-0.000 000 000 000 014 378 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 014 378 6(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
-0.000 000 000 000 014 378 6(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:

|-0.000 000 000 000 014 378 6| = 0.000 000 000 000 014 378 6


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: 0.000 000 000 000 014 378 6.

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) 0.000 000 000 000 014 378 6 × 2 = 0 + 0.000 000 000 000 028 757 2;
  • 2) 0.000 000 000 000 028 757 2 × 2 = 0 + 0.000 000 000 000 057 514 4;
  • 3) 0.000 000 000 000 057 514 4 × 2 = 0 + 0.000 000 000 000 115 028 8;
  • 4) 0.000 000 000 000 115 028 8 × 2 = 0 + 0.000 000 000 000 230 057 6;
  • 5) 0.000 000 000 000 230 057 6 × 2 = 0 + 0.000 000 000 000 460 115 2;
  • 6) 0.000 000 000 000 460 115 2 × 2 = 0 + 0.000 000 000 000 920 230 4;
  • 7) 0.000 000 000 000 920 230 4 × 2 = 0 + 0.000 000 000 001 840 460 8;
  • 8) 0.000 000 000 001 840 460 8 × 2 = 0 + 0.000 000 000 003 680 921 6;
  • 9) 0.000 000 000 003 680 921 6 × 2 = 0 + 0.000 000 000 007 361 843 2;
  • 10) 0.000 000 000 007 361 843 2 × 2 = 0 + 0.000 000 000 014 723 686 4;
  • 11) 0.000 000 000 014 723 686 4 × 2 = 0 + 0.000 000 000 029 447 372 8;
  • 12) 0.000 000 000 029 447 372 8 × 2 = 0 + 0.000 000 000 058 894 745 6;
  • 13) 0.000 000 000 058 894 745 6 × 2 = 0 + 0.000 000 000 117 789 491 2;
  • 14) 0.000 000 000 117 789 491 2 × 2 = 0 + 0.000 000 000 235 578 982 4;
  • 15) 0.000 000 000 235 578 982 4 × 2 = 0 + 0.000 000 000 471 157 964 8;
  • 16) 0.000 000 000 471 157 964 8 × 2 = 0 + 0.000 000 000 942 315 929 6;
  • 17) 0.000 000 000 942 315 929 6 × 2 = 0 + 0.000 000 001 884 631 859 2;
  • 18) 0.000 000 001 884 631 859 2 × 2 = 0 + 0.000 000 003 769 263 718 4;
  • 19) 0.000 000 003 769 263 718 4 × 2 = 0 + 0.000 000 007 538 527 436 8;
  • 20) 0.000 000 007 538 527 436 8 × 2 = 0 + 0.000 000 015 077 054 873 6;
  • 21) 0.000 000 015 077 054 873 6 × 2 = 0 + 0.000 000 030 154 109 747 2;
  • 22) 0.000 000 030 154 109 747 2 × 2 = 0 + 0.000 000 060 308 219 494 4;
  • 23) 0.000 000 060 308 219 494 4 × 2 = 0 + 0.000 000 120 616 438 988 8;
  • 24) 0.000 000 120 616 438 988 8 × 2 = 0 + 0.000 000 241 232 877 977 6;
  • 25) 0.000 000 241 232 877 977 6 × 2 = 0 + 0.000 000 482 465 755 955 2;
  • 26) 0.000 000 482 465 755 955 2 × 2 = 0 + 0.000 000 964 931 511 910 4;
  • 27) 0.000 000 964 931 511 910 4 × 2 = 0 + 0.000 001 929 863 023 820 8;
  • 28) 0.000 001 929 863 023 820 8 × 2 = 0 + 0.000 003 859 726 047 641 6;
  • 29) 0.000 003 859 726 047 641 6 × 2 = 0 + 0.000 007 719 452 095 283 2;
  • 30) 0.000 007 719 452 095 283 2 × 2 = 0 + 0.000 015 438 904 190 566 4;
  • 31) 0.000 015 438 904 190 566 4 × 2 = 0 + 0.000 030 877 808 381 132 8;
  • 32) 0.000 030 877 808 381 132 8 × 2 = 0 + 0.000 061 755 616 762 265 6;
  • 33) 0.000 061 755 616 762 265 6 × 2 = 0 + 0.000 123 511 233 524 531 2;
  • 34) 0.000 123 511 233 524 531 2 × 2 = 0 + 0.000 247 022 467 049 062 4;
  • 35) 0.000 247 022 467 049 062 4 × 2 = 0 + 0.000 494 044 934 098 124 8;
