-0.000 000 000 000 014 369 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 014 369 3(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
-0.000 000 000 000 014 369 3(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:

|-0.000 000 000 000 014 369 3| = 0.000 000 000 000 014 369 3


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: 0.000 000 000 000 014 369 3.

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) 0.000 000 000 000 014 369 3 × 2 = 0 + 0.000 000 000 000 028 738 6;
  • 2) 0.000 000 000 000 028 738 6 × 2 = 0 + 0.000 000 000 000 057 477 2;
  • 3) 0.000 000 000 000 057 477 2 × 2 = 0 + 0.000 000 000 000 114 954 4;
  • 4) 0.000 000 000 000 114 954 4 × 2 = 0 + 0.000 000 000 000 229 908 8;
  • 5) 0.000 000 000 000 229 908 8 × 2 = 0 + 0.000 000 000 000 459 817 6;
  • 6) 0.000 000 000 000 459 817 6 × 2 = 0 + 0.000 000 000 000 919 635 2;
  • 7) 0.000 000 000 000 919 635 2 × 2 = 0 + 0.000 000 000 001 839 270 4;
  • 8) 0.000 000 000 001 839 270 4 × 2 = 0 + 0.000 000 000 003 678 540 8;
  • 9) 0.000 000 000 003 678 540 8 × 2 = 0 + 0.000 000 000 007 357 081 6;
  • 10) 0.000 000 000 007 357 081 6 × 2 = 0 + 0.000 000 000 014 714 163 2;
  • 11) 0.000 000 000 014 714 163 2 × 2 = 0 + 0.000 000 000 029 428 326 4;
  • 12) 0.000 000 000 029 428 326 4 × 2 = 0 + 0.000 000 000 058 856 652 8;
  • 13) 0.000 000 000 058 856 652 8 × 2 = 0 + 0.000 000 000 117 713 305 6;
  • 14) 0.000 000 000 117 713 305 6 × 2 = 0 + 0.000 000 000 235 426 611 2;
  • 15) 0.000 000 000 235 426 611 2 × 2 = 0 + 0.000 000 000 470 853 222 4;
  • 16) 0.000 000 000 470 853 222 4 × 2 = 0 + 0.000 000 000 941 706 444 8;
  • 17) 0.000 000 000 941 706 444 8 × 2 = 0 + 0.000 000 001 883 412 889 6;
  • 18) 0.000 000 001 883 412 889 6 × 2 = 0 + 0.000 000 003 766 825 779 2;
  • 19) 0.000 000 003 766 825 779 2 × 2 = 0 + 0.000 000 007 533 651 558 4;
  • 20) 0.000 000 007 533 651 558 4 × 2 = 0 + 0.000 000 015 067 303 116 8;
  • 21) 0.000 000 015 067 303 116 8 × 2 = 0 + 0.000 000 030 134 606 233 6;
  • 22) 0.000 000 030 134 606 233 6 × 2 = 0 + 0.000 000 060 269 212 467 2;
  • 23) 0.000 000 060 269 212 467 2 × 2 = 0 + 0.000 000 120 538 424 934 4;
  • 24) 0.000 000 120 538 424 934 4 × 2 = 0 + 0.000 000 241 076 849 868 8;
  • 25) 0.000 000 241 076 849 868 8 × 2 = 0 + 0.000 000 482 153 699 737 6;
  • 26) 0.000 000 482 153 699 737 6 × 2 = 0 + 0.000 000 964 307 399 475 2;
  • 27) 0.000 000 964 307 399 475 2 × 2 = 0 + 0.000 001 928 614 798 950 4;
  • 28) 0.000 001 928 614 798 950 4 × 2 = 0 + 0.000 003 857 229 597 900 8;
  • 29) 0.000 003 857 229 597 900 8 × 2 = 0 + 0.000 007 714 459 195 801 6;
  • 30) 0.000 007 714 459 195 801 6 × 2 = 0 + 0.000 015 428 918 391 603 2;
  • 31) 0.000 015 428 918 391 603 2 × 2 = 0 + 0.000 030 857 836 783 206 4;
  • 32) 0.000 030 857 836 783 206 4 × 2 = 0 + 0.000 061 715 673 566 412 8;
  • 33) 0.000 061 715 673 566 412 8 × 2 = 0 + 0.000 123 431 347 132 825 6;
  • 34) 0.000 123 431 347 132 825 6 × 2 = 0 + 0.000 246 862 694 265 651 2;
  • 35) 0.000 246 862 694 265 651 2 × 2 = 0 + 0.000 493 725 388 531 302 4;
  • 36) 0.000 493 725 388 531 302 4 × 2 = 0 + 0.000 987 450 777 062 604 8;
  • 37) 0.000 987 450 777 062 604 8 × 2 = 0 + 0.001 974 901 554 125 209 6;
  • 38) 0.001 974 901 554 125 209 6 × 2 = 0 + 0.003 949 803 108 250 419 2;
  • 39) 0.003 949 803 108 250 419 2 × 2 = 0 + 0.007 899 606 216 500 838 4;
  • 40) 0.007 899 606 216 500 838 4 × 2 = 0 + 0.015 799 212 433 001 676 8;
  • 41) 0.015 799 212 433 001 676 8 × 2 = 0 + 0.031 598 424 866 003 353 6;
  • 42) 0.031 598 424 866 003 353 6 × 2 = 0 + 0.063 196 849 732 006 707 2;
  • 43) 0.063 196 849 732 006 707 2 × 2 = 0 + 0.126 393 699 464 013 414 4;
  • 44) 0.126 393 699 464 013 414 4 × 2 = 0 + 0.252 787 398 928 026 828 8;
  • 45) 0.252 787 398 928 026 828 8 × 2 = 0 + 0.505 574 797 856 053 657 6;
  • 46) 0.505 574 797 856 053 657 6 × 2 = 1 + 0.011 149 595 712 107 315 2;
