-0.000 000 000 000 014 385 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 014 385 9(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
-0.000 000 000 000 014 385 9(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:

|-0.000 000 000 000 014 385 9| = 0.000 000 000 000 014 385 9


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: 0.000 000 000 000 014 385 9.

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) 0.000 000 000 000 014 385 9 × 2 = 0 + 0.000 000 000 000 028 771 8;
  • 2) 0.000 000 000 000 028 771 8 × 2 = 0 + 0.000 000 000 000 057 543 6;
  • 3) 0.000 000 000 000 057 543 6 × 2 = 0 + 0.000 000 000 000 115 087 2;
  • 4) 0.000 000 000 000 115 087 2 × 2 = 0 + 0.000 000 000 000 230 174 4;
  • 5) 0.000 000 000 000 230 174 4 × 2 = 0 + 0.000 000 000 000 460 348 8;
  • 6) 0.000 000 000 000 460 348 8 × 2 = 0 + 0.000 000 000 000 920 697 6;
  • 7) 0.000 000 000 000 920 697 6 × 2 = 0 + 0.000 000 000 001 841 395 2;
  • 8) 0.000 000 000 001 841 395 2 × 2 = 0 + 0.000 000 000 003 682 790 4;
  • 9) 0.000 000 000 003 682 790 4 × 2 = 0 + 0.000 000 000 007 365 580 8;
  • 10) 0.000 000 000 007 365 580 8 × 2 = 0 + 0.000 000 000 014 731 161 6;
  • 11) 0.000 000 000 014 731 161 6 × 2 = 0 + 0.000 000 000 029 462 323 2;
  • 12) 0.000 000 000 029 462 323 2 × 2 = 0 + 0.000 000 000 058 924 646 4;
  • 13) 0.000 000 000 058 924 646 4 × 2 = 0 + 0.000 000 000 117 849 292 8;
  • 14) 0.000 000 000 117 849 292 8 × 2 = 0 + 0.000 000 000 235 698 585 6;
  • 15) 0.000 000 000 235 698 585 6 × 2 = 0 + 0.000 000 000 471 397 171 2;
  • 16) 0.000 000 000 471 397 171 2 × 2 = 0 + 0.000 000 000 942 794 342 4;
  • 17) 0.000 000 000 942 794 342 4 × 2 = 0 + 0.000 000 001 885 588 684 8;
  • 18) 0.000 000 001 885 588 684 8 × 2 = 0 + 0.000 000 003 771 177 369 6;
  • 19) 0.000 000 003 771 177 369 6 × 2 = 0 + 0.000 000 007 542 354 739 2;
  • 20) 0.000 000 007 542 354 739 2 × 2 = 0 + 0.000 000 015 084 709 478 4;
  • 21) 0.000 000 015 084 709 478 4 × 2 = 0 + 0.000 000 030 169 418 956 8;
  • 22) 0.000 000 030 169 418 956 8 × 2 = 0 + 0.000 000 060 338 837 913 6;
  • 23) 0.000 000 060 338 837 913 6 × 2 = 0 + 0.000 000 120 677 675 827 2;
  • 24) 0.000 000 120 677 675 827 2 × 2 = 0 + 0.000 000 241 355 351 654 4;
  • 25) 0.000 000 241 355 351 654 4 × 2 = 0 + 0.000 000 482 710 703 308 8;
  • 26) 0.000 000 482 710 703 308 8 × 2 = 0 + 0.000 000 965 421 406 617 6;
  • 27) 0.000 000 965 421 406 617 6 × 2 = 0 + 0.000 001 930 842 813 235 2;
  • 28) 0.000 001 930 842 813 235 2 × 2 = 0 + 0.000 003 861 685 626 470 4;
  • 29) 0.000 003 861 685 626 470 4 × 2 = 0 + 0.000 007 723 371 252 940 8;
  • 30) 0.000 007 723 371 252 940 8 × 2 = 0 + 0.000 015 446 742 505 881 6;
  • 31) 0.000 015 446 742 505 881 6 × 2 = 0 + 0.000 030 893 485 011 763 2;
  • 32) 0.000 030 893 485 011 763 2 × 2 = 0 + 0.000 061 786 970 023 526 4;
  • 33) 0.000 061 786 970 023 526 4 × 2 = 0 + 0.000 123 573 940 047 052 8;
  • 34) 0.000 123 573 940 047 052 8 × 2 = 0 + 0.000 247 147 880 094 105 6;
  • 35) 0.000 247 147 880 094 105 6 × 2 = 0 + 0.000 494 295 760 188 211 2;
