-2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16(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
-2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16(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:

|-2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16| = 2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16


2. First, convert to binary (in base 2) the integer part: 2.
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;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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.

2(10) =

10(2)


4. Convert to binary (base 2) the fractional part: 0.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16.

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) 0.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16 × 2 = 0 + 0.423 658 104 766 716 600 239 097 323 399 306 560 531 840 659 677 982 330 320 32;
  • 2) 0.423 658 104 766 716 600 239 097 323 399 306 560 531 840 659 677 982 330 320 32 × 2 = 0 + 0.847 316 209 533 433 200 478 194 646 798 613 121 063 681 319 355 964 660 640 64;
  • 3) 0.847 316 209 533 433 200 478 194 646 798 613 121 063 681 319 355 964 660 640 64 × 2 = 1 + 0.694 632 419 066 866 400 956 389 293 597 226 242 127 362 638 711 929 321 281 28;
  • 4) 0.694 632 419 066 866 400 956 389 293 597 226 242 127 362 638 711 929 321 281 28 × 2 = 1 + 0.389 264 838 133 732 801 912 778 587 194 452 484 254 725 277 423 858 642 562 56;
  • 5) 0.389 264 838 133 732 801 912 778 587 194 452 484 254 725 277 423 858 642 562 56 × 2 = 0 + 0.778 529 676 267 465 603 825 557 174 388 904 968 509 450 554 847 717 285 125 12;
  • 6) 0.778 529 676 267 465 603 825 557 174 388 904 968 509 450 554 847 717 285 125 12 × 2 = 1 + 0.557 059 352 534 931 207 651 114 348 777 809 937 018 901 109 695 434 570 250 24;
  • 7) 0.557 059 352 534 931 207 651 114 348 777 809 937 018 901 109 695 434 570 250 24 × 2 = 1 + 0.114 118 705 069 862 415 302 228 697 555 619 874 037 802 219 390 869 140 500 48;
  • 8) 0.114 118 705 069 862 415 302 228 697 555 619 874 037 802 219 390 869 140 500 48 × 2 = 0 + 0.228 237 410 139 724 830 604 457 395 111 239 748 075 604 438 781 738 281 000 96;
  • 9) 0.228 237 410 139 724 830 604 457 395 111 239 748 075 604 438 781 738 281 000 96 × 2 = 0 + 0.456 474 820 279 449 661 208 914 790 222 479 496 151 208 877 563 476 562 001 92;
  • 10) 0.456 474 820 279 449 661 208 914 790 222 479 496 151 208 877 563 476 562 001 92 × 2 = 0 + 0.912 949 640 558 899 322 417 829 580 444 958 992 302 417 755 126 953 124 003 84;
  • 11) 0.912 949 640 558 899 322 417 829 580 444 958 992 302 417 755 126 953 124 003 84 × 2 = 1 + 0.825 899 281 117 798 644 835 659 160 889 917 984 604 835 510 253 906 248 007 68;
  • 12) 0.825 899 281 117 798 644 835 659 160 889 917 984 604 835 510 253 906 248 007 68 × 2 = 1 + 0.651 798 562 235 597 289 671 318 321 779 835 969 209 671 020 507 812 496 015 36;
  • 13) 0.651 798 562 235 597 289 671 318 321 779 835 969 209 671 020 507 812 496 015 36 × 2 = 1 + 0.303 597 124 471 194 579 342 636 643 559 671 938 419 342 041 015 624 992 030 72;
  • 14) 0.303 597 124 471 194 579 342 636 643 559 671 938 419 342 041 015 624 992 030 72 × 2 = 0 + 0.607 194 248 942 389 158 685 273 287 119 343 876 838 684 082 031 249 984 061 44;
  • 15) 0.607 194 248 942 389 158 685 273 287 119 343 876 838 684 082 031 249 984 061 44 × 2 = 1 + 0.214 388 497 884 778 317 370 546 574 238 687 753 677 368 164 062 499 968 122 88;
  • 16) 0.214 388 497 884 778 317 370 546 574 238 687 753 677 368 164 062 499 968 122 88 × 2 = 0 + 0.428 776 995 769 556 634 741 093 148 477 375 507 354 736 328 124 999 936 245 76;
  • 17) 0.428 776 995 769 556 634 741 093 148 477 375 507 354 736 328 124 999 936 245 76 × 2 = 0 + 0.857 553 991 539 113 269 482 186 296 954 751 014 709 472 656 249 999 872 491 52;
  • 18) 0.857 553 991 539 113 269 482 186 296 954 751 014 709 472 656 249 999 872 491 52 × 2 = 1 + 0.715 107 983 078 226 538 964 372 593 909 502 029 418 945 312 499 999 744 983 04;
