64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 3.141 592 653 589 793 115 997 963 468 544 185 161 590 55 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 3.141 592 653 589 793 115 997 963 468 544 185 161 590 55(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. 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: 0.141 592 653 589 793 115 997 963 468 544 185 161 590 55.

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) 0.141 592 653 589 793 115 997 963 468 544 185 161 590 55 × 2 = 0 + 0.283 185 307 179 586 231 995 926 937 088 370 323 181 1;
  • 2) 0.283 185 307 179 586 231 995 926 937 088 370 323 181 1 × 2 = 0 + 0.566 370 614 359 172 463 991 853 874 176 740 646 362 2;
  • 3) 0.566 370 614 359 172 463 991 853 874 176 740 646 362 2 × 2 = 1 + 0.132 741 228 718 344 927 983 707 748 353 481 292 724 4;
  • 4) 0.132 741 228 718 344 927 983 707 748 353 481 292 724 4 × 2 = 0 + 0.265 482 457 436 689 855 967 415 496 706 962 585 448 8;
  • 5) 0.265 482 457 436 689 855 967 415 496 706 962 585 448 8 × 2 = 0 + 0.530 964 914 873 379 711 934 830 993 413 925 170 897 6;
  • 6) 0.530 964 914 873 379 711 934 830 993 413 925 170 897 6 × 2 = 1 + 0.061 929 829 746 759 423 869 661 986 827 850 341 795 2;
  • 7) 0.061 929 829 746 759 423 869 661 986 827 850 341 795 2 × 2 = 0 + 0.123 859 659 493 518 847 739 323 973 655 700 683 590 4;
  • 8) 0.123 859 659 493 518 847 739 323 973 655 700 683 590 4 × 2 = 0 + 0.247 719 318 987 037 695 478 647 947 311 401 367 180 8;
  • 9) 0.247 719 318 987 037 695 478 647 947 311 401 367 180 8 × 2 = 0 + 0.495 438 637 974 075 390 957 295 894 622 802 734 361 6;
  • 10) 0.495 438 637 974 075 390 957 295 894 622 802 734 361 6 × 2 = 0 + 0.990 877 275 948 150 781 914 591 789 245 605 468 723 2;
  • 11) 0.990 877 275 948 150 781 914 591 789 245 605 468 723 2 × 2 = 1 + 0.981 754 551 896 301 563 829 183 578 491 210 937 446 4;
  • 12) 0.981 754 551 896 301 563 829 183 578 491 210 937 446 4 × 2 = 1 + 0.963 509 103 792 603 127 658 367 156 982 421 874 892 8;
  • 13) 0.963 509 103 792 603 127 658 367 156 982 421 874 892 8 × 2 = 1 + 0.927 018 207 585 206 255 316 734 313 964 843 749 785 6;
  • 14) 0.927 018 207 585 206 255 316 734 313 964 843 749 785 6 × 2 = 1 + 0.854 036 415 170 412 510 633 468 627 929 687 499 571 2;
  • 15) 0.854 036 415 170 412 510 633 468 627 929 687 499 571 2 × 2 = 1 + 0.708 072 830 340 825 021 266 937 255 859 374 999 142 4;
  • 16) 0.708 072 830 340 825 021 266 937 255 859 374 999 142 4 × 2 = 1 + 0.416 145 660 681 650 042 533 874 511 718 749 998 284 8;
  • 17) 0.416 145 660 681 650 042 533 874 511 718 749 998 284 8 × 2 = 0 + 0.832 291 321 363 300 085 067 749 023 437 499 996 569 6;
  • 18) 0.832 291 321 363 300 085 067 749 023 437 499 996 569 6 × 2 = 1 + 0.664 582 642 726 600 170 135 498 046 874 999 993 139 2;
  • 19) 0.664 582 642 726 600 170 135 498 046 874 999 993 139 2 × 2 = 1 + 0.329 165 285 453 200 340 270 996 093 749 999 986 278 4;
  • 20) 0.329 165 285 453 200 340 270 996 093 749 999 986 278 4 × 2 = 0 + 0.658 330 570 906 400 680 541 992 187 499 999 972 556 8;
  • 21) 0.658 330 570 906 400 680 541 992 187 499 999 972 556 8 × 2 = 1 + 0.316 661 141 812 801 361 083 984 374 999 999 945 113 6;
  • 22) 0.316 661 141 812 801 361 083 984 374 999 999 945 113 6 × 2 = 0 + 0.633 322 283 625 602 722 167 968 749 999 999 890 227 2;
  • 23) 0.633 322 283 625 602 722 167 968 749 999 999 890 227 2 × 2 = 1 + 0.266 644 567 251 205 444 335 937 499 999 999 780 454 4;
  • 24) 0.266 644 567 251 205 444 335 937 499 999 999 780 454 4 × 2 = 0 + 0.533 289 134 502 410 888 671 874 999 999 999 560 908 8;
  • 25) 0.533 289 134 502 410 888 671 874 999 999 999 560 908 8 × 2 = 1 + 0.066 578 269 004 821 777 343 749 999 999 999 121 817 6;
  • 26) 0.066 578 269 004 821 777 343 749 999 999 999 121 817 6 × 2 = 0 + 0.133 156 538 009 643 554 687 499 999 999 998 243 635 2;
  • 27) 0.133 156 538 009 643 554 687 499 999 999 998 243 635 2 × 2 = 0 + 0.266 313 076 019 287 109 374 999 999 999 996 487 270 4;
  • 28) 0.266 313 076 019 287 109 374 999 999 999 996 487 270 4 × 2 = 0 + 0.532 626 152 038 574 218 749 999 999 999 992 974 540 8;
  • 29) 0.532 626 152 038 574 218 749 999 999 999 992 974 540 8 × 2 = 1 + 0.065 252 304 077 148 437 499 999 999 999 985 949 081 6;
  • 30) 0.065 252 304 077 148 437 499 999 999 999 985 949 081 6 × 2 = 0 + 0.130 504 608 154 296 874 999 999 999 999 971 898 163 2;
