18.269 999 999 999 999 573 674 358 543 939 888 477 325 439 397 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 18.269 999 999 999 999 573 674 358 543 939 888 477 325 439 397(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
18.269 999 999 999 999 573 674 358 543 939 888 477 325 439 397(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. First, convert to binary (in base 2) the integer part: 18.
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;
  • 18 ÷ 2 = 9 + 0;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. 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.

18(10) =

1 0010(2)


3. Convert to binary (base 2) the fractional part: 0.269 999 999 999 999 573 674 358 543 939 888 477 325 439 397.

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) 0.269 999 999 999 999 573 674 358 543 939 888 477 325 439 397 × 2 = 0 + 0.539 999 999 999 999 147 348 717 087 879 776 954 650 878 794;
  • 2) 0.539 999 999 999 999 147 348 717 087 879 776 954 650 878 794 × 2 = 1 + 0.079 999 999 999 998 294 697 434 175 759 553 909 301 757 588;
  • 3) 0.079 999 999 999 998 294 697 434 175 759 553 909 301 757 588 × 2 = 0 + 0.159 999 999 999 996 589 394 868 351 519 107 818 603 515 176;
  • 4) 0.159 999 999 999 996 589 394 868 351 519 107 818 603 515 176 × 2 = 0 + 0.319 999 999 999 993 178 789 736 703 038 215 637 207 030 352;
  • 5) 0.319 999 999 999 993 178 789 736 703 038 215 637 207 030 352 × 2 = 0 + 0.639 999 999 999 986 357 579 473 406 076 431 274 414 060 704;
  • 6) 0.639 999 999 999 986 357 579 473 406 076 431 274 414 060 704 × 2 = 1 + 0.279 999 999 999 972 715 158 946 812 152 862 548 828 121 408;
  • 7) 0.279 999 999 999 972 715 158 946 812 152 862 548 828 121 408 × 2 = 0 + 0.559 999 999 999 945 430 317 893 624 305 725 097 656 242 816;
  • 8) 0.559 999 999 999 945 430 317 893 624 305 725 097 656 242 816 × 2 = 1 + 0.119 999 999 999 890 860 635 787 248 611 450 195 312 485 632;
  • 9) 0.119 999 999 999 890 860 635 787 248 611 450 195 312 485 632 × 2 = 0 + 0.239 999 999 999 781 721 271 574 497 222 900 390 624 971 264;
  • 10) 0.239 999 999 999 781 721 271 574 497 222 900 390 624 971 264 × 2 = 0 + 0.479 999 999 999 563 442 543 148 994 445 800 781 249 942 528;
  • 11) 0.479 999 999 999 563 442 543 148 994 445 800 781 249 942 528 × 2 = 0 + 0.959 999 999 999 126 885 086 297 988 891 601 562 499 885 056;
  • 12) 0.959 999 999 999 126 885 086 297 988 891 601 562 499 885 056 × 2 = 1 + 0.919 999 999 998 253 770 172 595 977 783 203 124 999 770 112;
  • 13) 0.919 999 999 998 253 770 172 595 977 783 203 124 999 770 112 × 2 = 1 + 0.839 999 999 996 507 540 345 191 955 566 406 249 999 540 224;
  • 14) 0.839 999 999 996 507 540 345 191 955 566 406 249 999 540 224 × 2 = 1 + 0.679 999 999 993 015 080 690 383 911 132 812 499 999 080 448;
  • 15) 0.679 999 999 993 015 080 690 383 911 132 812 499 999 080 448 × 2 = 1 + 0.359 999 999 986 030 161 380 767 822 265 624 999 998 160 896;
  • 16) 0.359 999 999 986 030 161 380 767 822 265 624 999 998 160 896 × 2 = 0 + 0.719 999 999 972 060 322 761 535 644 531 249 999 996 321 792;
  • 17) 0.719 999 999 972 060 322 761 535 644 531 249 999 996 321 792 × 2 = 1 + 0.439 999 999 944 120 645 523 071 289 062 499 999 992 643 584;
  • 18) 0.439 999 999 944 120 645 523 071 289 062 499 999 992 643 584 × 2 = 0 + 0.879 999 999 888 241 291 046 142 578 124 999 999 985 287 168;
  • 19) 0.879 999 999 888 241 291 046 142 578 124 999 999 985 287 168 × 2 = 1 + 0.759 999 999 776 482 582 092 285 156 249 999 999 970 574 336;
  • 20) 0.759 999 999 776 482 582 092 285 156 249 999 999 970 574 336 × 2 = 1 + 0.519 999 999 552 965 164 184 570 312 499 999 999 941 148 672;
  • 21) 0.519 999 999 552 965 164 184 570 312 499 999 999 941 148 672 × 2 = 1 + 0.039 999 999 105 930 328 369 140 624 999 999 999 882 297 344;
  • 22) 0.039 999 999 105 930 328 369 140 624 999 999 999 882 297 344 × 2 = 0 + 0.079 999 998 211 860 656 738 281 249 999 999 999 764 594 688;
  • 23) 0.079 999 998 211 860 656 738 281 249 999 999 999 764 594 688 × 2 = 0 + 0.159 999 996 423 721 313 476 562 499 999 999 999 529 189 376;
  • 24) 0.159 999 996 423 721 313 476 562 499 999 999 999 529 189 376 × 2 = 0 + 0.319 999 992 847 442 626 953 124 999 999 999 999 058 378 752;