  • 19) 0.715 107 983 078 226 538 964 372 593 909 502 029 418 945 312 499 999 744 983 04 × 2 = 1 + 0.430 215 966 156 453 077 928 745 187 819 004 058 837 890 624 999 999 489 966 08;
  • 20) 0.430 215 966 156 453 077 928 745 187 819 004 058 837 890 624 999 999 489 966 08 × 2 = 0 + 0.860 431 932 312 906 155 857 490 375 638 008 117 675 781 249 999 998 979 932 16;
  • 21) 0.860 431 932 312 906 155 857 490 375 638 008 117 675 781 249 999 998 979 932 16 × 2 = 1 + 0.720 863 864 625 812 311 714 980 751 276 016 235 351 562 499 999 997 959 864 32;
  • 22) 0.720 863 864 625 812 311 714 980 751 276 016 235 351 562 499 999 997 959 864 32 × 2 = 1 + 0.441 727 729 251 624 623 429 961 502 552 032 470 703 124 999 999 995 919 728 64;
  • 23) 0.441 727 729 251 624 623 429 961 502 552 032 470 703 124 999 999 995 919 728 64 × 2 = 0 + 0.883 455 458 503 249 246 859 923 005 104 064 941 406 249 999 999 991 839 457 28;
  • 24) 0.883 455 458 503 249 246 859 923 005 104 064 941 406 249 999 999 991 839 457 28 × 2 = 1 + 0.766 910 917 006 498 493 719 846 010 208 129 882 812 499 999 999 983 678 914 56;
  • 25) 0.766 910 917 006 498 493 719 846 010 208 129 882 812 499 999 999 983 678 914 56 × 2 = 1 + 0.533 821 834 012 996 987 439 692 020 416 259 765 624 999 999 999 967 357 829 12;
  • 26) 0.533 821 834 012 996 987 439 692 020 416 259 765 624 999 999 999 967 357 829 12 × 2 = 1 + 0.067 643 668 025 993 974 879 384 040 832 519 531 249 999 999 999 934 715 658 24;
  • 27) 0.067 643 668 025 993 974 879 384 040 832 519 531 249 999 999 999 934 715 658 24 × 2 = 0 + 0.135 287 336 051 987 949 758 768 081 665 039 062 499 999 999 999 869 431 316 48;
  • 28) 0.135 287 336 051 987 949 758 768 081 665 039 062 499 999 999 999 869 431 316 48 × 2 = 0 + 0.270 574 672 103 975 899 517 536 163 330 078 124 999 999 999 999 738 862 632 96;
  • 29) 0.270 574 672 103 975 899 517 536 163 330 078 124 999 999 999 999 738 862 632 96 × 2 = 0 + 0.541 149 344 207 951 799 035 072 326 660 156 249 999 999 999 999 477 725 265 92;
  • 30) 0.541 149 344 207 951 799 035 072 326 660 156 249 999 999 999 999 477 725 265 92 × 2 = 1 + 0.082 298 688 415 903 598 070 144 653 320 312 499 999 999 999 998 955 450 531 84;
  • 31) 0.082 298 688 415 903 598 070 144 653 320 312 499 999 999 999 998 955 450 531 84 × 2 = 0 + 0.164 597 376 831 807 196 140 289 306 640 624 999 999 999 999 997 910 901 063 68;
  • 32) 0.164 597 376 831 807 196 140 289 306 640 624 999 999 999 999 997 910 901 063 68 × 2 = 0 + 0.329 194 753 663 614 392 280 578 613 281 249 999 999 999 999 995 821 802 127 36;
  • 33) 0.329 194 753 663 614 392 280 578 613 281 249 999 999 999 999 995 821 802 127 36 × 2 = 0 + 0.658 389 507 327 228 784 561 157 226 562 499 999 999 999 999 991 643 604 254 72;
  • 34) 0.658 389 507 327 228 784 561 157 226 562 499 999 999 999 999 991 643 604 254 72 × 2 = 1 + 0.316 779 014 654 457 569 122 314 453 124 999 999 999 999 999 983 287 208 509 44;
  • 35) 0.316 779 014 654 457 569 122 314 453 124 999 999 999 999 999 983 287 208 509 44 × 2 = 0 + 0.633 558 029 308 915 138 244 628 906 249 999 999 999 999 999 966 574 417 018 88;
  • 36) 0.633 558 029 308 915 138 244 628 906 249 999 999 999 999 999 966 574 417 018 88 × 2 = 1 + 0.267 116 058 617 830 276 489 257 812 499 999 999 999 999 999 933 148 834 037 76;
  • 37) 0.267 116 058 617 830 276 489 257 812 499 999 999 999 999 999 933 148 834 037 76 × 2 = 0 + 0.534 232 117 235 660 552 978 515 624 999 999 999 999 999 999 866 297 668 075 52;
  • 38) 0.534 232 117 235 660 552 978 515 624 999 999 999 999 999 999 866 297 668 075 52 × 2 = 1 + 0.068 464 234 471 321 105 957 031 249 999 999 999 999 999 999 732 595 336 151 04;
  • 39) 0.068 464 234 471 321 105 957 031 249 999 999 999 999 999 999 732 595 336 151 04 × 2 = 0 + 0.136 928 468 942 642 211 914 062 499 999 999 999 999 999 999 465 190 672 302 08;
  • 40) 0.136 928 468 942 642 211 914 062 499 999 999 999 999 999 999 465 190 672 302 08 × 2 = 0 + 0.273 856 937 885 284 423 828 124 999 999 999 999 999 999 998 930 381 344 604 16;
  • 41) 0.273 856 937 885 284 423 828 124 999 999 999 999 999 999 998 930 381 344 604 16 × 2 = 0 + 0.547 713 875 770 568 847 656 249 999 999 999 999 999 999 997 860 762 689 208 32;
