Convert 400 000 000 000 000 000 000 000 000 000 000 000 000 to 64 Bit Double Precision IEEE 754 Binary Floating Point Standard, From a Number in Base 10 Decimal System

How to convert the decimal number 400 000 000 000 000 000 000 000 000 000 000 000 000(10)
to
64 bit double precision IEEE 754 binary floating point
(1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. 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;
  • 400 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 200 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 200 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 100 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 100 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 50 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 50 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 25 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 25 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 12 500 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 12 500 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 6 250 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 6 250 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 3 125 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 3 125 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 1 562 500 000 000 000 000 000 000 000 000 000 000 + 0;
  • 1 562 500 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 781 250 000 000 000 000 000 000 000 000 000 000 + 0;
  • 781 250 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 390 625 000 000 000 000 000 000 000 000 000 000 + 0;
  • 390 625 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 195 312 500 000 000 000 000 000 000 000 000 000 + 0;
  • 195 312 500 000 000 000 000 000 000 000 000 000 ÷ 2 = 97 656 250 000 000 000 000 000 000 000 000 000 + 0;
  • 97 656 250 000 000 000 000 000 000 000 000 000 ÷ 2 = 48 828 125 000 000 000 000 000 000 000 000 000 + 0;
  • 48 828 125 000 000 000 000 000 000 000 000 000 ÷ 2 = 24 414 062 500 000 000 000 000 000 000 000 000 + 0;
  • 24 414 062 500 000 000 000 000 000 000 000 000 ÷ 2 = 12 207 031 250 000 000 000 000 000 000 000 000 + 0;
  • 12 207 031 250 000 000 000 000 000 000 000 000 ÷ 2 = 6 103 515 625 000 000 000 000 000 000 000 000 + 0;
  • 6 103 515 625 000 000 000 000 000 000 000 000 ÷ 2 = 3 051 757 812 500 000 000 000 000 000 000 000 + 0;
  • 3 051 757 812 500 000 000 000 000 000 000 000 ÷ 2 = 1 525 878 906 250 000 000 000 000 000 000 000 + 0;
  • 1 525 878 906 250 000 000 000 000 000 000 000 ÷ 2 = 762 939 453 125 000 000 000 000 000 000 000 + 0;
  • 762 939 453 125 000 000 000 000 000 000 000 ÷ 2 = 381 469 726 562 500 000 000 000 000 000 000 + 0;
  • 381 469 726 562 500 000 000 000 000 000 000 ÷ 2 = 190 734 863 281 250 000 000 000 000 000 000 + 0;
  • 190 734 863 281 250 000 000 000 000 000 000 ÷ 2 = 95 367 431 640 625 000 000 000 000 000 000 + 0;
  • 95 367 431 640 625 000 000 000 000 000 000 ÷ 2 = 47 683 715 820 312 500 000 000 000 000 000 + 0;
  • 47 683 715 820 312 500 000 000 000 000 000 ÷ 2 = 23 841 857 910 156 250 000 000 000 000 000 + 0;
