-0.000 000 000 000 014 392 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 014 392 2(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
-0.000 000 000 000 014 392 2(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:

|-0.000 000 000 000 014 392 2| = 0.000 000 000 000 014 392 2


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: 0.000 000 000 000 014 392 2.

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) 0.000 000 000 000 014 392 2 × 2 = 0 + 0.000 000 000 000 028 784 4;
  • 2) 0.000 000 000 000 028 784 4 × 2 = 0 + 0.000 000 000 000 057 568 8;
  • 3) 0.000 000 000 000 057 568 8 × 2 = 0 + 0.000 000 000 000 115 137 6;
  • 4) 0.000 000 000 000 115 137 6 × 2 = 0 + 0.000 000 000 000 230 275 2;
  • 5) 0.000 000 000 000 230 275 2 × 2 = 0 + 0.000 000 000 000 460 550 4;
  • 6) 0.000 000 000 000 460 550 4 × 2 = 0 + 0.000 000 000 000 921 100 8;
  • 7) 0.000 000 000 000 921 100 8 × 2 = 0 + 0.000 000 000 001 842 201 6;
  • 8) 0.000 000 000 001 842 201 6 × 2 = 0 + 0.000 000 000 003 684 403 2;
  • 9) 0.000 000 000 003 684 403 2 × 2 = 0 + 0.000 000 000 007 368 806 4;
  • 10) 0.000 000 000 007 368 806 4 × 2 = 0 + 0.000 000 000 014 737 612 8;
  • 11) 0.000 000 000 014 737 612 8 × 2 = 0 + 0.000 000 000 029 475 225 6;
  • 12) 0.000 000 000 029 475 225 6 × 2 = 0 + 0.000 000 000 058 950 451 2;
  • 13) 0.000 000 000 058 950 451 2 × 2 = 0 + 0.000 000 000 117 900 902 4;
  • 14) 0.000 000 000 117 900 902 4 × 2 = 0 + 0.000 000 000 235 801 804 8;
  • 15) 0.000 000 000 235 801 804 8 × 2 = 0 + 0.000 000 000 471 603 609 6;
  • 16) 0.000 000 000 471 603 609 6 × 2 = 0 + 0.000 000 000 943 207 219 2;
  • 17) 0.000 000 000 943 207 219 2 × 2 = 0 + 0.000 000 001 886 414 438 4;
  • 18) 0.000 000 001 886 414 438 4 × 2 = 0 + 0.000 000 003 772 828 876 8;
  • 19) 0.000 000 003 772 828 876 8 × 2 = 0 + 0.000 000 007 545 657 753 6;
  • 20) 0.000 000 007 545 657 753 6 × 2 = 0 + 0.000 000 015 091 315 507 2;
  • 21) 0.000 000 015 091 315 507 2 × 2 = 0 + 0.000 000 030 182 631 014 4;
  • 22) 0.000 000 030 182 631 014 4 × 2 = 0 + 0.000 000 060 365 262 028 8;
  • 23) 0.000 000 060 365 262 028 8 × 2 = 0 + 0.000 000 120 730 524 057 6;
  • 24) 0.000 000 120 730 524 057 6 × 2 = 0 + 0.000 000 241 461 048 115 2;
  • 25) 0.000 000 241 461 048 115 2 × 2 = 0 + 0.000 000 482 922 096 230 4;
  • 26) 0.000 000 482 922 096 230 4 × 2 = 0 + 0.000 000 965 844 192 460 8;
  • 27) 0.000 000 965 844 192 460 8 × 2 = 0 + 0.000 001 931 688 384 921 6;
  • 28) 0.000 001 931 688 384 921 6 × 2 = 0 + 0.000 003 863 376 769 843 2;
  • 29) 0.000 003 863 376 769 843 2 × 2 = 0 + 0.000 007 726 753 539 686 4;
  • 30) 0.000 007 726 753 539 686 4 × 2 = 0 + 0.000 015 453 507 079 372 8;
  • 31) 0.000 015 453 507 079 372 8 × 2 = 0 + 0.000 030 907 014 158 745 6;
  • 32) 0.000 030 907 014 158 745 6 × 2 = 0 + 0.000 061 814 028 317 491 2;
  • 33) 0.000 061 814 028 317 491 2 × 2 = 0 + 0.000 123 628 056 634 982 4;
  • 34) 0.000 123 628 056 634 982 4 × 2 = 0 + 0.000 247 256 113 269 964 8;
  • 35) 0.000 247 256 113 269 964 8 × 2 = 0 + 0.000 494 512 226 539 929 6;
  • 36) 0.000 494 512 226 539 929 6 × 2 = 0 + 0.000 989 024 453 079 859 2;
  • 37) 0.000 989 024 453 079 859 2 × 2 = 0 + 0.001 978 048 906 159 718 4;
  • 38) 0.001 978 048 906 159 718 4 × 2 = 0 + 0.003 956 097 812 319 436 8;
  • 39) 0.003 956 097 812 319 436 8 × 2 = 0 + 0.007 912 195 624 638 873 6;
  • 40) 0.007 912 195 624 638 873 6 × 2 = 0 + 0.015 824 391 249 277 747 2;
  • 41) 0.015 824 391 249 277 747 2 × 2 = 0 + 0.031 648 782 498 555 494 4;
  • 42) 0.031 648 782 498 555 494 4 × 2 = 0 + 0.063 297 564 997 110 988 8;
  • 43) 0.063 297 564 997 110 988 8 × 2 = 0 + 0.126 595 129 994 221 977 6;
  • 44) 0.126 595 129 994 221 977 6 × 2 = 0 + 0.253 190 259 988 443 955 2;
  • 45) 0.253 190 259 988 443 955 2 × 2 = 0 + 0.506 380 519 976 887 910 4;
  • 46) 0.506 380 519 976 887 910 4 × 2 = 1 + 0.012 761 039 953 775 820 8;
  • 47) 0.012 761 039 953 775 820 8 × 2 = 0 + 0.025 522 079 907 551 641 6;
