3.141 592 653 589 793 238 462 643 383 279 502 883 42 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.141 592 653 589 793 238 462 643 383 279 502 883 42(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
3.141 592 653 589 793 238 462 643 383 279 502 883 42(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: 3.
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;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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.

3(10) =

11(2)


3. Convert to binary (base 2) the fractional part: 0.141 592 653 589 793 238 462 643 383 279 502 883 42.

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) 0.141 592 653 589 793 238 462 643 383 279 502 883 42 × 2 = 0 + 0.283 185 307 179 586 476 925 286 766 559 005 766 84;
  • 2) 0.283 185 307 179 586 476 925 286 766 559 005 766 84 × 2 = 0 + 0.566 370 614 359 172 953 850 573 533 118 011 533 68;
  • 3) 0.566 370 614 359 172 953 850 573 533 118 011 533 68 × 2 = 1 + 0.132 741 228 718 345 907 701 147 066 236 023 067 36;
  • 4) 0.132 741 228 718 345 907 701 147 066 236 023 067 36 × 2 = 0 + 0.265 482 457 436 691 815 402 294 132 472 046 134 72;
  • 5) 0.265 482 457 436 691 815 402 294 132 472 046 134 72 × 2 = 0 + 0.530 964 914 873 383 630 804 588 264 944 092 269 44;
  • 6) 0.530 964 914 873 383 630 804 588 264 944 092 269 44 × 2 = 1 + 0.061 929 829 746 767 261 609 176 529 888 184 538 88;
  • 7) 0.061 929 829 746 767 261 609 176 529 888 184 538 88 × 2 = 0 + 0.123 859 659 493 534 523 218 353 059 776 369 077 76;
  • 8) 0.123 859 659 493 534 523 218 353 059 776 369 077 76 × 2 = 0 + 0.247 719 318 987 069 046 436 706 119 552 738 155 52;
  • 9) 0.247 719 318 987 069 046 436 706 119 552 738 155 52 × 2 = 0 + 0.495 438 637 974 138 092 873 412 239 105 476 311 04;
  • 10) 0.495 438 637 974 138 092 873 412 239 105 476 311 04 × 2 = 0 + 0.990 877 275 948 276 185 746 824 478 210 952 622 08;
  • 11) 0.990 877 275 948 276 185 746 824 478 210 952 622 08 × 2 = 1 + 0.981 754 551 896 552 371 493 648 956 421 905 244 16;
  • 12) 0.981 754 551 896 552 371 493 648 956 421 905 244 16 × 2 = 1 + 0.963 509 103 793 104 742 987 297 912 843 810 488 32;
  • 13) 0.963 509 103 793 104 742 987 297 912 843 810 488 32 × 2 = 1 + 0.927 018 207 586 209 485 974 595 825 687 620 976 64;
  • 14) 0.927 018 207 586 209 485 974 595 825 687 620 976 64 × 2 = 1 + 0.854 036 415 172 418 971 949 191 651 375 241 953 28;
  • 15) 0.854 036 415 172 418 971 949 191 651 375 241 953 28 × 2 = 1 + 0.708 072 830 344 837 943 898 383 302 750 483 906 56;
  • 16) 0.708 072 830 344 837 943 898 383 302 750 483 906 56 × 2 = 1 + 0.416 145 660 689 675 887 796 766 605 500 967 813 12;
  • 17) 0.416 145 660 689 675 887 796 766 605 500 967 813 12 × 2 = 0 + 0.832 291 321 379 351 775 593 533 211 001 935 626 24;
  • 18) 0.832 291 321 379 351 775 593 533 211 001 935 626 24 × 2 = 1 + 0.664 582 642 758 703 551 187 066 422 003 871 252 48;
  • 19) 0.664 582 642 758 703 551 187 066 422 003 871 252 48 × 2 = 1 + 0.329 165 285 517 407 102 374 132 844 007 742 504 96;
  • 20) 0.329 165 285 517 407 102 374 132 844 007 742 504 96 × 2 = 0 + 0.658 330 571 034 814 204 748 265 688 015 485 009 92;
  • 21) 0.658 330 571 034 814 204 748 265 688 015 485 009 92 × 2 = 1 + 0.316 661 142 069 628 409 496 531 376 030 970 019 84;
  • 22) 0.316 661 142 069 628 409 496 531 376 030 970 019 84 × 2 = 0 + 0.633 322 284 139 256 818 993 062 752 061 940 039 68;
  • 23) 0.633 322 284 139 256 818 993 062 752 061 940 039 68 × 2 = 1 + 0.266 644 568 278 513 637 986 125 504 123 880 079 36;
  • 24) 0.266 644 568 278 513 637 986 125 504 123 880 079 36 × 2 = 0 + 0.533 289 136 557 027 275 972 251 008 247 760 158 72;
  • 25) 0.533 289 136 557 027 275 972 251 008 247 760 158 72 × 2 = 1 + 0.066 578 273 114 054 551 944 502 016 495 520 317 44;
  • 26) 0.066 578 273 114 054 551 944 502 016 495 520 317 44 × 2 = 0 + 0.133 156 546 228 109 103 889 004 032 991 040 634 88;
  • 27) 0.133 156 546 228 109 103 889 004 032 991 040 634 88 × 2 = 0 + 0.266 313 092 456 218 207 778 008 065 982 081 269 76;
  • 28) 0.266 313 092 456 218 207 778 008 065 982 081 269 76 × 2 = 0 + 0.532 626 184 912 436 415 556 016 131 964 162 539 52;
  • 29) 0.532 626 184 912 436 415 556 016 131 964 162 539 52 × 2 = 1 + 0.065 252 369 824 872 831 112 032 263 928 325 079 04;
  • 30) 0.065 252 369 824 872 831 112 032 263 928 325 079 04 × 2 = 0 + 0.130 504 739 649 745 662 224 064 527 856 650 158 08;