  • 23 841 857 910 156 250 000 000 000 000 000 ÷ 2 = 11 920 928 955 078 125 000 000 000 000 000 + 0;
  • 11 920 928 955 078 125 000 000 000 000 000 ÷ 2 = 5 960 464 477 539 062 500 000 000 000 000 + 0;
  • 5 960 464 477 539 062 500 000 000 000 000 ÷ 2 = 2 980 232 238 769 531 250 000 000 000 000 + 0;
  • 2 980 232 238 769 531 250 000 000 000 000 ÷ 2 = 1 490 116 119 384 765 625 000 000 000 000 + 0;
  • 1 490 116 119 384 765 625 000 000 000 000 ÷ 2 = 745 058 059 692 382 812 500 000 000 000 + 0;
  • 745 058 059 692 382 812 500 000 000 000 ÷ 2 = 372 529 029 846 191 406 250 000 000 000 + 0;
  • 372 529 029 846 191 406 250 000 000 000 ÷ 2 = 186 264 514 923 095 703 125 000 000 000 + 0;
  • 186 264 514 923 095 703 125 000 000 000 ÷ 2 = 93 132 257 461 547 851 562 500 000 000 + 0;
  • 93 132 257 461 547 851 562 500 000 000 ÷ 2 = 46 566 128 730 773 925 781 250 000 000 + 0;
  • 46 566 128 730 773 925 781 250 000 000 ÷ 2 = 23 283 064 365 386 962 890 625 000 000 + 0;
  • 23 283 064 365 386 962 890 625 000 000 ÷ 2 = 11 641 532 182 693 481 445 312 500 000 + 0;
  • 11 641 532 182 693 481 445 312 500 000 ÷ 2 = 5 820 766 091 346 740 722 656 250 000 + 0;
  • 5 820 766 091 346 740 722 656 250 000 ÷ 2 = 2 910 383 045 673 370 361 328 125 000 + 0;
  • 2 910 383 045 673 370 361 328 125 000 ÷ 2 = 1 455 191 522 836 685 180 664 062 500 + 0;
  • 1 455 191 522 836 685 180 664 062 500 ÷ 2 = 727 595 761 418 342 590 332 031 250 + 0;
  • 727 595 761 418 342 590 332 031 250 ÷ 2 = 363 797 880 709 171 295 166 015 625 + 0;
  • 363 797 880 709 171 295 166 015 625 ÷ 2 = 181 898 940 354 585 647 583 007 812 + 1;
  • 181 898 940 354 585 647 583 007 812 ÷ 2 = 90 949 470 177 292 823 791 503 906 + 0;
  • 90 949 470 177 292 823 791 503 906 ÷ 2 = 45 474 735 088 646 411 895 751 953 + 0;
  • 45 474 735 088 646 411 895 751 953 ÷ 2 = 22 737 367 544 323 205 947 875 976 + 1;
  • 22 737 367 544 323 205 947 875 976 ÷ 2 = 11 368 683 772 161 602 973 937 988 + 0;
  • 11 368 683 772 161 602 973 937 988 ÷ 2 = 5 684 341 886 080 801 486 968 994 + 0;
  • 5 684 341 886 080 801 486 968 994 ÷ 2 = 2 842 170 943 040 400 743 484 497 + 0;
  • 2 842 170 943 040 400 743 484 497 ÷ 2 = 1 421 085 471 520 200 371 742 248 + 1;
  • 1 421 085 471 520 200 371 742 248 ÷ 2 = 710 542 735 760 100 185 871 124 + 0;
  • 710 542 735 760 100 185 871 124 ÷ 2 = 355 271 367 880 050 092 935 562 + 0;
  • 355 271 367 880 050 092 935 562 ÷ 2 = 177 635 683 940 025 046 467 781 + 0;
  • 177 635 683 940 025 046 467 781 ÷ 2 = 88 817 841 970 012 523 233 890 + 1;
  • 88 817 841 970 012 523 233 890 ÷ 2 = 44 408 920 985 006 261 616 945 + 0;
  • 44 408 920 985 006 261 616 945 ÷ 2 = 22 204 460 492 503 130 808 472 + 1;
  • 22 204 460 492 503 130 808 472 ÷ 2 = 11 102 230 246 251 565 404 236 + 0;
  • 11 102 230 246 251 565 404 236 ÷ 2 = 5 551 115 123 125 782 702 118 + 0;
  • 5 551 115 123 125 782 702 118 ÷ 2 = 2 775 557 561 562 891 351 059 + 0;
  • 2 775 557 561 562 891 351 059 ÷ 2 = 1 387 778 780 781 445 675 529 + 1;
  • 1 387 778 780 781 445 675 529 ÷ 2 = 693 889 390 390 722 837 764 + 1;