  • 48) 0.025 522 079 907 551 641 6 × 2 = 0 + 0.051 044 159 815 103 283 2;
  • 49) 0.051 044 159 815 103 283 2 × 2 = 0 + 0.102 088 319 630 206 566 4;
  • 50) 0.102 088 319 630 206 566 4 × 2 = 0 + 0.204 176 639 260 413 132 8;
  • 51) 0.204 176 639 260 413 132 8 × 2 = 0 + 0.408 353 278 520 826 265 6;
  • 52) 0.408 353 278 520 826 265 6 × 2 = 0 + 0.816 706 557 041 652 531 2;
  • 53) 0.816 706 557 041 652 531 2 × 2 = 1 + 0.633 413 114 083 305 062 4;
  • 54) 0.633 413 114 083 305 062 4 × 2 = 1 + 0.266 826 228 166 610 124 8;
  • 55) 0.266 826 228 166 610 124 8 × 2 = 0 + 0.533 652 456 333 220 249 6;
  • 56) 0.533 652 456 333 220 249 6 × 2 = 1 + 0.067 304 912 666 440 499 2;
  • 57) 0.067 304 912 666 440 499 2 × 2 = 0 + 0.134 609 825 332 880 998 4;
  • 58) 0.134 609 825 332 880 998 4 × 2 = 0 + 0.269 219 650 665 761 996 8;
  • 59) 0.269 219 650 665 761 996 8 × 2 = 0 + 0.538 439 301 331 523 993 6;
  • 60) 0.538 439 301 331 523 993 6 × 2 = 1 + 0.076 878 602 663 047 987 2;
  • 61) 0.076 878 602 663 047 987 2 × 2 = 0 + 0.153 757 205 326 095 974 4;
  • 62) 0.153 757 205 326 095 974 4 × 2 = 0 + 0.307 514 410 652 191 948 8;
  • 63) 0.307 514 410 652 191 948 8 × 2 = 0 + 0.615 028 821 304 383 897 6;
  • 64) 0.615 028 821 304 383 897 6 × 2 = 1 + 0.230 057 642 608 767 795 2;
  • 65) 0.230 057 642 608 767 795 2 × 2 = 0 + 0.460 115 285 217 535 590 4;
  • 66) 0.460 115 285 217 535 590 4 × 2 = 0 + 0.920 230 570 435 071 180 8;
  • 67) 0.920 230 570 435 071 180 8 × 2 = 1 + 0.840 461 140 870 142 361 6;
  • 68) 0.840 461 140 870 142 361 6 × 2 = 1 + 0.680 922 281 740 284 723 2;
  • 69) 0.680 922 281 740 284 723 2 × 2 = 1 + 0.361 844 563 480 569 446 4;
  • 70) 0.361 844 563 480 569 446 4 × 2 = 0 + 0.723 689 126 961 138 892 8;
  • 71) 0.723 689 126 961 138 892 8 × 2 = 1 + 0.447 378 253 922 277 785 6;
  • 72) 0.447 378 253 922 277 785 6 × 2 = 0 + 0.894 756 507 844 555 571 2;
  • 73) 0.894 756 507 844 555 571 2 × 2 = 1 + 0.789 513 015 689 111 142 4;
  • 74) 0.789 513 015 689 111 142 4 × 2 = 1 + 0.579 026 031 378 222 284 8;
  • 75) 0.579 026 031 378 222 284 8 × 2 = 1 + 0.158 052 062 756 444 569 6;
  • 76) 0.158 052 062 756 444 569 6 × 2 = 0 + 0.316 104 125 512 889 139 2;
  • 77) 0.316 104 125 512 889 139 2 × 2 = 0 + 0.632 208 251 025 778 278 4;
  • 78) 0.632 208 251 025 778 278 4 × 2 = 1 + 0.264 416 502 051 556 556 8;
  • 79) 0.264 416 502 051 556 556 8 × 2 = 0 + 0.528 833 004 103 113 113 6;
  • 80) 0.528 833 004 103 113 113 6 × 2 = 1 + 0.057 666 008 206 226 227 2;
  • 81) 0.057 666 008 206 226 227 2 × 2 = 0 + 0.115 332 016 412 452 454 4;
  • 82) 0.115 332 016 412 452 454 4 × 2 = 0 + 0.230 664 032 824 904 908 8;
  • 83) 0.230 664 032 824 904 908 8 × 2 = 0 + 0.461 328 065 649 809 817 6;
  • 84) 0.461 328 065 649 809 817 6 × 2 = 0 + 0.922 656 131 299 619 635 2;
  • 85) 0.922 656 131 299 619 635 2 × 2 = 1 + 0.845 312 262 599 239 270 4;
  • 86) 0.845 312 262 599 239 270 4 × 2 = 1 + 0.690 624 525 198 478 540 8;
  • 87) 0.690 624 525 198 478 540 8 × 2 = 1 + 0.381 249 050 396 957 081 6;
  • 88) 0.381 249 050 396 957 081 6 × 2 = 0 + 0.762 498 100 793 914 163 2;
  • 89) 0.762 498 100 793 914 163 2 × 2 = 1 + 0.524 996 201 587 828 326 4;
  • 90) 0.524 996 201 587 828 326 4 × 2 = 1 + 0.049 992 403 175 656 652 8;
  • 91) 0.049 992 403 175 656 652 8 × 2 = 0 + 0.099 984 806 351 313 305 6;
  • 92) 0.099 984 806 351 313 305 6 × 2 = 0 + 0.199 969 612 702 626 611 2;
  • 93) 0.199 969 612 702 626 611 2 × 2 = 0 + 0.399 939 225 405 253 222 4;
  • 94) 0.399 939 225 405 253 222 4 × 2 = 0 + 0.799 878 450 810 506 444 8;
  • 95) 0.799 878 450 810 506 444 8 × 2 = 1 + 0.599 756 901 621 012 889 6;
  • 96) 0.599 756 901 621 012 889 6 × 2 = 1 + 0.199 513 803 242 025 779 2;
  • 97) 0.199 513 803 242 025 779 2 × 2 = 0 + 0.399 027 606 484 051 558 4;
  • 98) 0.399 027 606 484 051 558 4 × 2 = 0 + 0.798 055 212 968 103 116 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:


0.000 000 000 000 014 392 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1101 0001 0001 0011 1010 1110 0101 0000 1110 1100 0011 00(2)

6. Positive number before normalization:

0.000 000 000 000 014 392 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1101 0001 0001 0011 1010 1110 0101 0000 1110 1100 0011 00(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 46 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 014 392 2(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1101 0001 0001 0011 1010 1110 0101 0000 1110 1100 0011 00(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1101 0001 0001 0011 1010 1110 0101 0000 1110 1100 0011 00(2) × 20 =

1.0000 0011 0100 0100 0100 1110 1011 1001 0100 0011 1011 0000 1100(2) × 2-46


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): -46

Mantissa (not normalized):
1.0000 0011 0100 0100 0100 1110 1011 1001 0100 0011 1011 0000 1100


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-46 + 2(11-1) - 1 =

(-46 + 1 023)(10) =

977(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;
  • 977 ÷ 2 = 488 + 1;
  • 488 ÷ 2 = 244 + 0;
  • 244 ÷ 2 = 122 + 0;
  • 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) =

977(10) =

011 1101 0001(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. 0000 0011 0100 0100 0100 1110 1011 1001 0100 0011 1011 0000 1100 =

0000 0011 0100 0100 0100 1110 1011 1001 0100 0011 1011 0000 1100


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 0001

Mantissa (52 bits) =
0000 0011 0100 0100 0100 1110 1011 1001 0100 0011 1011 0000 1100


Decimal number -0.000 000 000 000 014 392 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1101 0001 - 0000 0011 0100 0100 0100 1110 1011 1001 0100 0011 1011 0000 1100

» Convert decimal -0.000 000 000 000 014 400 9 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 09, 2026 [September]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: September - by our visitors

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