convert decimal -0.000 000 000 000 176 558 03 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 06, 2026 [June]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: June - by our visitors

» Improve your social skills: Your greatest rival, your most powerful ally. NotebookLM YouTube Video of 7.54 minutes

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