  • 31) 0.130 504 739 649 745 662 224 064 527 856 650 158 08 × 2 = 0 + 0.261 009 479 299 491 324 448 129 055 713 300 316 16;
  • 32) 0.261 009 479 299 491 324 448 129 055 713 300 316 16 × 2 = 0 + 0.522 018 958 598 982 648 896 258 111 426 600 632 32;
  • 33) 0.522 018 958 598 982 648 896 258 111 426 600 632 32 × 2 = 1 + 0.044 037 917 197 965 297 792 516 222 853 201 264 64;
  • 34) 0.044 037 917 197 965 297 792 516 222 853 201 264 64 × 2 = 0 + 0.088 075 834 395 930 595 585 032 445 706 402 529 28;
  • 35) 0.088 075 834 395 930 595 585 032 445 706 402 529 28 × 2 = 0 + 0.176 151 668 791 861 191 170 064 891 412 805 058 56;
  • 36) 0.176 151 668 791 861 191 170 064 891 412 805 058 56 × 2 = 0 + 0.352 303 337 583 722 382 340 129 782 825 610 117 12;
  • 37) 0.352 303 337 583 722 382 340 129 782 825 610 117 12 × 2 = 0 + 0.704 606 675 167 444 764 680 259 565 651 220 234 24;
  • 38) 0.704 606 675 167 444 764 680 259 565 651 220 234 24 × 2 = 1 + 0.409 213 350 334 889 529 360 519 131 302 440 468 48;
  • 39) 0.409 213 350 334 889 529 360 519 131 302 440 468 48 × 2 = 0 + 0.818 426 700 669 779 058 721 038 262 604 880 936 96;
  • 40) 0.818 426 700 669 779 058 721 038 262 604 880 936 96 × 2 = 1 + 0.636 853 401 339 558 117 442 076 525 209 761 873 92;
  • 41) 0.636 853 401 339 558 117 442 076 525 209 761 873 92 × 2 = 1 + 0.273 706 802 679 116 234 884 153 050 419 523 747 84;
  • 42) 0.273 706 802 679 116 234 884 153 050 419 523 747 84 × 2 = 0 + 0.547 413 605 358 232 469 768 306 100 839 047 495 68;
  • 43) 0.547 413 605 358 232 469 768 306 100 839 047 495 68 × 2 = 1 + 0.094 827 210 716 464 939 536 612 201 678 094 991 36;
  • 44) 0.094 827 210 716 464 939 536 612 201 678 094 991 36 × 2 = 0 + 0.189 654 421 432 929 879 073 224 403 356 189 982 72;
  • 45) 0.189 654 421 432 929 879 073 224 403 356 189 982 72 × 2 = 0 + 0.379 308 842 865 859 758 146 448 806 712 379 965 44;
  • 46) 0.379 308 842 865 859 758 146 448 806 712 379 965 44 × 2 = 0 + 0.758 617 685 731 719 516 292 897 613 424 759 930 88;
  • 47) 0.758 617 685 731 719 516 292 897 613 424 759 930 88 × 2 = 1 + 0.517 235 371 463 439 032 585 795 226 849 519 861 76;
  • 48) 0.517 235 371 463 439 032 585 795 226 849 519 861 76 × 2 = 1 + 0.034 470 742 926 878 065 171 590 453 699 039 723 52;
  • 49) 0.034 470 742 926 878 065 171 590 453 699 039 723 52 × 2 = 0 + 0.068 941 485 853 756 130 343 180 907 398 079 447 04;
  • 50) 0.068 941 485 853 756 130 343 180 907 398 079 447 04 × 2 = 0 + 0.137 882 971 707 512 260 686 361 814 796 158 894 08;
  • 51) 0.137 882 971 707 512 260 686 361 814 796 158 894 08 × 2 = 0 + 0.275 765 943 415 024 521 372 723 629 592 317 788 16;
  • 52) 0.275 765 943 415 024 521 372 723 629 592 317 788 16 × 2 = 0 + 0.551 531 886 830 049 042 745 447 259 184 635 576 32;
  • 53) 0.551 531 886 830 049 042 745 447 259 184 635 576 32 × 2 = 1 + 0.103 063 773 660 098 085 490 894 518 369 271 152 64;

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:


0.141 592 653 589 793 238 462 643 383 279 502 883 42(10) =

0.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2)

5. Positive number before normalization:

3.141 592 653 589 793 238 462 643 383 279 502 883 42(10) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2)

6. 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:


3.141 592 653 589 793 238 462 643 383 279 502 883 42(10) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2) × 20 =

1.1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 01(2) × 21


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): 1

Mantissa (not normalized):
1.1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 01


8. 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)


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;
  • 1 024 ÷ 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;

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) =

1024(10) =

100 0000 0000(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, 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. 1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 01 =

1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000


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) =
100 0000 0000

Mantissa (52 bits) =
1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000


Decimal number 3.141 592 653 589 793 238 462 643 383 279 502 883 42 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000

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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