  • 47) 0.011 149 595 712 107 315 2 × 2 = 0 + 0.022 299 191 424 214 630 4;
  • 48) 0.022 299 191 424 214 630 4 × 2 = 0 + 0.044 598 382 848 429 260 8;
  • 49) 0.044 598 382 848 429 260 8 × 2 = 0 + 0.089 196 765 696 858 521 6;
  • 50) 0.089 196 765 696 858 521 6 × 2 = 0 + 0.178 393 531 393 717 043 2;
  • 51) 0.178 393 531 393 717 043 2 × 2 = 0 + 0.356 787 062 787 434 086 4;
  • 52) 0.356 787 062 787 434 086 4 × 2 = 0 + 0.713 574 125 574 868 172 8;
  • 53) 0.713 574 125 574 868 172 8 × 2 = 1 + 0.427 148 251 149 736 345 6;
  • 54) 0.427 148 251 149 736 345 6 × 2 = 0 + 0.854 296 502 299 472 691 2;
  • 55) 0.854 296 502 299 472 691 2 × 2 = 1 + 0.708 593 004 598 945 382 4;
  • 56) 0.708 593 004 598 945 382 4 × 2 = 1 + 0.417 186 009 197 890 764 8;
  • 57) 0.417 186 009 197 890 764 8 × 2 = 0 + 0.834 372 018 395 781 529 6;
  • 58) 0.834 372 018 395 781 529 6 × 2 = 1 + 0.668 744 036 791 563 059 2;
  • 59) 0.668 744 036 791 563 059 2 × 2 = 1 + 0.337 488 073 583 126 118 4;
  • 60) 0.337 488 073 583 126 118 4 × 2 = 0 + 0.674 976 147 166 252 236 8;
  • 61) 0.674 976 147 166 252 236 8 × 2 = 1 + 0.349 952 294 332 504 473 6;
  • 62) 0.349 952 294 332 504 473 6 × 2 = 0 + 0.699 904 588 665 008 947 2;
  • 63) 0.699 904 588 665 008 947 2 × 2 = 1 + 0.399 809 177 330 017 894 4;
  • 64) 0.399 809 177 330 017 894 4 × 2 = 0 + 0.799 618 354 660 035 788 8;
  • 65) 0.799 618 354 660 035 788 8 × 2 = 1 + 0.599 236 709 320 071 577 6;
  • 66) 0.599 236 709 320 071 577 6 × 2 = 1 + 0.198 473 418 640 143 155 2;
  • 67) 0.198 473 418 640 143 155 2 × 2 = 0 + 0.396 946 837 280 286 310 4;
  • 68) 0.396 946 837 280 286 310 4 × 2 = 0 + 0.793 893 674 560 572 620 8;
  • 69) 0.793 893 674 560 572 620 8 × 2 = 1 + 0.587 787 349 121 145 241 6;
  • 70) 0.587 787 349 121 145 241 6 × 2 = 1 + 0.175 574 698 242 290 483 2;
  • 71) 0.175 574 698 242 290 483 2 × 2 = 0 + 0.351 149 396 484 580 966 4;
  • 72) 0.351 149 396 484 580 966 4 × 2 = 0 + 0.702 298 792 969 161 932 8;
  • 73) 0.702 298 792 969 161 932 8 × 2 = 1 + 0.404 597 585 938 323 865 6;
  • 74) 0.404 597 585 938 323 865 6 × 2 = 0 + 0.809 195 171 876 647 731 2;
  • 75) 0.809 195 171 876 647 731 2 × 2 = 1 + 0.618 390 343 753 295 462 4;
  • 76) 0.618 390 343 753 295 462 4 × 2 = 1 + 0.236 780 687 506 590 924 8;
  • 77) 0.236 780 687 506 590 924 8 × 2 = 0 + 0.473 561 375 013 181 849 6;
  • 78) 0.473 561 375 013 181 849 6 × 2 = 0 + 0.947 122 750 026 363 699 2;
  • 79) 0.947 122 750 026 363 699 2 × 2 = 1 + 0.894 245 500 052 727 398 4;
  • 80) 0.894 245 500 052 727 398 4 × 2 = 1 + 0.788 491 000 105 454 796 8;
  • 81) 0.788 491 000 105 454 796 8 × 2 = 1 + 0.576 982 000 210 909 593 6;
  • 82) 0.576 982 000 210 909 593 6 × 2 = 1 + 0.153 964 000 421 819 187 2;
  • 83) 0.153 964 000 421 819 187 2 × 2 = 0 + 0.307 928 000 843 638 374 4;
  • 84) 0.307 928 000 843 638 374 4 × 2 = 0 + 0.615 856 001 687 276 748 8;
  • 85) 0.615 856 001 687 276 748 8 × 2 = 1 + 0.231 712 003 374 553 497 6;
  • 86) 0.231 712 003 374 553 497 6 × 2 = 0 + 0.463 424 006 749 106 995 2;
  • 87) 0.463 424 006 749 106 995 2 × 2 = 0 + 0.926 848 013 498 213 990 4;
  • 88) 0.926 848 013 498 213 990 4 × 2 = 1 + 0.853 696 026 996 427 980 8;
  • 89) 0.853 696 026 996 427 980 8 × 2 = 1 + 0.707 392 053 992 855 961 6;
  • 90) 0.707 392 053 992 855 961 6 × 2 = 1 + 0.414 784 107 985 711 923 2;
  • 91) 0.414 784 107 985 711 923 2 × 2 = 0 + 0.829 568 215 971 423 846 4;
  • 92) 0.829 568 215 971 423 846 4 × 2 = 1 + 0.659 136 431 942 847 692 8;
  • 93) 0.659 136 431 942 847 692 8 × 2 = 1 + 0.318 272 863 885 695 385 6;
  • 94) 0.318 272 863 885 695 385 6 × 2 = 0 + 0.636 545 727 771 390 771 2;
  • 95) 0.636 545 727 771 390 771 2 × 2 = 1 + 0.273 091 455 542 781 542 4;
  • 96) 0.273 091 455 542 781 542 4 × 2 = 0 + 0.546 182 911 085 563 084 8;
  • 97) 0.546 182 911 085 563 084 8 × 2 = 1 + 0.092 365 822 171 126 169 6;
  • 98) 0.092 365 822 171 126 169 6 × 2 = 0 + 0.184 731 644 342 252 339 2;

