5.919 999 999 999 999 928 945 726 423 976 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 5.919 999 999 999 999 928 945 726 423 976 3(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
5.919 999 999 999 999 928 945 726 423 976 3(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: 5.
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;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 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.

5(10) =

101(2)


3. Convert to binary (base 2) the fractional part: 0.919 999 999 999 999 928 945 726 423 976 3.

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) 0.919 999 999 999 999 928 945 726 423 976 3 × 2 = 1 + 0.839 999 999 999 999 857 891 452 847 952 6;
  • 2) 0.839 999 999 999 999 857 891 452 847 952 6 × 2 = 1 + 0.679 999 999 999 999 715 782 905 695 905 2;
  • 3) 0.679 999 999 999 999 715 782 905 695 905 2 × 2 = 1 + 0.359 999 999 999 999 431 565 811 391 810 4;
  • 4) 0.359 999 999 999 999 431 565 811 391 810 4 × 2 = 0 + 0.719 999 999 999 998 863 131 622 783 620 8;
  • 5) 0.719 999 999 999 998 863 131 622 783 620 8 × 2 = 1 + 0.439 999 999 999 997 726 263 245 567 241 6;
  • 6) 0.439 999 999 999 997 726 263 245 567 241 6 × 2 = 0 + 0.879 999 999 999 995 452 526 491 134 483 2;
  • 7) 0.879 999 999 999 995 452 526 491 134 483 2 × 2 = 1 + 0.759 999 999 999 990 905 052 982 268 966 4;
  • 8) 0.759 999 999 999 990 905 052 982 268 966 4 × 2 = 1 + 0.519 999 999 999 981 810 105 964 537 932 8;
  • 9) 0.519 999 999 999 981 810 105 964 537 932 8 × 2 = 1 + 0.039 999 999 999 963 620 211 929 075 865 6;
  • 10) 0.039 999 999 999 963 620 211 929 075 865 6 × 2 = 0 + 0.079 999 999 999 927 240 423 858 151 731 2;
  • 11) 0.079 999 999 999 927 240 423 858 151 731 2 × 2 = 0 + 0.159 999 999 999 854 480 847 716 303 462 4;
  • 12) 0.159 999 999 999 854 480 847 716 303 462 4 × 2 = 0 + 0.319 999 999 999 708 961 695 432 606 924 8;
  • 13) 0.319 999 999 999 708 961 695 432 606 924 8 × 2 = 0 + 0.639 999 999 999 417 923 390 865 213 849 6;
  • 14) 0.639 999 999 999 417 923 390 865 213 849 6 × 2 = 1 + 0.279 999 999 998 835 846 781 730 427 699 2;
  • 15) 0.279 999 999 998 835 846 781 730 427 699 2 × 2 = 0 + 0.559 999 999 997 671 693 563 460 855 398 4;
  • 16) 0.559 999 999 997 671 693 563 460 855 398 4 × 2 = 1 + 0.119 999 999 995 343 387 126 921 710 796 8;
  • 17) 0.119 999 999 995 343 387 126 921 710 796 8 × 2 = 0 + 0.239 999 999 990 686 774 253 843 421 593 6;
  • 18) 0.239 999 999 990 686 774 253 843 421 593 6 × 2 = 0 + 0.479 999 999 981 373 548 507 686 843 187 2;
  • 19) 0.479 999 999 981 373 548 507 686 843 187 2 × 2 = 0 + 0.959 999 999 962 747 097 015 373 686 374 4;
  • 20) 0.959 999 999 962 747 097 015 373 686 374 4 × 2 = 1 + 0.919 999 999 925 494 194 030 747 372 748 8;
  • 21) 0.919 999 999 925 494 194 030 747 372 748 8 × 2 = 1 + 0.839 999 999 850 988 388 061 494 745 497 6;
  • 22) 0.839 999 999 850 988 388 061 494 745 497 6 × 2 = 1 + 0.679 999 999 701 976 776 122 989 490 995 2;
  • 23) 0.679 999 999 701 976 776 122 989 490 995 2 × 2 = 1 + 0.359 999 999 403 953 552 245 978 981 990 4;
  • 24) 0.359 999 999 403 953 552 245 978 981 990 4 × 2 = 0 + 0.719 999 998 807 907 104 491 957 963 980 8;
  • 25) 0.719 999 998 807 907 104 491 957 963 980 8 × 2 = 1 + 0.439 999 997 615 814 208 983 915 927 961 6;
  • 26) 0.439 999 997 615 814 208 983 915 927 961 6 × 2 = 0 + 0.879 999 995 231 628 417 967 831 855 923 2;
  • 27) 0.879 999 995 231 628 417 967 831 855 923 2 × 2 = 1 + 0.759 999 990 463 256 835 935 663 711 846 4;
  • 28) 0.759 999 990 463 256 835 935 663 711 846 4 × 2 = 1 + 0.519 999 980 926 513 671 871 327 423 692 8;
  • 29) 0.519 999 980 926 513 671 871 327 423 692 8 × 2 = 1 + 0.039 999 961 853 027 343 742 654 847 385 6;
  • 30) 0.039 999 961 853 027 343 742 654 847 385 6 × 2 = 0 + 0.079 999 923 706 054 687 485 309 694 771 2;
  • 31) 0.079 999 923 706 054 687 485 309 694 771 2 × 2 = 0 + 0.159 999 847 412 109 374 970 619 389 542 4;
  • 32) 0.159 999 847 412 109 374 970 619 389 542 4 × 2 = 0 + 0.319 999 694 824 218 749 941 238 779 084 8;
