-0.000 000 000 000 000 004 573 04 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 004 573 04(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
-0.000 000 000 000 000 004 573 04(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:

|-0.000 000 000 000 000 004 573 04| = 0.000 000 000 000 000 004 573 04


2. 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;

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.

0(10) =

0(2)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 004 573 04.

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) 0.000 000 000 000 000 004 573 04 × 2 = 0 + 0.000 000 000 000 000 009 146 08;
  • 2) 0.000 000 000 000 000 009 146 08 × 2 = 0 + 0.000 000 000 000 000 018 292 16;
  • 3) 0.000 000 000 000 000 018 292 16 × 2 = 0 + 0.000 000 000 000 000 036 584 32;
  • 4) 0.000 000 000 000 000 036 584 32 × 2 = 0 + 0.000 000 000 000 000 073 168 64;
  • 5) 0.000 000 000 000 000 073 168 64 × 2 = 0 + 0.000 000 000 000 000 146 337 28;
  • 6) 0.000 000 000 000 000 146 337 28 × 2 = 0 + 0.000 000 000 000 000 292 674 56;
  • 7) 0.000 000 000 000 000 292 674 56 × 2 = 0 + 0.000 000 000 000 000 585 349 12;
  • 8) 0.000 000 000 000 000 585 349 12 × 2 = 0 + 0.000 000 000 000 001 170 698 24;
  • 9) 0.000 000 000 000 001 170 698 24 × 2 = 0 + 0.000 000 000 000 002 341 396 48;
  • 10) 0.000 000 000 000 002 341 396 48 × 2 = 0 + 0.000 000 000 000 004 682 792 96;
  • 11) 0.000 000 000 000 004 682 792 96 × 2 = 0 + 0.000 000 000 000 009 365 585 92;
  • 12) 0.000 000 000 000 009 365 585 92 × 2 = 0 + 0.000 000 000 000 018 731 171 84;
  • 13) 0.000 000 000 000 018 731 171 84 × 2 = 0 + 0.000 000 000 000 037 462 343 68;
  • 14) 0.000 000 000 000 037 462 343 68 × 2 = 0 + 0.000 000 000 000 074 924 687 36;
  • 15) 0.000 000 000 000 074 924 687 36 × 2 = 0 + 0.000 000 000 000 149 849 374 72;
  • 16) 0.000 000 000 000 149 849 374 72 × 2 = 0 + 0.000 000 000 000 299 698 749 44;
  • 17) 0.000 000 000 000 299 698 749 44 × 2 = 0 + 0.000 000 000 000 599 397 498 88;
  • 18) 0.000 000 000 000 599 397 498 88 × 2 = 0 + 0.000 000 000 001 198 794 997 76;
  • 19) 0.000 000 000 001 198 794 997 76 × 2 = 0 + 0.000 000 000 002 397 589 995 52;
  • 20) 0.000 000 000 002 397 589 995 52 × 2 = 0 + 0.000 000 000 004 795 179 991 04;
  • 21) 0.000 000 000 004 795 179 991 04 × 2 = 0 + 0.000 000 000 009 590 359 982 08;
  • 22) 0.000 000 000 009 590 359 982 08 × 2 = 0 + 0.000 000 000 019 180 719 964 16;
  • 23) 0.000 000 000 019 180 719 964 16 × 2 = 0 + 0.000 000 000 038 361 439 928 32;
  • 24) 0.000 000 000 038 361 439 928 32 × 2 = 0 + 0.000 000 000 076 722 879 856 64;
  • 25) 0.000 000 000 076 722 879 856 64 × 2 = 0 + 0.000 000 000 153 445 759 713 28;
  • 26) 0.000 000 000 153 445 759 713 28 × 2 = 0 + 0.000 000 000 306 891 519 426 56;
  • 27) 0.000 000 000 306 891 519 426 56 × 2 = 0 + 0.000 000 000 613 783 038 853 12;
  • 28) 0.000 000 000 613 783 038 853 12 × 2 = 0 + 0.000 000 001 227 566 077 706 24;
  • 29) 0.000 000 001 227 566 077 706 24 × 2 = 0 + 0.000 000 002 455 132 155 412 48;
  • 30) 0.000 000 002 455 132 155 412 48 × 2 = 0 + 0.000 000 004 910 264 310 824 96;
  • 31) 0.000 000 004 910 264 310 824 96 × 2 = 0 + 0.000 000 009 820 528 621 649 92;
  • 32) 0.000 000 009 820 528 621 649 92 × 2 = 0 + 0.000 000 019 641 057 243 299 84;
