64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: -0.000 000 000 000 038 460 874 543 480 73 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number -0.000 000 000 000 038 460 874 543 480 73(10) converted and written in 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:

|-0.000 000 000 000 038 460 874 543 480 73| = 0.000 000 000 000 038 460 874 543 480 73

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: 0.000 000 000 000 038 460 874 543 480 73.

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) 0.000 000 000 000 038 460 874 543 480 73 × 2 = 0 + 0.000 000 000 000 076 921 749 086 961 46;
  • 2) 0.000 000 000 000 076 921 749 086 961 46 × 2 = 0 + 0.000 000 000 000 153 843 498 173 922 92;
  • 3) 0.000 000 000 000 153 843 498 173 922 92 × 2 = 0 + 0.000 000 000 000 307 686 996 347 845 84;
  • 4) 0.000 000 000 000 307 686 996 347 845 84 × 2 = 0 + 0.000 000 000 000 615 373 992 695 691 68;
  • 5) 0.000 000 000 000 615 373 992 695 691 68 × 2 = 0 + 0.000 000 000 001 230 747 985 391 383 36;
  • 6) 0.000 000 000 001 230 747 985 391 383 36 × 2 = 0 + 0.000 000 000 002 461 495 970 782 766 72;
  • 7) 0.000 000 000 002 461 495 970 782 766 72 × 2 = 0 + 0.000 000 000 004 922 991 941 565 533 44;
  • 8) 0.000 000 000 004 922 991 941 565 533 44 × 2 = 0 + 0.000 000 000 009 845 983 883 131 066 88;
  • 9) 0.000 000 000 009 845 983 883 131 066 88 × 2 = 0 + 0.000 000 000 019 691 967 766 262 133 76;
  • 10) 0.000 000 000 019 691 967 766 262 133 76 × 2 = 0 + 0.000 000 000 039 383 935 532 524 267 52;
  • 11) 0.000 000 000 039 383 935 532 524 267 52 × 2 = 0 + 0.000 000 000 078 767 871 065 048 535 04;
  • 12) 0.000 000 000 078 767 871 065 048 535 04 × 2 = 0 + 0.000 000 000 157 535 742 130 097 070 08;
  • 13) 0.000 000 000 157 535 742 130 097 070 08 × 2 = 0 + 0.000 000 000 315 071 484 260 194 140 16;
  • 14) 0.000 000 000 315 071 484 260 194 140 16 × 2 = 0 + 0.000 000 000 630 142 968 520 388 280 32;
  • 15) 0.000 000 000 630 142 968 520 388 280 32 × 2 = 0 + 0.000 000 001 260 285 937 040 776 560 64;
  • 16) 0.000 000 001 260 285 937 040 776 560 64 × 2 = 0 + 0.000 000 002 520 571 874 081 553 121 28;
  • 17) 0.000 000 002 520 571 874 081 553 121 28 × 2 = 0 + 0.000 000 005 041 143 748 163 106 242 56;
  • 18) 0.000 000 005 041 143 748 163 106 242 56 × 2 = 0 + 0.000 000 010 082 287 496 326 212 485 12;
  • 19) 0.000 000 010 082 287 496 326 212 485 12 × 2 = 0 + 0.000 000 020 164 574 992 652 424 970 24;
  • 20) 0.000 000 020 164 574 992 652 424 970 24 × 2 = 0 + 0.000 000 040 329 149 985 304 849 940 48;
  • 21) 0.000 000 040 329 149 985 304 849 940 48 × 2 = 0 + 0.000 000 080 658 299 970 609 699 880 96;
  • 22) 0.000 000 080 658 299 970 609 699 880 96 × 2 = 0 + 0.000 000 161 316 599 941 219 399 761 92;
  • 23) 0.000 000 161 316 599 941 219 399 761 92 × 2 = 0 + 0.000 000 322 633 199 882 438 799 523 84;
  • 24) 0.000 000 322 633 199 882 438 799 523 84 × 2 = 0 + 0.000 000 645 266 399 764 877 599 047 68;
  • 25) 0.000 000 645 266 399 764 877 599 047 68 × 2 = 0 + 0.000 001 290 532 799 529 755 198 095 36;
  • 26) 0.000 001 290 532 799 529 755 198 095 36 × 2 = 0 + 0.000 002 581 065 599 059 510 396 190 72;
  • 27) 0.000 002 581 065 599 059 510 396 190 72 × 2 = 0 + 0.000 005 162 131 198 119 020 792 381 44;
  • 28) 0.000 005 162 131 198 119 020 792 381 44 × 2 = 0 + 0.000 010 324 262 396 238 041 584 762 88;
  • 29) 0.000 010 324 262 396 238 041 584 762 88 × 2 = 0 + 0.000 020 648 524 792 476 083 169 525 76;
