0.010 000 000 000 000 000 208 166 817 117 216 851 329 430 791 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.010 000 000 000 000 000 208 166 817 117 216 851 329 430 791(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
0.010 000 000 000 000 000 208 166 817 117 216 851 329 430 791(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: 0.010 000 000 000 000 000 208 166 817 117 216 851 329 430 791.

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) 0.010 000 000 000 000 000 208 166 817 117 216 851 329 430 791 × 2 = 0 + 0.020 000 000 000 000 000 416 333 634 234 433 702 658 861 582;
  • 2) 0.020 000 000 000 000 000 416 333 634 234 433 702 658 861 582 × 2 = 0 + 0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 164;
  • 3) 0.040 000 000 000 000 000 832 667 268 468 867 405 317 723 164 × 2 = 0 + 0.080 000 000 000 000 001 665 334 536 937 734 810 635 446 328;
  • 4) 0.080 000 000 000 000 001 665 334 536 937 734 810 635 446 328 × 2 = 0 + 0.160 000 000 000 000 003 330 669 073 875 469 621 270 892 656;
  • 5) 0.160 000 000 000 000 003 330 669 073 875 469 621 270 892 656 × 2 = 0 + 0.320 000 000 000 000 006 661 338 147 750 939 242 541 785 312;
  • 6) 0.320 000 000 000 000 006 661 338 147 750 939 242 541 785 312 × 2 = 0 + 0.640 000 000 000 000 013 322 676 295 501 878 485 083 570 624;
  • 7) 0.640 000 000 000 000 013 322 676 295 501 878 485 083 570 624 × 2 = 1 + 0.280 000 000 000 000 026 645 352 591 003 756 970 167 141 248;
  • 8) 0.280 000 000 000 000 026 645 352 591 003 756 970 167 141 248 × 2 = 0 + 0.560 000 000 000 000 053 290 705 182 007 513 940 334 282 496;
  • 9) 0.560 000 000 000 000 053 290 705 182 007 513 940 334 282 496 × 2 = 1 + 0.120 000 000 000 000 106 581 410 364 015 027 880 668 564 992;
  • 10) 0.120 000 000 000 000 106 581 410 364 015 027 880 668 564 992 × 2 = 0 + 0.240 000 000 000 000 213 162 820 728 030 055 761 337 129 984;
  • 11) 0.240 000 000 000 000 213 162 820 728 030 055 761 337 129 984 × 2 = 0 + 0.480 000 000 000 000 426 325 641 456 060 111 522 674 259 968;
  • 12) 0.480 000 000 000 000 426 325 641 456 060 111 522 674 259 968 × 2 = 0 + 0.960 000 000 000 000 852 651 282 912 120 223 045 348 519 936;
  • 13) 0.960 000 000 000 000 852 651 282 912 120 223 045 348 519 936 × 2 = 1 + 0.920 000 000 000 001 705 302 565 824 240 446 090 697 039 872;
  • 14) 0.920 000 000 000 001 705 302 565 824 240 446 090 697 039 872 × 2 = 1 + 0.840 000 000 000 003 410 605 131 648 480 892 181 394 079 744;
  • 15) 0.840 000 000 000 003 410 605 131 648 480 892 181 394 079 744 × 2 = 1 + 0.680 000 000 000 006 821 210 263 296 961 784 362 788 159 488;
  • 16) 0.680 000 000 000 006 821 210 263 296 961 784 362 788 159 488 × 2 = 1 + 0.360 000 000 000 013 642 420 526 593 923 568 725 576 318 976;
  • 17) 0.360 000 000 000 013 642 420 526 593 923 568 725 576 318 976 × 2 = 0 + 0.720 000 000 000 027 284 841 053 187 847 137 451 152 637 952;
  • 18) 0.720 000 000 000 027 284 841 053 187 847 137 451 152 637 952 × 2 = 1 + 0.440 000 000 000 054 569 682 106 375 694 274 902 305 275 904;
  • 19) 0.440 000 000 000 054 569 682 106 375 694 274 902 305 275 904 × 2 = 0 + 0.880 000 000 000 109 139 364 212 751 388 549 804 610 551 808;
  • 20) 0.880 000 000 000 109 139 364 212 751 388 549 804 610 551 808 × 2 = 1 + 0.760 000 000 000 218 278 728 425 502 777 099 609 221 103 616;
  • 21) 0.760 000 000 000 218 278 728 425 502 777 099 609 221 103 616 × 2 = 1 + 0.520 000 000 000 436 557 456 851 005 554 199 218 442 207 232;
  • 22) 0.520 000 000 000 436 557 456 851 005 554 199 218 442 207 232 × 2 = 1 + 0.040 000 000 000 873 114 913 702 011 108 398 436 884 414 464;
  • 23) 0.040 000 000 000 873 114 913 702 011 108 398 436 884 414 464 × 2 = 0 + 0.080 000 000 001 746 229 827 404 022 216 796 873 768 828 928;
  • 24) 0.080 000 000 001 746 229 827 404 022 216 796 873 768 828 928 × 2 = 0 + 0.160 000 000 003 492 459 654 808 044 433 593 747 537 657 856;
  • 25) 0.160 000 000 003 492 459 654 808 044 433 593 747 537 657 856 × 2 = 0 + 0.320 000 000 006 984 919 309 616 088 867 187 495 075 315 712;
  • 26) 0.320 000 000 006 984 919 309 616 088 867 187 495 075 315 712 × 2 = 0 + 0.640 000 000 013 969 838 619 232 177 734 374 990 150 631 424;
  • 27) 0.640 000 000 013 969 838 619 232 177 734 374 990 150 631 424 × 2 = 1 + 0.280 000 000 027 939 677 238 464 355 468 749 980 301 262 848;
  • 28) 0.280 000 000 027 939 677 238 464 355 468 749 980 301 262 848 × 2 = 0 + 0.560 000 000 055 879 354 476 928 710 937 499 960 602 525 696;