  • 693 889 390 390 722 837 764 ÷ 2 = 346 944 695 195 361 418 882 + 0;
  • 346 944 695 195 361 418 882 ÷ 2 = 173 472 347 597 680 709 441 + 0;
  • 173 472 347 597 680 709 441 ÷ 2 = 86 736 173 798 840 354 720 + 1;
  • 86 736 173 798 840 354 720 ÷ 2 = 43 368 086 899 420 177 360 + 0;
  • 43 368 086 899 420 177 360 ÷ 2 = 21 684 043 449 710 088 680 + 0;
  • 21 684 043 449 710 088 680 ÷ 2 = 10 842 021 724 855 044 340 + 0;
  • 10 842 021 724 855 044 340 ÷ 2 = 5 421 010 862 427 522 170 + 0;
  • 5 421 010 862 427 522 170 ÷ 2 = 2 710 505 431 213 761 085 + 0;
  • 2 710 505 431 213 761 085 ÷ 2 = 1 355 252 715 606 880 542 + 1;
  • 1 355 252 715 606 880 542 ÷ 2 = 677 626 357 803 440 271 + 0;
  • 677 626 357 803 440 271 ÷ 2 = 338 813 178 901 720 135 + 1;
  • 338 813 178 901 720 135 ÷ 2 = 169 406 589 450 860 067 + 1;
  • 169 406 589 450 860 067 ÷ 2 = 84 703 294 725 430 033 + 1;
  • 84 703 294 725 430 033 ÷ 2 = 42 351 647 362 715 016 + 1;
  • 42 351 647 362 715 016 ÷ 2 = 21 175 823 681 357 508 + 0;
  • 21 175 823 681 357 508 ÷ 2 = 10 587 911 840 678 754 + 0;
  • 10 587 911 840 678 754 ÷ 2 = 5 293 955 920 339 377 + 0;
  • 5 293 955 920 339 377 ÷ 2 = 2 646 977 960 169 688 + 1;
  • 2 646 977 960 169 688 ÷ 2 = 1 323 488 980 084 844 + 0;
  • 1 323 488 980 084 844 ÷ 2 = 661 744 490 042 422 + 0;
  • 661 744 490 042 422 ÷ 2 = 330 872 245 021 211 + 0;
  • 330 872 245 021 211 ÷ 2 = 165 436 122 510 605 + 1;
  • 165 436 122 510 605 ÷ 2 = 82 718 061 255 302 + 1;
  • 82 718 061 255 302 ÷ 2 = 41 359 030 627 651 + 0;
  • 41 359 030 627 651 ÷ 2 = 20 679 515 313 825 + 1;
  • 20 679 515 313 825 ÷ 2 = 10 339 757 656 912 + 1;
  • 10 339 757 656 912 ÷ 2 = 5 169 878 828 456 + 0;
  • 5 169 878 828 456 ÷ 2 = 2 584 939 414 228 + 0;
  • 2 584 939 414 228 ÷ 2 = 1 292 469 707 114 + 0;
  • 1 292 469 707 114 ÷ 2 = 646 234 853 557 + 0;
  • 646 234 853 557 ÷ 2 = 323 117 426 778 + 1;
  • 323 117 426 778 ÷ 2 = 161 558 713 389 + 0;
  • 161 558 713 389 ÷ 2 = 80 779 356 694 + 1;
  • 80 779 356 694 ÷ 2 = 40 389 678 347 + 0;
  • 40 389 678 347 ÷ 2 = 20 194 839 173 + 1;
  • 20 194 839 173 ÷ 2 = 10 097 419 586 + 1;
  • 10 097 419 586 ÷ 2 = 5 048 709 793 + 0;
  • 5 048 709 793 ÷ 2 = 2 524 354 896 + 1;
  • 2 524 354 896 ÷ 2 = 1 262 177 448 + 0;
  • 1 262 177 448 ÷ 2 = 631 088 724 + 0;
  • 631 088 724 ÷ 2 = 315 544 362 + 0;
  • 315 544 362 ÷ 2 = 157 772 181 + 0;
  • 157 772 181 ÷ 2 = 78 886 090 + 1;
  • 78 886 090 ÷ 2 = 39 443 045 + 0;
  • 39 443 045 ÷ 2 = 19 721 522 + 1;
  • 19 721 522 ÷ 2 = 9 860 761 + 0;
  • 9 860 761 ÷ 2 = 4 930 380 + 1;
  • 4 930 380 ÷ 2 = 2 465 190 + 0;
  • 2 465 190 ÷ 2 = 1 232 595 + 0;
  • 1 232 595 ÷ 2 = 616 297 + 1;
  • 616 297 ÷ 2 = 308 148 + 1;
  • 308 148 ÷ 2 = 154 074 + 0;
  • 154 074 ÷ 2 = 77 037 + 0;
  • 77 037 ÷ 2 = 38 518 + 1;
  • 38 518 ÷ 2 = 19 259 + 0;
  • 19 259 ÷ 2 = 9 629 + 1;
  • 9 629 ÷ 2 = 4 814 + 1;
  • 4 814 ÷ 2 = 2 407 + 0;
  • 2 407 ÷ 2 = 1 203 + 1;
  • 1 203 ÷ 2 = 601 + 1;
  • 601 ÷ 2 = 300 + 1;
  • 300 ÷ 2 = 150 + 0;
  • 150 ÷ 2 = 75 + 0;
  • 75 ÷ 2 = 37 + 1;
  • 37 ÷ 2 = 18 + 1;
  • 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 positive number.