  • 30) 0.000 020 648 524 792 476 083 169 525 76 × 2 = 0 + 0.000 041 297 049 584 952 166 339 051 52;
  • 31) 0.000 041 297 049 584 952 166 339 051 52 × 2 = 0 + 0.000 082 594 099 169 904 332 678 103 04;
  • 32) 0.000 082 594 099 169 904 332 678 103 04 × 2 = 0 + 0.000 165 188 198 339 808 665 356 206 08;
  • 33) 0.000 165 188 198 339 808 665 356 206 08 × 2 = 0 + 0.000 330 376 396 679 617 330 712 412 16;
  • 34) 0.000 330 376 396 679 617 330 712 412 16 × 2 = 0 + 0.000 660 752 793 359 234 661 424 824 32;
  • 35) 0.000 660 752 793 359 234 661 424 824 32 × 2 = 0 + 0.001 321 505 586 718 469 322 849 648 64;
  • 36) 0.001 321 505 586 718 469 322 849 648 64 × 2 = 0 + 0.002 643 011 173 436 938 645 699 297 28;
  • 37) 0.002 643 011 173 436 938 645 699 297 28 × 2 = 0 + 0.005 286 022 346 873 877 291 398 594 56;
  • 38) 0.005 286 022 346 873 877 291 398 594 56 × 2 = 0 + 0.010 572 044 693 747 754 582 797 189 12;
  • 39) 0.010 572 044 693 747 754 582 797 189 12 × 2 = 0 + 0.021 144 089 387 495 509 165 594 378 24;
  • 40) 0.021 144 089 387 495 509 165 594 378 24 × 2 = 0 + 0.042 288 178 774 991 018 331 188 756 48;
  • 41) 0.042 288 178 774 991 018 331 188 756 48 × 2 = 0 + 0.084 576 357 549 982 036 662 377 512 96;
  • 42) 0.084 576 357 549 982 036 662 377 512 96 × 2 = 0 + 0.169 152 715 099 964 073 324 755 025 92;
  • 43) 0.169 152 715 099 964 073 324 755 025 92 × 2 = 0 + 0.338 305 430 199 928 146 649 510 051 84;
  • 44) 0.338 305 430 199 928 146 649 510 051 84 × 2 = 0 + 0.676 610 860 399 856 293 299 020 103 68;
  • 45) 0.676 610 860 399 856 293 299 020 103 68 × 2 = 1 + 0.353 221 720 799 712 586 598 040 207 36;
  • 46) 0.353 221 720 799 712 586 598 040 207 36 × 2 = 0 + 0.706 443 441 599 425 173 196 080 414 72;
  • 47) 0.706 443 441 599 425 173 196 080 414 72 × 2 = 1 + 0.412 886 883 198 850 346 392 160 829 44;
  • 48) 0.412 886 883 198 850 346 392 160 829 44 × 2 = 0 + 0.825 773 766 397 700 692 784 321 658 88;
  • 49) 0.825 773 766 397 700 692 784 321 658 88 × 2 = 1 + 0.651 547 532 795 401 385 568 643 317 76;
  • 50) 0.651 547 532 795 401 385 568 643 317 76 × 2 = 1 + 0.303 095 065 590 802 771 137 286 635 52;
  • 51) 0.303 095 065 590 802 771 137 286 635 52 × 2 = 0 + 0.606 190 131 181 605 542 274 573 271 04;
  • 52) 0.606 190 131 181 605 542 274 573 271 04 × 2 = 1 + 0.212 380 262 363 211 084 549 146 542 08;
  • 53) 0.212 380 262 363 211 084 549 146 542 08 × 2 = 0 + 0.424 760 524 726 422 169 098 293 084 16;
  • 54) 0.424 760 524 726 422 169 098 293 084 16 × 2 = 0 + 0.849 521 049 452 844 338 196 586 168 32;
  • 55) 0.849 521 049 452 844 338 196 586 168 32 × 2 = 1 + 0.699 042 098 905 688 676 393 172 336 64;
  • 56) 0.699 042 098 905 688 676 393 172 336 64 × 2 = 1 + 0.398 084 197 811 377 352 786 344 673 28;
  • 57) 0.398 084 197 811 377 352 786 344 673 28 × 2 = 0 + 0.796 168 395 622 754 705 572 689 346 56;
  • 58) 0.796 168 395 622 754 705 572 689 346 56 × 2 = 1 + 0.592 336 791 245 509 411 145 378 693 12;
  • 59) 0.592 336 791 245 509 411 145 378 693 12 × 2 = 1 + 0.184 673 582 491 018 822 290 757 386 24;
  • 60) 0.184 673 582 491 018 822 290 757 386 24 × 2 = 0 + 0.369 347 164 982 037 644 581 514 772 48;
  • 61) 0.369 347 164 982 037 644 581 514 772 48 × 2 = 0 + 0.738 694 329 964 075 289 163 029 544 96;
  • 62) 0.738 694 329 964 075 289 163 029 544 96 × 2 = 1 + 0.477 388 659 928 150 578 326 059 089 92;
  • 63) 0.477 388 659 928 150 578 326 059 089 92 × 2 = 0 + 0.954 777 319 856 301 156 652 118 179 84;