  • 42) 0.547 713 875 770 568 847 656 249 999 999 999 999 999 999 997 860 762 689 208 32 × 2 = 1 + 0.095 427 751 541 137 695 312 499 999 999 999 999 999 999 995 721 525 378 416 64;
  • 43) 0.095 427 751 541 137 695 312 499 999 999 999 999 999 999 995 721 525 378 416 64 × 2 = 0 + 0.190 855 503 082 275 390 624 999 999 999 999 999 999 999 991 443 050 756 833 28;
  • 44) 0.190 855 503 082 275 390 624 999 999 999 999 999 999 999 991 443 050 756 833 28 × 2 = 0 + 0.381 711 006 164 550 781 249 999 999 999 999 999 999 999 982 886 101 513 666 56;
  • 45) 0.381 711 006 164 550 781 249 999 999 999 999 999 999 999 982 886 101 513 666 56 × 2 = 0 + 0.763 422 012 329 101 562 499 999 999 999 999 999 999 999 965 772 203 027 333 12;
  • 46) 0.763 422 012 329 101 562 499 999 999 999 999 999 999 999 965 772 203 027 333 12 × 2 = 1 + 0.526 844 024 658 203 124 999 999 999 999 999 999 999 999 931 544 406 054 666 24;
  • 47) 0.526 844 024 658 203 124 999 999 999 999 999 999 999 999 931 544 406 054 666 24 × 2 = 1 + 0.053 688 049 316 406 249 999 999 999 999 999 999 999 999 863 088 812 109 332 48;
  • 48) 0.053 688 049 316 406 249 999 999 999 999 999 999 999 999 863 088 812 109 332 48 × 2 = 0 + 0.107 376 098 632 812 499 999 999 999 999 999 999 999 999 726 177 624 218 664 96;
  • 49) 0.107 376 098 632 812 499 999 999 999 999 999 999 999 999 726 177 624 218 664 96 × 2 = 0 + 0.214 752 197 265 624 999 999 999 999 999 999 999 999 999 452 355 248 437 329 92;
  • 50) 0.214 752 197 265 624 999 999 999 999 999 999 999 999 999 452 355 248 437 329 92 × 2 = 0 + 0.429 504 394 531 249 999 999 999 999 999 999 999 999 998 904 710 496 874 659 84;
  • 51) 0.429 504 394 531 249 999 999 999 999 999 999 999 999 998 904 710 496 874 659 84 × 2 = 0 + 0.859 008 789 062 499 999 999 999 999 999 999 999 999 997 809 420 993 749 319 68;
  • 52) 0.859 008 789 062 499 999 999 999 999 999 999 999 999 997 809 420 993 749 319 68 × 2 = 1 + 0.718 017 578 124 999 999 999 999 999 999 999 999 999 995 618 841 987 498 639 36;
  • 53) 0.718 017 578 124 999 999 999 999 999 999 999 999 999 995 618 841 987 498 639 36 × 2 = 1 + 0.436 035 156 249 999 999 999 999 999 999 999 999 999 991 237 683 974 997 278 72;

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:


0.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16(10) =

0.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2)

6. Positive number before normalization:

2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16(10) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the left, so that only one non zero digit remains to the left of it:


2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16(10) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2) =

10.0011 0110 0011 1010 0110 1101 1100 0100 0101 0100 0100 0110 0001 1(2) × 20 =

1.0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11(2) × 21


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): 1

Mantissa (not normalized):
1.0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

1 + 2(11-1) - 1 =

(1 + 1 023)(10) =

1 024(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;
  • 1 024 ÷ 2 = 512 + 0;
  • 512 ÷ 2 = 256 + 0;
  • 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;

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) =

1024(10) =

100 0000 0000(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, 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. 0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000 11 =

0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000


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) =
100 0000 0000

Mantissa (52 bits) =
0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000


Decimal number -2.211 829 052 383 358 300 119 548 661 699 653 280 265 920 329 838 991 165 160 16 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 0000 - 0001 1011 0001 1101 0011 0110 1110 0010 0010 1010 0010 0011 0000

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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