  • 33) 0.319 999 694 824 218 749 941 238 779 084 8 × 2 = 0 + 0.639 999 389 648 437 499 882 477 558 169 6;
  • 34) 0.639 999 389 648 437 499 882 477 558 169 6 × 2 = 1 + 0.279 998 779 296 874 999 764 955 116 339 2;
  • 35) 0.279 998 779 296 874 999 764 955 116 339 2 × 2 = 0 + 0.559 997 558 593 749 999 529 910 232 678 4;
  • 36) 0.559 997 558 593 749 999 529 910 232 678 4 × 2 = 1 + 0.119 995 117 187 499 999 059 820 465 356 8;
  • 37) 0.119 995 117 187 499 999 059 820 465 356 8 × 2 = 0 + 0.239 990 234 374 999 998 119 640 930 713 6;
  • 38) 0.239 990 234 374 999 998 119 640 930 713 6 × 2 = 0 + 0.479 980 468 749 999 996 239 281 861 427 2;
  • 39) 0.479 980 468 749 999 996 239 281 861 427 2 × 2 = 0 + 0.959 960 937 499 999 992 478 563 722 854 4;
  • 40) 0.959 960 937 499 999 992 478 563 722 854 4 × 2 = 1 + 0.919 921 874 999 999 984 957 127 445 708 8;
  • 41) 0.919 921 874 999 999 984 957 127 445 708 8 × 2 = 1 + 0.839 843 749 999 999 969 914 254 891 417 6;
  • 42) 0.839 843 749 999 999 969 914 254 891 417 6 × 2 = 1 + 0.679 687 499 999 999 939 828 509 782 835 2;
  • 43) 0.679 687 499 999 999 939 828 509 782 835 2 × 2 = 1 + 0.359 374 999 999 999 879 657 019 565 670 4;
  • 44) 0.359 374 999 999 999 879 657 019 565 670 4 × 2 = 0 + 0.718 749 999 999 999 759 314 039 131 340 8;
  • 45) 0.718 749 999 999 999 759 314 039 131 340 8 × 2 = 1 + 0.437 499 999 999 999 518 628 078 262 681 6;
  • 46) 0.437 499 999 999 999 518 628 078 262 681 6 × 2 = 0 + 0.874 999 999 999 999 037 256 156 525 363 2;
  • 47) 0.874 999 999 999 999 037 256 156 525 363 2 × 2 = 1 + 0.749 999 999 999 998 074 512 313 050 726 4;
  • 48) 0.749 999 999 999 998 074 512 313 050 726 4 × 2 = 1 + 0.499 999 999 999 996 149 024 626 101 452 8;
  • 49) 0.499 999 999 999 996 149 024 626 101 452 8 × 2 = 0 + 0.999 999 999 999 992 298 049 252 202 905 6;
  • 50) 0.999 999 999 999 992 298 049 252 202 905 6 × 2 = 1 + 0.999 999 999 999 984 596 098 504 405 811 2;
  • 51) 0.999 999 999 999 984 596 098 504 405 811 2 × 2 = 1 + 0.999 999 999 999 969 192 197 008 811 622 4;
  • 52) 0.999 999 999 999 969 192 197 008 811 622 4 × 2 = 1 + 0.999 999 999 999 938 384 394 017 623 244 8;
  • 53) 0.999 999 999 999 938 384 394 017 623 244 8 × 2 = 1 + 0.999 999 999 999 876 768 788 035 246 489 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 - 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:


0.919 999 999 999 999 928 945 726 423 976 3(10) =

0.1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 0111 1(2)

5. Positive number before normalization:

5.919 999 999 999 999 928 945 726 423 976 3(10) =

101.1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 0111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 2 positions to the left, so that only one non zero digit remains to the left of it:


5.919 999 999 999 999 928 945 726 423 976 3(10) =

101.1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 0111 1(2) =

101.1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 0111 1(2) × 20 =

1.0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1101 111(2) × 22


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): 2

Mantissa (not normalized):
1.0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1101 111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

2 + 2(11-1) - 1 =

(2 + 1 023)(10) =

1 025(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;
  • 1 025 ÷ 2 = 512 + 1;
  • 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) =

1025(10) =

100 0000 0001(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. 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1101 111 =

0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1101


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 0001

Mantissa (52 bits) =
0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1101


Decimal number 5.919 999 999 999 999 928 945 726 423 976 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0001 - 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1101

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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