  • 64) 0.954 777 319 856 301 156 652 118 179 84 × 2 = 1 + 0.909 554 639 712 602 313 304 236 359 68;
  • 65) 0.909 554 639 712 602 313 304 236 359 68 × 2 = 1 + 0.819 109 279 425 204 626 608 472 719 36;
  • 66) 0.819 109 279 425 204 626 608 472 719 36 × 2 = 1 + 0.638 218 558 850 409 253 216 945 438 72;
  • 67) 0.638 218 558 850 409 253 216 945 438 72 × 2 = 1 + 0.276 437 117 700 818 506 433 890 877 44;
  • 68) 0.276 437 117 700 818 506 433 890 877 44 × 2 = 0 + 0.552 874 235 401 637 012 867 781 754 88;
  • 69) 0.552 874 235 401 637 012 867 781 754 88 × 2 = 1 + 0.105 748 470 803 274 025 735 563 509 76;
  • 70) 0.105 748 470 803 274 025 735 563 509 76 × 2 = 0 + 0.211 496 941 606 548 051 471 127 019 52;
  • 71) 0.211 496 941 606 548 051 471 127 019 52 × 2 = 0 + 0.422 993 883 213 096 102 942 254 039 04;
  • 72) 0.422 993 883 213 096 102 942 254 039 04 × 2 = 0 + 0.845 987 766 426 192 205 884 508 078 08;
  • 73) 0.845 987 766 426 192 205 884 508 078 08 × 2 = 1 + 0.691 975 532 852 384 411 769 016 156 16;
  • 74) 0.691 975 532 852 384 411 769 016 156 16 × 2 = 1 + 0.383 951 065 704 768 823 538 032 312 32;
  • 75) 0.383 951 065 704 768 823 538 032 312 32 × 2 = 0 + 0.767 902 131 409 537 647 076 064 624 64;
  • 76) 0.767 902 131 409 537 647 076 064 624 64 × 2 = 1 + 0.535 804 262 819 075 294 152 129 249 28;
  • 77) 0.535 804 262 819 075 294 152 129 249 28 × 2 = 1 + 0.071 608 525 638 150 588 304 258 498 56;
  • 78) 0.071 608 525 638 150 588 304 258 498 56 × 2 = 0 + 0.143 217 051 276 301 176 608 516 997 12;
  • 79) 0.143 217 051 276 301 176 608 516 997 12 × 2 = 0 + 0.286 434 102 552 602 353 217 033 994 24;
  • 80) 0.286 434 102 552 602 353 217 033 994 24 × 2 = 0 + 0.572 868 205 105 204 706 434 067 988 48;
  • 81) 0.572 868 205 105 204 706 434 067 988 48 × 2 = 1 + 0.145 736 410 210 409 412 868 135 976 96;
  • 82) 0.145 736 410 210 409 412 868 135 976 96 × 2 = 0 + 0.291 472 820 420 818 825 736 271 953 92;
  • 83) 0.291 472 820 420 818 825 736 271 953 92 × 2 = 0 + 0.582 945 640 841 637 651 472 543 907 84;
  • 84) 0.582 945 640 841 637 651 472 543 907 84 × 2 = 1 + 0.165 891 281 683 275 302 945 087 815 68;
  • 85) 0.165 891 281 683 275 302 945 087 815 68 × 2 = 0 + 0.331 782 563 366 550 605 890 175 631 36;
  • 86) 0.331 782 563 366 550 605 890 175 631 36 × 2 = 0 + 0.663 565 126 733 101 211 780 351 262 72;
  • 87) 0.663 565 126 733 101 211 780 351 262 72 × 2 = 1 + 0.327 130 253 466 202 423 560 702 525 44;
  • 88) 0.327 130 253 466 202 423 560 702 525 44 × 2 = 0 + 0.654 260 506 932 404 847 121 405 050 88;
  • 89) 0.654 260 506 932 404 847 121 405 050 88 × 2 = 1 + 0.308 521 013 864 809 694 242 810 101 76;
  • 90) 0.308 521 013 864 809 694 242 810 101 76 × 2 = 0 + 0.617 042 027 729 619 388 485 620 203 52;
  • 91) 0.617 042 027 729 619 388 485 620 203 52 × 2 = 1 + 0.234 084 055 459 238 776 971 240 407 04;
  • 92) 0.234 084 055 459 238 776 971 240 407 04 × 2 = 0 + 0.468 168 110 918 477 553 942 480 814 08;
  • 93) 0.468 168 110 918 477 553 942 480 814 08 × 2 = 0 + 0.936 336 221 836 955 107 884 961 628 16;
  • 94) 0.936 336 221 836 955 107 884 961 628 16 × 2 = 1 + 0.872 672 443 673 910 215 769 923 256 32;
  • 95) 0.872 672 443 673 910 215 769 923 256 32 × 2 = 1 + 0.745 344 887 347 820 431 539 846 512 64;
  • 96) 0.745 344 887 347 820 431 539 846 512 64 × 2 = 1 + 0.490 689 774 695 640 863 079 693 025 28;
  • 97) 0.490 689 774 695 640 863 079 693 025 28 × 2 = 0 + 0.981 379 549 391 281 726 159 386 050 56;