  • 31) 0.130 504 608 154 296 874 999 999 999 999 971 898 163 2 × 2 = 0 + 0.261 009 216 308 593 749 999 999 999 999 943 796 326 4;
  • 32) 0.261 009 216 308 593 749 999 999 999 999 943 796 326 4 × 2 = 0 + 0.522 018 432 617 187 499 999 999 999 999 887 592 652 8;
  • 33) 0.522 018 432 617 187 499 999 999 999 999 887 592 652 8 × 2 = 1 + 0.044 036 865 234 374 999 999 999 999 999 775 185 305 6;
  • 34) 0.044 036 865 234 374 999 999 999 999 999 775 185 305 6 × 2 = 0 + 0.088 073 730 468 749 999 999 999 999 999 550 370 611 2;
  • 35) 0.088 073 730 468 749 999 999 999 999 999 550 370 611 2 × 2 = 0 + 0.176 147 460 937 499 999 999 999 999 999 100 741 222 4;
  • 36) 0.176 147 460 937 499 999 999 999 999 999 100 741 222 4 × 2 = 0 + 0.352 294 921 874 999 999 999 999 999 998 201 482 444 8;
  • 37) 0.352 294 921 874 999 999 999 999 999 998 201 482 444 8 × 2 = 0 + 0.704 589 843 749 999 999 999 999 999 996 402 964 889 6;
  • 38) 0.704 589 843 749 999 999 999 999 999 996 402 964 889 6 × 2 = 1 + 0.409 179 687 499 999 999 999 999 999 992 805 929 779 2;
  • 39) 0.409 179 687 499 999 999 999 999 999 992 805 929 779 2 × 2 = 0 + 0.818 359 374 999 999 999 999 999 999 985 611 859 558 4;
  • 40) 0.818 359 374 999 999 999 999 999 999 985 611 859 558 4 × 2 = 1 + 0.636 718 749 999 999 999 999 999 999 971 223 719 116 8;
  • 41) 0.636 718 749 999 999 999 999 999 999 971 223 719 116 8 × 2 = 1 + 0.273 437 499 999 999 999 999 999 999 942 447 438 233 6;
  • 42) 0.273 437 499 999 999 999 999 999 999 942 447 438 233 6 × 2 = 0 + 0.546 874 999 999 999 999 999 999 999 884 894 876 467 2;
  • 43) 0.546 874 999 999 999 999 999 999 999 884 894 876 467 2 × 2 = 1 + 0.093 749 999 999 999 999 999 999 999 769 789 752 934 4;
  • 44) 0.093 749 999 999 999 999 999 999 999 769 789 752 934 4 × 2 = 0 + 0.187 499 999 999 999 999 999 999 999 539 579 505 868 8;
  • 45) 0.187 499 999 999 999 999 999 999 999 539 579 505 868 8 × 2 = 0 + 0.374 999 999 999 999 999 999 999 999 079 159 011 737 6;
  • 46) 0.374 999 999 999 999 999 999 999 999 079 159 011 737 6 × 2 = 0 + 0.749 999 999 999 999 999 999 999 998 158 318 023 475 2;
  • 47) 0.749 999 999 999 999 999 999 999 998 158 318 023 475 2 × 2 = 1 + 0.499 999 999 999 999 999 999 999 996 316 636 046 950 4;
  • 48) 0.499 999 999 999 999 999 999 999 996 316 636 046 950 4 × 2 = 0 + 0.999 999 999 999 999 999 999 999 992 633 272 093 900 8;
  • 49) 0.999 999 999 999 999 999 999 999 992 633 272 093 900 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 985 266 544 187 801 6;
  • 50) 0.999 999 999 999 999 999 999 999 985 266 544 187 801 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 970 533 088 375 603 2;
  • 51) 0.999 999 999 999 999 999 999 999 970 533 088 375 603 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 941 066 176 751 206 4;
  • 52) 0.999 999 999 999 999 999 999 999 941 066 176 751 206 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 882 132 353 502 412 8;
  • 53) 0.999 999 999 999 999 999 999 999 882 132 353 502 412 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 764 264 707 004 825 6;

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...)


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:


0.141 592 653 589 793 115 997 963 468 544 185 161 590 55(10) =


0.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0010 1111 1(2)


5. Positive number before normalization:

3.141 592 653 589 793 115 997 963 468 544 185 161 590 55(10) =


11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0010 1111 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:


3.141 592 653 589 793 115 997 963 468 544 185 161 590 55(10) =


11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0010 1111 1(2) =


11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0010 1111 1(2) × 20 =


1.1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 0111 11(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 0111 11


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 0111 11 =


1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 0111


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 0111


The base ten decimal number 3.141 592 653 589 793 115 997 963 468 544 185 161 590 55 converted and written in 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 0111

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