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:


0.000 000 000 000 014 369 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1011 0110 1010 1100 1100 1011 0011 1100 1001 1101 1010 10(2)

6. Positive number before normalization:

0.000 000 000 000 014 369 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1011 0110 1010 1100 1100 1011 0011 1100 1001 1101 1010 10(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:


0.000 000 000 000 014 369 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1011 0110 1010 1100 1100 1011 0011 1100 1001 1101 1010 10(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1011 0110 1010 1100 1100 1011 0011 1100 1001 1101 1010 10(2) × 20 =

1.0000 0010 1101 1010 1011 0011 0010 1100 1111 0010 0111 0110 1010(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 0010 1101 1010 1011 0011 0010 1100 1111 0010 0111 0110 1010


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 0010 1101 1010 1011 0011 0010 1100 1111 0010 0111 0110 1010 =

0000 0010 1101 1010 1011 0011 0010 1100 1111 0010 0111 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 1101 0001

Mantissa (52 bits) =
0000 0010 1101 1010 1011 0011 0010 1100 1111 0010 0111 0110 1010


Decimal number -0.000 000 000 000 014 369 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1101 0001 - 0000 0010 1101 1010 1011 0011 0010 1100 1111 0010 0111 0110 1010

» Convert decimal -0.000 000 000 000 014 366 4 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