0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 751 068 102 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 751 068 102(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
0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 751 068 102(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: 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;

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.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 751 068 102.

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) 0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 751 068 102 × 2 = 0 + 0.080 000 000 000 000 001 665 334 536 937 734 810 635 447 502 136 204;
  • 2) 0.080 000 000 000 000 001 665 334 536 937 734 810 635 447 502 136 204 × 2 = 0 + 0.160 000 000 000 000 003 330 669 073 875 469 621 270 895 004 272 408;
  • 3) 0.160 000 000 000 000 003 330 669 073 875 469 621 270 895 004 272 408 × 2 = 0 + 0.320 000 000 000 000 006 661 338 147 750 939 242 541 790 008 544 816;
  • 4) 0.320 000 000 000 000 006 661 338 147 750 939 242 541 790 008 544 816 × 2 = 0 + 0.640 000 000 000 000 013 322 676 295 501 878 485 083 580 017 089 632;
  • 5) 0.640 000 000 000 000 013 322 676 295 501 878 485 083 580 017 089 632 × 2 = 1 + 0.280 000 000 000 000 026 645 352 591 003 756 970 167 160 034 179 264;
  • 6) 0.280 000 000 000 000 026 645 352 591 003 756 970 167 160 034 179 264 × 2 = 0 + 0.560 000 000 000 000 053 290 705 182 007 513 940 334 320 068 358 528;
  • 7) 0.560 000 000 000 000 053 290 705 182 007 513 940 334 320 068 358 528 × 2 = 1 + 0.120 000 000 000 000 106 581 410 364 015 027 880 668 640 136 717 056;
  • 8) 0.120 000 000 000 000 106 581 410 364 015 027 880 668 640 136 717 056 × 2 = 0 + 0.240 000 000 000 000 213 162 820 728 030 055 761 337 280 273 434 112;
  • 9) 0.240 000 000 000 000 213 162 820 728 030 055 761 337 280 273 434 112 × 2 = 0 + 0.480 000 000 000 000 426 325 641 456 060 111 522 674 560 546 868 224;
  • 10) 0.480 000 000 000 000 426 325 641 456 060 111 522 674 560 546 868 224 × 2 = 0 + 0.960 000 000 000 000 852 651 282 912 120 223 045 349 121 093 736 448;
  • 11) 0.960 000 000 000 000 852 651 282 912 120 223 045 349 121 093 736 448 × 2 = 1 + 0.920 000 000 000 001 705 302 565 824 240 446 090 698 242 187 472 896;
  • 12) 0.920 000 000 000 001 705 302 565 824 240 446 090 698 242 187 472 896 × 2 = 1 + 0.840 000 000 000 003 410 605 131 648 480 892 181 396 484 374 945 792;
  • 13) 0.840 000 000 000 003 410 605 131 648 480 892 181 396 484 374 945 792 × 2 = 1 + 0.680 000 000 000 006 821 210 263 296 961 784 362 792 968 749 891 584;
  • 14) 0.680 000 000 000 006 821 210 263 296 961 784 362 792 968 749 891 584 × 2 = 1 + 0.360 000 000 000 013 642 420 526 593 923 568 725 585 937 499 783 168;
  • 15) 0.360 000 000 000 013 642 420 526 593 923 568 725 585 937 499 783 168 × 2 = 0 + 0.720 000 000 000 027 284 841 053 187 847 137 451 171 874 999 566 336;
  • 16) 0.720 000 000 000 027 284 841 053 187 847 137 451 171 874 999 566 336 × 2 = 1 + 0.440 000 000 000 054 569 682 106 375 694 274 902 343 749 999 132 672;
  • 17) 0.440 000 000 000 054 569 682 106 375 694 274 902 343 749 999 132 672 × 2 = 0 + 0.880 000 000 000 109 139 364 212 751 388 549 804 687 499 998 265 344;
  • 18) 0.880 000 000 000 109 139 364 212 751 388 549 804 687 499 998 265 344 × 2 = 1 + 0.760 000 000 000 218 278 728 425 502 777 099 609 374 999 996 530 688;
  • 19) 0.760 000 000 000 218 278 728 425 502 777 099 609 374 999 996 530 688 × 2 = 1 + 0.520 000 000 000 436 557 456 851 005 554 199 218 749 999 993 061 376;
  • 20) 0.520 000 000 000 436 557 456 851 005 554 199 218 749 999 993 061 376 × 2 = 1 + 0.040 000 000 000 873 114 913 702 011 108 398 437 499 999 986 122 752;
  • 21) 0.040 000 000 000 873 114 913 702 011 108 398 437 499 999 986 122 752 × 2 = 0 + 0.080 000 000 001 746 229 827 404 022 216 796 874 999 999 972 245 504;