  • 33) 0.000 000 019 641 057 243 299 84 × 2 = 0 + 0.000 000 039 282 114 486 599 68;
  • 34) 0.000 000 039 282 114 486 599 68 × 2 = 0 + 0.000 000 078 564 228 973 199 36;
  • 35) 0.000 000 078 564 228 973 199 36 × 2 = 0 + 0.000 000 157 128 457 946 398 72;
  • 36) 0.000 000 157 128 457 946 398 72 × 2 = 0 + 0.000 000 314 256 915 892 797 44;
  • 37) 0.000 000 314 256 915 892 797 44 × 2 = 0 + 0.000 000 628 513 831 785 594 88;
  • 38) 0.000 000 628 513 831 785 594 88 × 2 = 0 + 0.000 001 257 027 663 571 189 76;
  • 39) 0.000 001 257 027 663 571 189 76 × 2 = 0 + 0.000 002 514 055 327 142 379 52;
  • 40) 0.000 002 514 055 327 142 379 52 × 2 = 0 + 0.000 005 028 110 654 284 759 04;
  • 41) 0.000 005 028 110 654 284 759 04 × 2 = 0 + 0.000 010 056 221 308 569 518 08;
  • 42) 0.000 010 056 221 308 569 518 08 × 2 = 0 + 0.000 020 112 442 617 139 036 16;
  • 43) 0.000 020 112 442 617 139 036 16 × 2 = 0 + 0.000 040 224 885 234 278 072 32;
  • 44) 0.000 040 224 885 234 278 072 32 × 2 = 0 + 0.000 080 449 770 468 556 144 64;
  • 45) 0.000 080 449 770 468 556 144 64 × 2 = 0 + 0.000 160 899 540 937 112 289 28;
  • 46) 0.000 160 899 540 937 112 289 28 × 2 = 0 + 0.000 321 799 081 874 224 578 56;
  • 47) 0.000 321 799 081 874 224 578 56 × 2 = 0 + 0.000 643 598 163 748 449 157 12;
  • 48) 0.000 643 598 163 748 449 157 12 × 2 = 0 + 0.001 287 196 327 496 898 314 24;
  • 49) 0.001 287 196 327 496 898 314 24 × 2 = 0 + 0.002 574 392 654 993 796 628 48;
  • 50) 0.002 574 392 654 993 796 628 48 × 2 = 0 + 0.005 148 785 309 987 593 256 96;
  • 51) 0.005 148 785 309 987 593 256 96 × 2 = 0 + 0.010 297 570 619 975 186 513 92;
  • 52) 0.010 297 570 619 975 186 513 92 × 2 = 0 + 0.020 595 141 239 950 373 027 84;
  • 53) 0.020 595 141 239 950 373 027 84 × 2 = 0 + 0.041 190 282 479 900 746 055 68;
  • 54) 0.041 190 282 479 900 746 055 68 × 2 = 0 + 0.082 380 564 959 801 492 111 36;
  • 55) 0.082 380 564 959 801 492 111 36 × 2 = 0 + 0.164 761 129 919 602 984 222 72;
  • 56) 0.164 761 129 919 602 984 222 72 × 2 = 0 + 0.329 522 259 839 205 968 445 44;
  • 57) 0.329 522 259 839 205 968 445 44 × 2 = 0 + 0.659 044 519 678 411 936 890 88;
  • 58) 0.659 044 519 678 411 936 890 88 × 2 = 1 + 0.318 089 039 356 823 873 781 76;
  • 59) 0.318 089 039 356 823 873 781 76 × 2 = 0 + 0.636 178 078 713 647 747 563 52;
  • 60) 0.636 178 078 713 647 747 563 52 × 2 = 1 + 0.272 356 157 427 295 495 127 04;
  • 61) 0.272 356 157 427 295 495 127 04 × 2 = 0 + 0.544 712 314 854 590 990 254 08;
  • 62) 0.544 712 314 854 590 990 254 08 × 2 = 1 + 0.089 424 629 709 181 980 508 16;
  • 63) 0.089 424 629 709 181 980 508 16 × 2 = 0 + 0.178 849 259 418 363 961 016 32;
  • 64) 0.178 849 259 418 363 961 016 32 × 2 = 0 + 0.357 698 518 836 727 922 032 64;
  • 65) 0.357 698 518 836 727 922 032 64 × 2 = 0 + 0.715 397 037 673 455 844 065 28;
  • 66) 0.715 397 037 673 455 844 065 28 × 2 = 1 + 0.430 794 075 346 911 688 130 56;
  • 67) 0.430 794 075 346 911 688 130 56 × 2 = 0 + 0.861 588 150 693 823 376 261 12;
  • 68) 0.861 588 150 693 823 376 261 12 × 2 = 1 + 0.723 176 301 387 646 752 522 24;
  • 69) 0.723 176 301 387 646 752 522 24 × 2 = 1 + 0.446 352 602 775 293 505 044 48;
  • 70) 0.446 352 602 775 293 505 044 48 × 2 = 0 + 0.892 705 205 550 587 010 088 96;
  • 71) 0.892 705 205 550 587 010 088 96 × 2 = 1 + 0.785 410 411 101 174 020 177 92;