  • 36) 0.000 494 295 760 188 211 2 × 2 = 0 + 0.000 988 591 520 376 422 4;
  • 37) 0.000 988 591 520 376 422 4 × 2 = 0 + 0.001 977 183 040 752 844 8;
  • 38) 0.001 977 183 040 752 844 8 × 2 = 0 + 0.003 954 366 081 505 689 6;
  • 39) 0.003 954 366 081 505 689 6 × 2 = 0 + 0.007 908 732 163 011 379 2;
  • 40) 0.007 908 732 163 011 379 2 × 2 = 0 + 0.015 817 464 326 022 758 4;
  • 41) 0.015 817 464 326 022 758 4 × 2 = 0 + 0.031 634 928 652 045 516 8;
  • 42) 0.031 634 928 652 045 516 8 × 2 = 0 + 0.063 269 857 304 091 033 6;
  • 43) 0.063 269 857 304 091 033 6 × 2 = 0 + 0.126 539 714 608 182 067 2;
  • 44) 0.126 539 714 608 182 067 2 × 2 = 0 + 0.253 079 429 216 364 134 4;
  • 45) 0.253 079 429 216 364 134 4 × 2 = 0 + 0.506 158 858 432 728 268 8;
  • 46) 0.506 158 858 432 728 268 8 × 2 = 1 + 0.012 317 716 865 456 537 6;
  • 47) 0.012 317 716 865 456 537 6 × 2 = 0 + 0.024 635 433 730 913 075 2;
  • 48) 0.024 635 433 730 913 075 2 × 2 = 0 + 0.049 270 867 461 826 150 4;
  • 49) 0.049 270 867 461 826 150 4 × 2 = 0 + 0.098 541 734 923 652 300 8;
  • 50) 0.098 541 734 923 652 300 8 × 2 = 0 + 0.197 083 469 847 304 601 6;
  • 51) 0.197 083 469 847 304 601 6 × 2 = 0 + 0.394 166 939 694 609 203 2;
  • 52) 0.394 166 939 694 609 203 2 × 2 = 0 + 0.788 333 879 389 218 406 4;
  • 53) 0.788 333 879 389 218 406 4 × 2 = 1 + 0.576 667 758 778 436 812 8;
  • 54) 0.576 667 758 778 436 812 8 × 2 = 1 + 0.153 335 517 556 873 625 6;
  • 55) 0.153 335 517 556 873 625 6 × 2 = 0 + 0.306 671 035 113 747 251 2;
  • 56) 0.306 671 035 113 747 251 2 × 2 = 0 + 0.613 342 070 227 494 502 4;
  • 57) 0.613 342 070 227 494 502 4 × 2 = 1 + 0.226 684 140 454 989 004 8;
  • 58) 0.226 684 140 454 989 004 8 × 2 = 0 + 0.453 368 280 909 978 009 6;
  • 59) 0.453 368 280 909 978 009 6 × 2 = 0 + 0.906 736 561 819 956 019 2;
  • 60) 0.906 736 561 819 956 019 2 × 2 = 1 + 0.813 473 123 639 912 038 4;
  • 61) 0.813 473 123 639 912 038 4 × 2 = 1 + 0.626 946 247 279 824 076 8;
  • 62) 0.626 946 247 279 824 076 8 × 2 = 1 + 0.253 892 494 559 648 153 6;
  • 63) 0.253 892 494 559 648 153 6 × 2 = 0 + 0.507 784 989 119 296 307 2;
  • 64) 0.507 784 989 119 296 307 2 × 2 = 1 + 0.015 569 978 238 592 614 4;
  • 65) 0.015 569 978 238 592 614 4 × 2 = 0 + 0.031 139 956 477 185 228 8;
  • 66) 0.031 139 956 477 185 228 8 × 2 = 0 + 0.062 279 912 954 370 457 6;
  • 67) 0.062 279 912 954 370 457 6 × 2 = 0 + 0.124 559 825 908 740 915 2;
  • 68) 0.124 559 825 908 740 915 2 × 2 = 0 + 0.249 119 651 817 481 830 4;
  • 69) 0.249 119 651 817 481 830 4 × 2 = 0 + 0.498 239 303 634 963 660 8;
  • 70) 0.498 239 303 634 963 660 8 × 2 = 0 + 0.996 478 607 269 927 321 6;
  • 71) 0.996 478 607 269 927 321 6 × 2 = 1 + 0.992 957 214 539 854 643 2;
  • 72) 0.992 957 214 539 854 643 2 × 2 = 1 + 0.985 914 429 079 709 286 4;
  • 73) 0.985 914 429 079 709 286 4 × 2 = 1 + 0.971 828 858 159 418 572 8;
  • 74) 0.971 828 858 159 418 572 8 × 2 = 1 + 0.943 657 716 318 837 145 6;
  • 75) 0.943 657 716 318 837 145 6 × 2 = 1 + 0.887 315 432 637 674 291 2;
  • 76) 0.887 315 432 637 674 291 2 × 2 = 1 + 0.774 630 865 275 348 582 4;