  • 29) 0.560 000 000 055 879 354 476 928 710 937 499 960 602 525 696 × 2 = 1 + 0.120 000 000 111 758 708 953 857 421 874 999 921 205 051 392;
  • 30) 0.120 000 000 111 758 708 953 857 421 874 999 921 205 051 392 × 2 = 0 + 0.240 000 000 223 517 417 907 714 843 749 999 842 410 102 784;
  • 31) 0.240 000 000 223 517 417 907 714 843 749 999 842 410 102 784 × 2 = 0 + 0.480 000 000 447 034 835 815 429 687 499 999 684 820 205 568;
  • 32) 0.480 000 000 447 034 835 815 429 687 499 999 684 820 205 568 × 2 = 0 + 0.960 000 000 894 069 671 630 859 374 999 999 369 640 411 136;
  • 33) 0.960 000 000 894 069 671 630 859 374 999 999 369 640 411 136 × 2 = 1 + 0.920 000 001 788 139 343 261 718 749 999 998 739 280 822 272;
  • 34) 0.920 000 001 788 139 343 261 718 749 999 998 739 280 822 272 × 2 = 1 + 0.840 000 003 576 278 686 523 437 499 999 997 478 561 644 544;
  • 35) 0.840 000 003 576 278 686 523 437 499 999 997 478 561 644 544 × 2 = 1 + 0.680 000 007 152 557 373 046 874 999 999 994 957 123 289 088;
  • 36) 0.680 000 007 152 557 373 046 874 999 999 994 957 123 289 088 × 2 = 1 + 0.360 000 014 305 114 746 093 749 999 999 989 914 246 578 176;
  • 37) 0.360 000 014 305 114 746 093 749 999 999 989 914 246 578 176 × 2 = 0 + 0.720 000 028 610 229 492 187 499 999 999 979 828 493 156 352;
  • 38) 0.720 000 028 610 229 492 187 499 999 999 979 828 493 156 352 × 2 = 1 + 0.440 000 057 220 458 984 374 999 999 999 959 656 986 312 704;
  • 39) 0.440 000 057 220 458 984 374 999 999 999 959 656 986 312 704 × 2 = 0 + 0.880 000 114 440 917 968 749 999 999 999 919 313 972 625 408;
  • 40) 0.880 000 114 440 917 968 749 999 999 999 919 313 972 625 408 × 2 = 1 + 0.760 000 228 881 835 937 499 999 999 999 838 627 945 250 816;
  • 41) 0.760 000 228 881 835 937 499 999 999 999 838 627 945 250 816 × 2 = 1 + 0.520 000 457 763 671 874 999 999 999 999 677 255 890 501 632;
  • 42) 0.520 000 457 763 671 874 999 999 999 999 677 255 890 501 632 × 2 = 1 + 0.040 000 915 527 343 749 999 999 999 999 354 511 781 003 264;
  • 43) 0.040 000 915 527 343 749 999 999 999 999 354 511 781 003 264 × 2 = 0 + 0.080 001 831 054 687 499 999 999 999 998 709 023 562 006 528;
  • 44) 0.080 001 831 054 687 499 999 999 999 998 709 023 562 006 528 × 2 = 0 + 0.160 003 662 109 374 999 999 999 999 997 418 047 124 013 056;
  • 45) 0.160 003 662 109 374 999 999 999 999 997 418 047 124 013 056 × 2 = 0 + 0.320 007 324 218 749 999 999 999 999 994 836 094 248 026 112;
  • 46) 0.320 007 324 218 749 999 999 999 999 994 836 094 248 026 112 × 2 = 0 + 0.640 014 648 437 499 999 999 999 999 989 672 188 496 052 224;
  • 47) 0.640 014 648 437 499 999 999 999 999 989 672 188 496 052 224 × 2 = 1 + 0.280 029 296 874 999 999 999 999 999 979 344 376 992 104 448;
  • 48) 0.280 029 296 874 999 999 999 999 999 979 344 376 992 104 448 × 2 = 0 + 0.560 058 593 749 999 999 999 999 999 958 688 753 984 208 896;
  • 49) 0.560 058 593 749 999 999 999 999 999 958 688 753 984 208 896 × 2 = 1 + 0.120 117 187 499 999 999 999 999 999 917 377 507 968 417 792;
  • 50) 0.120 117 187 499 999 999 999 999 999 917 377 507 968 417 792 × 2 = 0 + 0.240 234 374 999 999 999 999 999 999 834 755 015 936 835 584;
  • 51) 0.240 234 374 999 999 999 999 999 999 834 755 015 936 835 584 × 2 = 0 + 0.480 468 749 999 999 999 999 999 999 669 510 031 873 671 168;
  • 52) 0.480 468 749 999 999 999 999 999 999 669 510 031 873 671 168 × 2 = 0 + 0.960 937 499 999 999 999 999 999 999 339 020 063 747 342 336;
  • 53) 0.960 937 499 999 999 999 999 999 999 339 020 063 747 342 336 × 2 = 1 + 0.921 874 999 999 999 999 999 999 998 678 040 127 494 684 672;
  • 54) 0.921 874 999 999 999 999 999 999 998 678 040 127 494 684 672 × 2 = 1 + 0.843 749 999 999 999 999 999 999 997 356 080 254 989 369 344;
  • 55) 0.843 749 999 999 999 999 999 999 997 356 080 254 989 369 344 × 2 = 1 + 0.687 499 999 999 999 999 999 999 994 712 160 509 978 738 688;
  • 56) 0.687 499 999 999 999 999 999 999 994 712 160 509 978 738 688 × 2 = 1 + 0.374 999 999 999 999 999 999 999 989 424 321 019 957 477 376;
  • 57) 0.374 999 999 999 999 999 999 999 989 424 321 019 957 477 376 × 2 = 0 + 0.749 999 999 999 999 999 999 999 978 848 642 039 914 954 752;
  • 58) 0.749 999 999 999 999 999 999 999 978 848 642 039 914 954 752 × 2 = 1 + 0.499 999 999 999 999 999 999 999 957 697 284 079 829 909 504;
  • 59) 0.499 999 999 999 999 999 999 999 957 697 284 079 829 909 504 × 2 = 0 + 0.999 999 999 999 999 999 999 999 915 394 568 159 659 819 008;