  • 22) 0.080 000 000 001 746 229 827 404 022 216 796 874 999 999 972 245 504 × 2 = 0 + 0.160 000 000 003 492 459 654 808 044 433 593 749 999 999 944 491 008;
  • 23) 0.160 000 000 003 492 459 654 808 044 433 593 749 999 999 944 491 008 × 2 = 0 + 0.320 000 000 006 984 919 309 616 088 867 187 499 999 999 888 982 016;
  • 24) 0.320 000 000 006 984 919 309 616 088 867 187 499 999 999 888 982 016 × 2 = 0 + 0.640 000 000 013 969 838 619 232 177 734 374 999 999 999 777 964 032;
  • 25) 0.640 000 000 013 969 838 619 232 177 734 374 999 999 999 777 964 032 × 2 = 1 + 0.280 000 000 027 939 677 238 464 355 468 749 999 999 999 555 928 064;
  • 26) 0.280 000 000 027 939 677 238 464 355 468 749 999 999 999 555 928 064 × 2 = 0 + 0.560 000 000 055 879 354 476 928 710 937 499 999 999 999 111 856 128;
  • 27) 0.560 000 000 055 879 354 476 928 710 937 499 999 999 999 111 856 128 × 2 = 1 + 0.120 000 000 111 758 708 953 857 421 874 999 999 999 998 223 712 256;
  • 28) 0.120 000 000 111 758 708 953 857 421 874 999 999 999 998 223 712 256 × 2 = 0 + 0.240 000 000 223 517 417 907 714 843 749 999 999 999 996 447 424 512;
  • 29) 0.240 000 000 223 517 417 907 714 843 749 999 999 999 996 447 424 512 × 2 = 0 + 0.480 000 000 447 034 835 815 429 687 499 999 999 999 992 894 849 024;
  • 30) 0.480 000 000 447 034 835 815 429 687 499 999 999 999 992 894 849 024 × 2 = 0 + 0.960 000 000 894 069 671 630 859 374 999 999 999 999 985 789 698 048;
  • 31) 0.960 000 000 894 069 671 630 859 374 999 999 999 999 985 789 698 048 × 2 = 1 + 0.920 000 001 788 139 343 261 718 749 999 999 999 999 971 579 396 096;
  • 32) 0.920 000 001 788 139 343 261 718 749 999 999 999 999 971 579 396 096 × 2 = 1 + 0.840 000 003 576 278 686 523 437 499 999 999 999 999 943 158 792 192;
  • 33) 0.840 000 003 576 278 686 523 437 499 999 999 999 999 943 158 792 192 × 2 = 1 + 0.680 000 007 152 557 373 046 874 999 999 999 999 999 886 317 584 384;
  • 34) 0.680 000 007 152 557 373 046 874 999 999 999 999 999 886 317 584 384 × 2 = 1 + 0.360 000 014 305 114 746 093 749 999 999 999 999 999 772 635 168 768;
  • 35) 0.360 000 014 305 114 746 093 749 999 999 999 999 999 772 635 168 768 × 2 = 0 + 0.720 000 028 610 229 492 187 499 999 999 999 999 999 545 270 337 536;
  • 36) 0.720 000 028 610 229 492 187 499 999 999 999 999 999 545 270 337 536 × 2 = 1 + 0.440 000 057 220 458 984 374 999 999 999 999 999 999 090 540 675 072;
  • 37) 0.440 000 057 220 458 984 374 999 999 999 999 999 999 090 540 675 072 × 2 = 0 + 0.880 000 114 440 917 968 749 999 999 999 999 999 998 181 081 350 144;
  • 38) 0.880 000 114 440 917 968 749 999 999 999 999 999 998 181 081 350 144 × 2 = 1 + 0.760 000 228 881 835 937 499 999 999 999 999 999 996 362 162 700 288;
  • 39) 0.760 000 228 881 835 937 499 999 999 999 999 999 996 362 162 700 288 × 2 = 1 + 0.520 000 457 763 671 874 999 999 999 999 999 999 992 724 325 400 576;
  • 40) 0.520 000 457 763 671 874 999 999 999 999 999 999 992 724 325 400 576 × 2 = 1 + 0.040 000 915 527 343 749 999 999 999 999 999 999 985 448 650 801 152;
  • 41) 0.040 000 915 527 343 749 999 999 999 999 999 999 985 448 650 801 152 × 2 = 0 + 0.080 001 831 054 687 499 999 999 999 999 999 999 970 897 301 602 304;
  • 42) 0.080 001 831 054 687 499 999 999 999 999 999 999 970 897 301 602 304 × 2 = 0 + 0.160 003 662 109 374 999 999 999 999 999 999 999 941 794 603 204 608;
  • 43) 0.160 003 662 109 374 999 999 999 999 999 999 999 941 794 603 204 608 × 2 = 0 + 0.320 007 324 218 749 999 999 999 999 999 999 999 883 589 206 409 216;
  • 44) 0.320 007 324 218 749 999 999 999 999 999 999 999 883 589 206 409 216 × 2 = 0 + 0.640 014 648 437 499 999 999 999 999 999 999 999 767 178 412 818 432;