  • 36) 0.000 494 044 934 098 124 8 × 2 = 0 + 0.000 988 089 868 196 249 6;
  • 37) 0.000 988 089 868 196 249 6 × 2 = 0 + 0.001 976 179 736 392 499 2;
  • 38) 0.001 976 179 736 392 499 2 × 2 = 0 + 0.003 952 359 472 784 998 4;
  • 39) 0.003 952 359 472 784 998 4 × 2 = 0 + 0.007 904 718 945 569 996 8;
  • 40) 0.007 904 718 945 569 996 8 × 2 = 0 + 0.015 809 437 891 139 993 6;
  • 41) 0.015 809 437 891 139 993 6 × 2 = 0 + 0.031 618 875 782 279 987 2;
  • 42) 0.031 618 875 782 279 987 2 × 2 = 0 + 0.063 237 751 564 559 974 4;
  • 43) 0.063 237 751 564 559 974 4 × 2 = 0 + 0.126 475 503 129 119 948 8;
  • 44) 0.126 475 503 129 119 948 8 × 2 = 0 + 0.252 951 006 258 239 897 6;
  • 45) 0.252 951 006 258 239 897 6 × 2 = 0 + 0.505 902 012 516 479 795 2;
  • 46) 0.505 902 012 516 479 795 2 × 2 = 1 + 0.011 804 025 032 959 590 4;
  • 47) 0.011 804 025 032 959 590 4 × 2 = 0 + 0.023 608 050 065 919 180 8;
  • 48) 0.023 608 050 065 919 180 8 × 2 = 0 + 0.047 216 100 131 838 361 6;
  • 49) 0.047 216 100 131 838 361 6 × 2 = 0 + 0.094 432 200 263 676 723 2;
  • 50) 0.094 432 200 263 676 723 2 × 2 = 0 + 0.188 864 400 527 353 446 4;
  • 51) 0.188 864 400 527 353 446 4 × 2 = 0 + 0.377 728 801 054 706 892 8;
  • 52) 0.377 728 801 054 706 892 8 × 2 = 0 + 0.755 457 602 109 413 785 6;
  • 53) 0.755 457 602 109 413 785 6 × 2 = 1 + 0.510 915 204 218 827 571 2;
  • 54) 0.510 915 204 218 827 571 2 × 2 = 1 + 0.021 830 408 437 655 142 4;
  • 55) 0.021 830 408 437 655 142 4 × 2 = 0 + 0.043 660 816 875 310 284 8;
  • 56) 0.043 660 816 875 310 284 8 × 2 = 0 + 0.087 321 633 750 620 569 6;
  • 57) 0.087 321 633 750 620 569 6 × 2 = 0 + 0.174 643 267 501 241 139 2;
  • 58) 0.174 643 267 501 241 139 2 × 2 = 0 + 0.349 286 535 002 482 278 4;
  • 59) 0.349 286 535 002 482 278 4 × 2 = 0 + 0.698 573 070 004 964 556 8;
  • 60) 0.698 573 070 004 964 556 8 × 2 = 1 + 0.397 146 140 009 929 113 6;
  • 61) 0.397 146 140 009 929 113 6 × 2 = 0 + 0.794 292 280 019 858 227 2;
  • 62) 0.794 292 280 019 858 227 2 × 2 = 1 + 0.588 584 560 039 716 454 4;
  • 63) 0.588 584 560 039 716 454 4 × 2 = 1 + 0.177 169 120 079 432 908 8;
  • 64) 0.177 169 120 079 432 908 8 × 2 = 0 + 0.354 338 240 158 865 817 6;
  • 65) 0.354 338 240 158 865 817 6 × 2 = 0 + 0.708 676 480 317 731 635 2;
  • 66) 0.708 676 480 317 731 635 2 × 2 = 1 + 0.417 352 960 635 463 270 4;
  • 67) 0.417 352 960 635 463 270 4 × 2 = 0 + 0.834 705 921 270 926 540 8;
  • 68) 0.834 705 921 270 926 540 8 × 2 = 1 + 0.669 411 842 541 853 081 6;
  • 69) 0.669 411 842 541 853 081 6 × 2 = 1 + 0.338 823 685 083 706 163 2;
  • 70) 0.338 823 685 083 706 163 2 × 2 = 0 + 0.677 647 370 167 412 326 4;
  • 71) 0.677 647 370 167 412 326 4 × 2 = 1 + 0.355 294 740 334 824 652 8;
  • 72) 0.355 294 740 334 824 652 8 × 2 = 0 + 0.710 589 480 669 649 305 6;
  • 73) 0.710 589 480 669 649 305 6 × 2 = 1 + 0.421 178 961 339 298 611 2;
  • 74) 0.421 178 961 339 298 611 2 × 2 = 0 + 0.842 357 922 678 597 222 4;
  • 75) 0.842 357 922 678 597 222 4 × 2 = 1 + 0.684 715 845 357 194 444 8;
  • 76) 0.684 715 845 357 194 444 8 × 2 = 1 + 0.369 431 690 714 388 889 6;