  • 72) 0.785 410 411 101 174 020 177 92 × 2 = 1 + 0.570 820 822 202 348 040 355 84;
  • 73) 0.570 820 822 202 348 040 355 84 × 2 = 1 + 0.141 641 644 404 696 080 711 68;
  • 74) 0.141 641 644 404 696 080 711 68 × 2 = 0 + 0.283 283 288 809 392 161 423 36;
  • 75) 0.283 283 288 809 392 161 423 36 × 2 = 0 + 0.566 566 577 618 784 322 846 72;
  • 76) 0.566 566 577 618 784 322 846 72 × 2 = 1 + 0.133 133 155 237 568 645 693 44;
  • 77) 0.133 133 155 237 568 645 693 44 × 2 = 0 + 0.266 266 310 475 137 291 386 88;
  • 78) 0.266 266 310 475 137 291 386 88 × 2 = 0 + 0.532 532 620 950 274 582 773 76;
  • 79) 0.532 532 620 950 274 582 773 76 × 2 = 1 + 0.065 065 241 900 549 165 547 52;
  • 80) 0.065 065 241 900 549 165 547 52 × 2 = 0 + 0.130 130 483 801 098 331 095 04;
  • 81) 0.130 130 483 801 098 331 095 04 × 2 = 0 + 0.260 260 967 602 196 662 190 08;
  • 82) 0.260 260 967 602 196 662 190 08 × 2 = 0 + 0.520 521 935 204 393 324 380 16;
  • 83) 0.520 521 935 204 393 324 380 16 × 2 = 1 + 0.041 043 870 408 786 648 760 32;
  • 84) 0.041 043 870 408 786 648 760 32 × 2 = 0 + 0.082 087 740 817 573 297 520 64;
  • 85) 0.082 087 740 817 573 297 520 64 × 2 = 0 + 0.164 175 481 635 146 595 041 28;
  • 86) 0.164 175 481 635 146 595 041 28 × 2 = 0 + 0.328 350 963 270 293 190 082 56;
  • 87) 0.328 350 963 270 293 190 082 56 × 2 = 0 + 0.656 701 926 540 586 380 165 12;
  • 88) 0.656 701 926 540 586 380 165 12 × 2 = 1 + 0.313 403 853 081 172 760 330 24;
  • 89) 0.313 403 853 081 172 760 330 24 × 2 = 0 + 0.626 807 706 162 345 520 660 48;
  • 90) 0.626 807 706 162 345 520 660 48 × 2 = 1 + 0.253 615 412 324 691 041 320 96;
  • 91) 0.253 615 412 324 691 041 320 96 × 2 = 0 + 0.507 230 824 649 382 082 641 92;
  • 92) 0.507 230 824 649 382 082 641 92 × 2 = 1 + 0.014 461 649 298 764 165 283 84;
  • 93) 0.014 461 649 298 764 165 283 84 × 2 = 0 + 0.028 923 298 597 528 330 567 68;
  • 94) 0.028 923 298 597 528 330 567 68 × 2 = 0 + 0.057 846 597 195 056 661 135 36;
  • 95) 0.057 846 597 195 056 661 135 36 × 2 = 0 + 0.115 693 194 390 113 322 270 72;
  • 96) 0.115 693 194 390 113 322 270 72 × 2 = 0 + 0.231 386 388 780 226 644 541 44;
  • 97) 0.231 386 388 780 226 644 541 44 × 2 = 0 + 0.462 772 777 560 453 289 082 88;
  • 98) 0.462 772 777 560 453 289 082 88 × 2 = 0 + 0.925 545 555 120 906 578 165 76;
  • 99) 0.925 545 555 120 906 578 165 76 × 2 = 1 + 0.851 091 110 241 813 156 331 52;
  • 100) 0.851 091 110 241 813 156 331 52 × 2 = 1 + 0.702 182 220 483 626 312 663 04;
  • 101) 0.702 182 220 483 626 312 663 04 × 2 = 1 + 0.404 364 440 967 252 625 326 08;
  • 102) 0.404 364 440 967 252 625 326 08 × 2 = 0 + 0.808 728 881 934 505 250 652 16;
  • 103) 0.808 728 881 934 505 250 652 16 × 2 = 1 + 0.617 457 763 869 010 501 304 32;
  • 104) 0.617 457 763 869 010 501 304 32 × 2 = 1 + 0.234 915 527 738 021 002 608 64;
  • 105) 0.234 915 527 738 021 002 608 64 × 2 = 0 + 0.469 831 055 476 042 005 217 28;
  • 106) 0.469 831 055 476 042 005 217 28 × 2 = 0 + 0.939 662 110 952 084 010 434 56;
  • 107) 0.939 662 110 952 084 010 434 56 × 2 = 1 + 0.879 324 221 904 168 020 869 12;
  • 108) 0.879 324 221 904 168 020 869 12 × 2 = 1 + 0.758 648 443 808 336 041 738 24;
  • 109) 0.758 648 443 808 336 041 738 24 × 2 = 1 + 0.517 296 887 616 672 083 476 48;
  • 110) 0.517 296 887 616 672 083 476 48 × 2 = 1 + 0.034 593 775 233 344 166 952 96;