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...)


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:


0.000 000 000 000 038 460 874 543 480 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 1101 0011 0110 0101 1110 1000 1101 1000 1001 0010 1010 0111 0(2)


6. Positive number before normalization:

0.000 000 000 000 038 460 874 543 480 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 1101 0011 0110 0101 1110 1000 1101 1000 1001 0010 1010 0111 0(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 45 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 038 460 874 543 480 73(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 1101 0011 0110 0101 1110 1000 1101 1000 1001 0010 1010 0111 0(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1010 1101 0011 0110 0101 1110 1000 1101 1000 1001 0010 1010 0111 0(2) × 20 =

1.0101 1010 0110 1100 1011 1101 0001 1011 0001 0010 0101 0100 1110(2) × 2-45


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): -45

Mantissa (not normalized):
1.0101 1010 0110 1100 1011 1101 0001 1011 0001 0010 0101 0100 1110


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-45 + 2(11-1) - 1 =

(-45 + 1 023)(10) =

978(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;
  • 978 ÷ 2 = 489 + 0;
  • 489 ÷ 2 = 244 + 1;
  • 244 ÷ 2 = 122 + 0;
  • 122 ÷ 2 = 61 + 0;
  • 61 ÷ 2 = 30 + 1;
  • 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) =

978(10) =

011 1101 0010(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 1010 0110 1100 1011 1101 0001 1011 0001 0010 0101 0100 1110 =

0101 1010 0110 1100 1011 1101 0001 1011 0001 0010 0101 0100 1110


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 1101 0010

Mantissa (52 bits) =
0101 1010 0110 1100 1011 1101 0001 1011 0001 0010 0101 0100 1110


The base ten decimal number -0.000 000 000 000 038 460 874 543 480 73 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
1 - 011 1101 0010 - 0101 1010 0110 1100 1011 1101 0001 1011 0001 0010 0101 0100 1110

More operations with decimal numbers converted to 64 bit double precision IEEE 754 binary floating point representation:

» Number -0.000 000 000 000 038 460 874 543 480 74 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

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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