  • 77) 0.774 630 865 275 348 582 4 × 2 = 1 + 0.549 261 730 550 697 164 8;
  • 78) 0.549 261 730 550 697 164 8 × 2 = 1 + 0.098 523 461 101 394 329 6;
  • 79) 0.098 523 461 101 394 329 6 × 2 = 0 + 0.197 046 922 202 788 659 2;
  • 80) 0.197 046 922 202 788 659 2 × 2 = 0 + 0.394 093 844 405 577 318 4;
  • 81) 0.394 093 844 405 577 318 4 × 2 = 0 + 0.788 187 688 811 154 636 8;
  • 82) 0.788 187 688 811 154 636 8 × 2 = 1 + 0.576 375 377 622 309 273 6;
  • 83) 0.576 375 377 622 309 273 6 × 2 = 1 + 0.152 750 755 244 618 547 2;
  • 84) 0.152 750 755 244 618 547 2 × 2 = 0 + 0.305 501 510 489 237 094 4;
  • 85) 0.305 501 510 489 237 094 4 × 2 = 0 + 0.611 003 020 978 474 188 8;
  • 86) 0.611 003 020 978 474 188 8 × 2 = 1 + 0.222 006 041 956 948 377 6;
  • 87) 0.222 006 041 956 948 377 6 × 2 = 0 + 0.444 012 083 913 896 755 2;
  • 88) 0.444 012 083 913 896 755 2 × 2 = 0 + 0.888 024 167 827 793 510 4;
  • 89) 0.888 024 167 827 793 510 4 × 2 = 1 + 0.776 048 335 655 587 020 8;
  • 90) 0.776 048 335 655 587 020 8 × 2 = 1 + 0.552 096 671 311 174 041 6;
  • 91) 0.552 096 671 311 174 041 6 × 2 = 1 + 0.104 193 342 622 348 083 2;
  • 92) 0.104 193 342 622 348 083 2 × 2 = 0 + 0.208 386 685 244 696 166 4;
  • 93) 0.208 386 685 244 696 166 4 × 2 = 0 + 0.416 773 370 489 392 332 8;
  • 94) 0.416 773 370 489 392 332 8 × 2 = 0 + 0.833 546 740 978 784 665 6;
  • 95) 0.833 546 740 978 784 665 6 × 2 = 1 + 0.667 093 481 957 569 331 2;
  • 96) 0.667 093 481 957 569 331 2 × 2 = 1 + 0.334 186 963 915 138 662 4;
  • 97) 0.334 186 963 915 138 662 4 × 2 = 0 + 0.668 373 927 830 277 324 8;
  • 98) 0.668 373 927 830 277 324 8 × 2 = 1 + 0.336 747 855 660 554 649 6;

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:


0.000 000 000 000 014 385 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1100 1001 1101 0000 0011 1111 1100 0110 0100 1110 0011 01(2)

6. Positive number before normalization:

0.000 000 000 000 014 385 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1100 1001 1101 0000 0011 1111 1100 0110 0100 1110 0011 01(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:


0.000 000 000 000 014 385 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1100 1001 1101 0000 0011 1111 1100 0110 0100 1110 0011 01(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1100 1001 1101 0000 0011 1111 1100 0110 0100 1110 0011 01(2) × 20 =

1.0000 0011 0010 0111 0100 0000 1111 1111 0001 1001 0011 1000 1101(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 0010 0111 0100 0000 1111 1111 0001 1001 0011 1000 1101


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 0010 0111 0100 0000 1111 1111 0001 1001 0011 1000 1101 =

0000 0011 0010 0111 0100 0000 1111 1111 0001 1001 0011 1000 1101


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 0010 0111 0100 0000 1111 1111 0001 1001 0011 1000 1101


Decimal number -0.000 000 000 000 014 385 9 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1101 0001 - 0000 0011 0010 0111 0100 0000 1111 1111 0001 1001 0011 1000 1101

» Convert decimal -0.000 000 000 000 014 389 8 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