  • 45) 0.640 014 648 437 499 999 999 999 999 999 999 999 767 178 412 818 432 × 2 = 1 + 0.280 029 296 874 999 999 999 999 999 999 999 999 534 356 825 636 864;
  • 46) 0.280 029 296 874 999 999 999 999 999 999 999 999 534 356 825 636 864 × 2 = 0 + 0.560 058 593 749 999 999 999 999 999 999 999 999 068 713 651 273 728;
  • 47) 0.560 058 593 749 999 999 999 999 999 999 999 999 068 713 651 273 728 × 2 = 1 + 0.120 117 187 499 999 999 999 999 999 999 999 998 137 427 302 547 456;
  • 48) 0.120 117 187 499 999 999 999 999 999 999 999 998 137 427 302 547 456 × 2 = 0 + 0.240 234 374 999 999 999 999 999 999 999 999 996 274 854 605 094 912;
  • 49) 0.240 234 374 999 999 999 999 999 999 999 999 996 274 854 605 094 912 × 2 = 0 + 0.480 468 749 999 999 999 999 999 999 999 999 992 549 709 210 189 824;
  • 50) 0.480 468 749 999 999 999 999 999 999 999 999 992 549 709 210 189 824 × 2 = 0 + 0.960 937 499 999 999 999 999 999 999 999 999 985 099 418 420 379 648;
  • 51) 0.960 937 499 999 999 999 999 999 999 999 999 985 099 418 420 379 648 × 2 = 1 + 0.921 874 999 999 999 999 999 999 999 999 999 970 198 836 840 759 296;
  • 52) 0.921 874 999 999 999 999 999 999 999 999 999 970 198 836 840 759 296 × 2 = 1 + 0.843 749 999 999 999 999 999 999 999 999 999 940 397 673 681 518 592;
  • 53) 0.843 749 999 999 999 999 999 999 999 999 999 940 397 673 681 518 592 × 2 = 1 + 0.687 499 999 999 999 999 999 999 999 999 999 880 795 347 363 037 184;
  • 54) 0.687 499 999 999 999 999 999 999 999 999 999 880 795 347 363 037 184 × 2 = 1 + 0.374 999 999 999 999 999 999 999 999 999 999 761 590 694 726 074 368;
  • 55) 0.374 999 999 999 999 999 999 999 999 999 999 761 590 694 726 074 368 × 2 = 0 + 0.749 999 999 999 999 999 999 999 999 999 999 523 181 389 452 148 736;
  • 56) 0.749 999 999 999 999 999 999 999 999 999 999 523 181 389 452 148 736 × 2 = 1 + 0.499 999 999 999 999 999 999 999 999 999 999 046 362 778 904 297 472;
  • 57) 0.499 999 999 999 999 999 999 999 999 999 999 046 362 778 904 297 472 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 998 092 725 557 808 594 944;

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:


0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 751 068 102(10) =

0.0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0(2)

5. Positive number before normalization:

0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 751 068 102(10) =

0.0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 5 positions to the right, so that only one non zero digit remains to the left of it:


0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 751 068 102(10) =

0.0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0(2) =

0.0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0(2) × 20 =

1.0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010(2) × 2-5


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): -5

Mantissa (not normalized):
1.0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-5 + 2(11-1) - 1 =

(-5 + 1 023)(10) =

1 018(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;
  • 1 018 ÷ 2 = 509 + 0;
  • 509 ÷ 2 = 254 + 1;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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) =

1018(10) =

011 1111 1010(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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 =

0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010


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) =
011 1111 1010

Mantissa (52 bits) =
0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010


Decimal number 0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 751 068 102 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1010 - 0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010

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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