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

400 000 000 000 000 000 000 000 000 000 000 000 000(10) =


1 0010 1100 1110 1101 0011 0010 1010 0001 0110 1010 0001 1011 0001 0001 1110 1000 0010 0110 0010 1000 1000 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 128 positions to the left so that only one non zero digit remains to the left of it:

400 000 000 000 000 000 000 000 000 000 000 000 000(10) =


1 0010 1100 1110 1101 0011 0010 1010 0001 0110 1010 0001 1011 0001 0001 1110 1000 0010 0110 0010 1000 1000 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) =


1 0010 1100 1110 1101 0011 0010 1010 0001 0110 1010 0001 1011 0001 0001 1110 1000 0010 0110 0010 1000 1000 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) × 20 =


1.0010 1100 1110 1101 0011 0010 1010 0001 0110 1010 0001 1011 0001 0001 1110 1000 0010 0110 0010 1000 1000 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) × 2128


4. 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): 128


Mantissa (not normalized):
1.0010 1100 1110 1101 0011 0010 1010 0001 0110 1010 0001 1011 0001 0001 1110 1000 0010 0110 0010 1000 1000 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000


5. Adjust the exponent.

Use the 11 bit excess/bias notation:

Exponent (adjusted) =


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


128 + 2(11-1) - 1 =


(128 + 1 023)(10) =


1 151(10)


6. 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;
  • 1 151 ÷ 2 = 575 + 1;
  • 575 ÷ 2 = 287 + 1;
  • 287 ÷ 2 = 143 + 1;
  • 143 ÷ 2 = 71 + 1;
  • 71 ÷ 2 = 35 + 1;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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

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

Exponent (adjusted) =


1151(10) =


100 0111 1111(2)


8. 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 1100 1110 1101 0011 0010 1010 0001 0110 1010 0001 1011 0001 0001 1110 1000 0010 0110 0010 1000 1000 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 =


0010 1100 1110 1101 0011 0010 1010 0001 0110 1010 0001 1011 0001


9. 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 0111 1111


Mantissa (52 bits) =
0010 1100 1110 1101 0011 0010 1010 0001 0110 1010 0001 1011 0001


Conclusion:

Number 400 000 000 000 000 000 000 000 000 000 000 000 000 converted from decimal system (base 10)
to
64 bit double precision IEEE 754 binary floating point:
0 - 100 0111 1111 - 0010 1100 1110 1101 0011 0010 1010 0001 0110 1010 0001 1011 0001

(64 bits IEEE 754)
  • Sign (1 bit):

    • 0

      63
  • Exponent (11 bits):

    • 1

      62
    • 0

      61
    • 0

      60
    • 0

      59
    • 1

      58
    • 1

      57
    • 1

      56
    • 1

      55
    • 1

      54
    • 1

      53
    • 1

      52
  • Mantissa (52 bits):

    • 0

      51
    • 0

      50
    • 1

      49
    • 0

      48
    • 1

      47
    • 1

      46
    • 0

      45
    • 0

      44
    • 1

      43
    • 1

      42
    • 1

      41
    • 0

      40
    • 1

      39
    • 1

      38
    • 0

      37
    • 1

      36
    • 0

      35
    • 0

      34
    • 1

      33
    • 1

      32
    • 0

      31
    • 0

      30
    • 1

      29
    • 0

      28
    • 1

      27
    • 0

      26
    • 1

      25
    • 0

      24
    • 0

      23
    • 0

      22
    • 0

      21
    • 1

      20
    • 0

      19
    • 1

      18
    • 1

      17
    • 0

      16
    • 1

      15
    • 0

      14
    • 1

      13
    • 0

      12
    • 0

      11
    • 0

      10
    • 0

      9
    • 1

      8
    • 1

      7
    • 0

      6
    • 1

      5
    • 1

      4
    • 0

      3
    • 0

      2
    • 0

      1
    • 1

      0

More operations of this kind:

399 999 999 999 999 999 999 999 999 999 999 999 999 = ? ... 400 000 000 000 000 000 000 000 000 000 000 000 001 = ?


Convert to 64 bit double precision IEEE 754 binary floating point standard

A number in 64 bit double precision IEEE 754 binary floating point standard representation requires three building elements: sign (it takes one bit and it's either 0 for positive or 1 for negative numbers), exponent (11 bits), mantissa (52 bits)

Latest decimal numbers converted from base ten to 64 bit double precision IEEE 754 floating point binary standard representation

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