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:


0.010 000 000 000 000 000 208 166 817 117 216 851 329 430 791(10) =

0.0000 0010 1000 1111 0101 1100 0010 1000 1111 0101 1100 0010 1000 1111 010(2)

5. Positive number before normalization:

0.010 000 000 000 000 000 208 166 817 117 216 851 329 430 791(10) =

0.0000 0010 1000 1111 0101 1100 0010 1000 1111 0101 1100 0010 1000 1111 010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 7 positions to the right, so that only one non zero digit remains to the left of it:


0.010 000 000 000 000 000 208 166 817 117 216 851 329 430 791(10) =

0.0000 0010 1000 1111 0101 1100 0010 1000 1111 0101 1100 0010 1000 1111 010(2) =

0.0000 0010 1000 1111 0101 1100 0010 1000 1111 0101 1100 0010 1000 1111 010(2) × 20 =

1.0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010(2) × 2-7


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): -7

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 =

-7 + 2(11-1) - 1 =

(-7 + 1 023)(10) =

1 016(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;
  • 1 016 ÷ 2 = 508 + 0;
  • 508 ÷ 2 = 254 + 0;
  • 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) =

1016(10) =

011 1111 1000(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 1000

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


Decimal number 0.010 000 000 000 000 000 208 166 817 117 216 851 329 430 791 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1000 - 0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010

» Convert decimal 0.010 000 000 000 000 000 208 166 817 117 216 851 329 430 796 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