  • 25) 0.319 999 992 847 442 626 953 124 999 999 999 999 058 378 752 × 2 = 0 + 0.639 999 985 694 885 253 906 249 999 999 999 998 116 757 504;
  • 26) 0.639 999 985 694 885 253 906 249 999 999 999 998 116 757 504 × 2 = 1 + 0.279 999 971 389 770 507 812 499 999 999 999 996 233 515 008;
  • 27) 0.279 999 971 389 770 507 812 499 999 999 999 996 233 515 008 × 2 = 0 + 0.559 999 942 779 541 015 624 999 999 999 999 992 467 030 016;
  • 28) 0.559 999 942 779 541 015 624 999 999 999 999 992 467 030 016 × 2 = 1 + 0.119 999 885 559 082 031 249 999 999 999 999 984 934 060 032;
  • 29) 0.119 999 885 559 082 031 249 999 999 999 999 984 934 060 032 × 2 = 0 + 0.239 999 771 118 164 062 499 999 999 999 999 969 868 120 064;
  • 30) 0.239 999 771 118 164 062 499 999 999 999 999 969 868 120 064 × 2 = 0 + 0.479 999 542 236 328 124 999 999 999 999 999 939 736 240 128;
  • 31) 0.479 999 542 236 328 124 999 999 999 999 999 939 736 240 128 × 2 = 0 + 0.959 999 084 472 656 249 999 999 999 999 999 879 472 480 256;
  • 32) 0.959 999 084 472 656 249 999 999 999 999 999 879 472 480 256 × 2 = 1 + 0.919 998 168 945 312 499 999 999 999 999 999 758 944 960 512;
  • 33) 0.919 998 168 945 312 499 999 999 999 999 999 758 944 960 512 × 2 = 1 + 0.839 996 337 890 624 999 999 999 999 999 999 517 889 921 024;
  • 34) 0.839 996 337 890 624 999 999 999 999 999 999 517 889 921 024 × 2 = 1 + 0.679 992 675 781 249 999 999 999 999 999 999 035 779 842 048;
  • 35) 0.679 992 675 781 249 999 999 999 999 999 999 035 779 842 048 × 2 = 1 + 0.359 985 351 562 499 999 999 999 999 999 998 071 559 684 096;
  • 36) 0.359 985 351 562 499 999 999 999 999 999 998 071 559 684 096 × 2 = 0 + 0.719 970 703 124 999 999 999 999 999 999 996 143 119 368 192;
  • 37) 0.719 970 703 124 999 999 999 999 999 999 996 143 119 368 192 × 2 = 1 + 0.439 941 406 249 999 999 999 999 999 999 992 286 238 736 384;
  • 38) 0.439 941 406 249 999 999 999 999 999 999 992 286 238 736 384 × 2 = 0 + 0.879 882 812 499 999 999 999 999 999 999 984 572 477 472 768;
  • 39) 0.879 882 812 499 999 999 999 999 999 999 984 572 477 472 768 × 2 = 1 + 0.759 765 624 999 999 999 999 999 999 999 969 144 954 945 536;
  • 40) 0.759 765 624 999 999 999 999 999 999 999 969 144 954 945 536 × 2 = 1 + 0.519 531 249 999 999 999 999 999 999 999 938 289 909 891 072;
  • 41) 0.519 531 249 999 999 999 999 999 999 999 938 289 909 891 072 × 2 = 1 + 0.039 062 499 999 999 999 999 999 999 999 876 579 819 782 144;
  • 42) 0.039 062 499 999 999 999 999 999 999 999 876 579 819 782 144 × 2 = 0 + 0.078 124 999 999 999 999 999 999 999 999 753 159 639 564 288;
  • 43) 0.078 124 999 999 999 999 999 999 999 999 753 159 639 564 288 × 2 = 0 + 0.156 249 999 999 999 999 999 999 999 999 506 319 279 128 576;
  • 44) 0.156 249 999 999 999 999 999 999 999 999 506 319 279 128 576 × 2 = 0 + 0.312 499 999 999 999 999 999 999 999 999 012 638 558 257 152;
  • 45) 0.312 499 999 999 999 999 999 999 999 999 012 638 558 257 152 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 998 025 277 116 514 304;
  • 46) 0.624 999 999 999 999 999 999 999 999 998 025 277 116 514 304 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 996 050 554 233 028 608;
  • 47) 0.249 999 999 999 999 999 999 999 999 996 050 554 233 028 608 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 992 101 108 466 057 216;
  • 48) 0.499 999 999 999 999 999 999 999 999 992 101 108 466 057 216 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 984 202 216 932 114 432;
  • 49) 0.999 999 999 999 999 999 999 999 999 984 202 216 932 114 432 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 968 404 433 864 228 864;
  • 50) 0.999 999 999 999 999 999 999 999 999 968 404 433 864 228 864 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 936 808 867 728 457 728;
  • 51) 0.999 999 999 999 999 999 999 999 999 936 808 867 728 457 728 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 873 617 735 456 915 456;
  • 52) 0.999 999 999 999 999 999 999 999 999 873 617 735 456 915 456 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 747 235 470 913 830 912;
  • 53) 0.999 999 999 999 999 999 999 999 999 747 235 470 913 830 912 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 494 470 941 827 661 824;