  • 77) 0.369 431 690 714 388 889 6 × 2 = 0 + 0.738 863 381 428 777 779 2;
  • 78) 0.738 863 381 428 777 779 2 × 2 = 1 + 0.477 726 762 857 555 558 4;
  • 79) 0.477 726 762 857 555 558 4 × 2 = 0 + 0.955 453 525 715 111 116 8;
  • 80) 0.955 453 525 715 111 116 8 × 2 = 1 + 0.910 907 051 430 222 233 6;
  • 81) 0.910 907 051 430 222 233 6 × 2 = 1 + 0.821 814 102 860 444 467 2;
  • 82) 0.821 814 102 860 444 467 2 × 2 = 1 + 0.643 628 205 720 888 934 4;
  • 83) 0.643 628 205 720 888 934 4 × 2 = 1 + 0.287 256 411 441 777 868 8;
  • 84) 0.287 256 411 441 777 868 8 × 2 = 0 + 0.574 512 822 883 555 737 6;
  • 85) 0.574 512 822 883 555 737 6 × 2 = 1 + 0.149 025 645 767 111 475 2;
  • 86) 0.149 025 645 767 111 475 2 × 2 = 0 + 0.298 051 291 534 222 950 4;
  • 87) 0.298 051 291 534 222 950 4 × 2 = 0 + 0.596 102 583 068 445 900 8;
  • 88) 0.596 102 583 068 445 900 8 × 2 = 1 + 0.192 205 166 136 891 801 6;
  • 89) 0.192 205 166 136 891 801 6 × 2 = 0 + 0.384 410 332 273 783 603 2;
  • 90) 0.384 410 332 273 783 603 2 × 2 = 0 + 0.768 820 664 547 567 206 4;
  • 91) 0.768 820 664 547 567 206 4 × 2 = 1 + 0.537 641 329 095 134 412 8;
  • 92) 0.537 641 329 095 134 412 8 × 2 = 1 + 0.075 282 658 190 268 825 6;
  • 93) 0.075 282 658 190 268 825 6 × 2 = 0 + 0.150 565 316 380 537 651 2;
  • 94) 0.150 565 316 380 537 651 2 × 2 = 0 + 0.301 130 632 761 075 302 4;
  • 95) 0.301 130 632 761 075 302 4 × 2 = 0 + 0.602 261 265 522 150 604 8;
  • 96) 0.602 261 265 522 150 604 8 × 2 = 1 + 0.204 522 531 044 301 209 6;
  • 97) 0.204 522 531 044 301 209 6 × 2 = 0 + 0.409 045 062 088 602 419 2;
  • 98) 0.409 045 062 088 602 419 2 × 2 = 0 + 0.818 090 124 177 204 838 4;

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:


0.000 000 000 000 014 378 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1100 0001 0110 0101 1010 1011 0101 1110 1001 0011 0001 00(2)

6. Positive number before normalization:

0.000 000 000 000 014 378 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1100 0001 0110 0101 1010 1011 0101 1110 1001 0011 0001 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:


0.000 000 000 000 014 378 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1100 0001 0110 0101 1010 1011 0101 1110 1001 0011 0001 00(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0000 1100 0001 0110 0101 1010 1011 0101 1110 1001 0011 0001 00(2) × 20 =

1.0000 0011 0000 0101 1001 0110 1010 1101 0111 1010 0100 1100 0100(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 0000 0101 1001 0110 1010 1101 0111 1010 0100 1100 0100


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 0000 0101 1001 0110 1010 1101 0111 1010 0100 1100 0100 =

0000 0011 0000 0101 1001 0110 1010 1101 0111 1010 0100 1100 0100


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 0000 0101 1001 0110 1010 1101 0111 1010 0100 1100 0100


Decimal number -0.000 000 000 000 014 378 6 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1101 0001 - 0000 0011 0000 0101 1001 0110 1010 1101 0111 1010 0100 1100 0100

» Convert decimal -0.000 000 000 000 014 384 2 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