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:


0.000 000 000 000 000 004 573 04(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1011 1001 0010 0010 0001 0101 0000 0011 1011 0011 11(2)

6. Positive number before normalization:

0.000 000 000 000 000 004 573 04(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1011 1001 0010 0010 0001 0101 0000 0011 1011 0011 11(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 58 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 004 573 04(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1011 1001 0010 0010 0001 0101 0000 0011 1011 0011 11(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 0101 1011 1001 0010 0010 0001 0101 0000 0011 1011 0011 11(2) × 20 =

1.0101 0001 0110 1110 0100 1000 1000 0101 0100 0000 1110 1100 1111(2) × 2-58


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): -58

Mantissa (not normalized):
1.0101 0001 0110 1110 0100 1000 1000 0101 0100 0000 1110 1100 1111


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-58 + 2(11-1) - 1 =

(-58 + 1 023)(10) =

965(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;
  • 965 ÷ 2 = 482 + 1;
  • 482 ÷ 2 = 241 + 0;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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) =

965(10) =

011 1100 0101(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, only if necessary (not the case here).


Mantissa (normalized) =

1. 0101 0001 0110 1110 0100 1000 1000 0101 0100 0000 1110 1100 1111 =

0101 0001 0110 1110 0100 1000 1000 0101 0100 0000 1110 1100 1111


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) =
011 1100 0101

Mantissa (52 bits) =
0101 0001 0110 1110 0100 1000 1000 0101 0100 0000 1110 1100 1111


Decimal number -0.000 000 000 000 000 004 573 04 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1100 0101 - 0101 0001 0110 1110 0100 1000 1000 0101 0100 0000 1110 1100 1111

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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