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).


4. 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:


0.269 999 999 999 999 573 674 358 543 939 888 477 325 439 397(10) =

0.0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0100 1111 1(2)

5. Positive number before normalization:

18.269 999 999 999 999 573 674 358 543 939 888 477 325 439 397(10) =

1 0010.0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0100 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 4 positions to the left, so that only one non zero digit remains to the left of it:


18.269 999 999 999 999 573 674 358 543 939 888 477 325 439 397(10) =

1 0010.0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0100 1111 1(2) =

1 0010.0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0100 1111 1(2) × 20 =

1.0010 0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0100 1111 1(2) × 24


7. 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 0 (a positive number)

Exponent (unadjusted): 4

Mantissa (not normalized):
1.0010 0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0100 1111 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

4 + 2(11-1) - 1 =

(4 + 1 023)(10) =

1 027(10)


9. 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;
  • 1 027 ÷ 2 = 513 + 1;
  • 513 ÷ 2 = 256 + 1;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1027(10) =

100 0000 0011(2)


11. 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =

1. 0010 0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0100 1 1111 =

0010 0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0100


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
100 0000 0011

Mantissa (52 bits) =
0010 0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0100


Decimal number 18.269 999 999 999 999 573 674 358 543 939 888 477 325 439 397 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0011 - 0010 0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 0100

» Convert decimal 18.269 999 999 999 999 573 674 358 543 939 888 477 325 439 38 to 64 bit double precision IEEE 754 binary floating point representation standard

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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