17.035 499 999 999 998 976 818 388 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 17.035 499 999 999 998 976 818 388(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
17.035 499 999 999 998 976 818 388(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: 17.
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;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 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.

17(10) =

1 0001(2)


3. Convert to binary (base 2) the fractional part: 0.035 499 999 999 998 976 818 388.

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) 0.035 499 999 999 998 976 818 388 × 2 = 0 + 0.070 999 999 999 997 953 636 776;
  • 2) 0.070 999 999 999 997 953 636 776 × 2 = 0 + 0.141 999 999 999 995 907 273 552;
  • 3) 0.141 999 999 999 995 907 273 552 × 2 = 0 + 0.283 999 999 999 991 814 547 104;
  • 4) 0.283 999 999 999 991 814 547 104 × 2 = 0 + 0.567 999 999 999 983 629 094 208;
  • 5) 0.567 999 999 999 983 629 094 208 × 2 = 1 + 0.135 999 999 999 967 258 188 416;
  • 6) 0.135 999 999 999 967 258 188 416 × 2 = 0 + 0.271 999 999 999 934 516 376 832;
  • 7) 0.271 999 999 999 934 516 376 832 × 2 = 0 + 0.543 999 999 999 869 032 753 664;
  • 8) 0.543 999 999 999 869 032 753 664 × 2 = 1 + 0.087 999 999 999 738 065 507 328;
  • 9) 0.087 999 999 999 738 065 507 328 × 2 = 0 + 0.175 999 999 999 476 131 014 656;
  • 10) 0.175 999 999 999 476 131 014 656 × 2 = 0 + 0.351 999 999 998 952 262 029 312;
  • 11) 0.351 999 999 998 952 262 029 312 × 2 = 0 + 0.703 999 999 997 904 524 058 624;
  • 12) 0.703 999 999 997 904 524 058 624 × 2 = 1 + 0.407 999 999 995 809 048 117 248;
  • 13) 0.407 999 999 995 809 048 117 248 × 2 = 0 + 0.815 999 999 991 618 096 234 496;
  • 14) 0.815 999 999 991 618 096 234 496 × 2 = 1 + 0.631 999 999 983 236 192 468 992;
  • 15) 0.631 999 999 983 236 192 468 992 × 2 = 1 + 0.263 999 999 966 472 384 937 984;
  • 16) 0.263 999 999 966 472 384 937 984 × 2 = 0 + 0.527 999 999 932 944 769 875 968;
  • 17) 0.527 999 999 932 944 769 875 968 × 2 = 1 + 0.055 999 999 865 889 539 751 936;
  • 18) 0.055 999 999 865 889 539 751 936 × 2 = 0 + 0.111 999 999 731 779 079 503 872;
  • 19) 0.111 999 999 731 779 079 503 872 × 2 = 0 + 0.223 999 999 463 558 159 007 744;
  • 20) 0.223 999 999 463 558 159 007 744 × 2 = 0 + 0.447 999 998 927 116 318 015 488;
  • 21) 0.447 999 998 927 116 318 015 488 × 2 = 0 + 0.895 999 997 854 232 636 030 976;
  • 22) 0.895 999 997 854 232 636 030 976 × 2 = 1 + 0.791 999 995 708 465 272 061 952;
  • 23) 0.791 999 995 708 465 272 061 952 × 2 = 1 + 0.583 999 991 416 930 544 123 904;
  • 24) 0.583 999 991 416 930 544 123 904 × 2 = 1 + 0.167 999 982 833 861 088 247 808;
  • 25) 0.167 999 982 833 861 088 247 808 × 2 = 0 + 0.335 999 965 667 722 176 495 616;
  • 26) 0.335 999 965 667 722 176 495 616 × 2 = 0 + 0.671 999 931 335 444 352 991 232;
  • 27) 0.671 999 931 335 444 352 991 232 × 2 = 1 + 0.343 999 862 670 888 705 982 464;
  • 28) 0.343 999 862 670 888 705 982 464 × 2 = 0 + 0.687 999 725 341 777 411 964 928;
  • 29) 0.687 999 725 341 777 411 964 928 × 2 = 1 + 0.375 999 450 683 554 823 929 856;
  • 30) 0.375 999 450 683 554 823 929 856 × 2 = 0 + 0.751 998 901 367 109 647 859 712;
  • 31) 0.751 998 901 367 109 647 859 712 × 2 = 1 + 0.503 997 802 734 219 295 719 424;
  • 32) 0.503 997 802 734 219 295 719 424 × 2 = 1 + 0.007 995 605 468 438 591 438 848;
  • 33) 0.007 995 605 468 438 591 438 848 × 2 = 0 + 0.015 991 210 936 877 182 877 696;
  • 34) 0.015 991 210 936 877 182 877 696 × 2 = 0 + 0.031 982 421 873 754 365 755 392;
  • 35) 0.031 982 421 873 754 365 755 392 × 2 = 0 + 0.063 964 843 747 508 731 510 784;
  • 36) 0.063 964 843 747 508 731 510 784 × 2 = 0 + 0.127 929 687 495 017 463 021 568;
  • 37) 0.127 929 687 495 017 463 021 568 × 2 = 0 + 0.255 859 374 990 034 926 043 136;
  • 38) 0.255 859 374 990 034 926 043 136 × 2 = 0 + 0.511 718 749 980 069 852 086 272;
  • 39) 0.511 718 749 980 069 852 086 272 × 2 = 1 + 0.023 437 499 960 139 704 172 544;
  • 40) 0.023 437 499 960 139 704 172 544 × 2 = 0 + 0.046 874 999 920 279 408 345 088;
  • 41) 0.046 874 999 920 279 408 345 088 × 2 = 0 + 0.093 749 999 840 558 816 690 176;
  • 42) 0.093 749 999 840 558 816 690 176 × 2 = 0 + 0.187 499 999 681 117 633 380 352;
  • 43) 0.187 499 999 681 117 633 380 352 × 2 = 0 + 0.374 999 999 362 235 266 760 704;
  • 44) 0.374 999 999 362 235 266 760 704 × 2 = 0 + 0.749 999 998 724 470 533 521 408;
  • 45) 0.749 999 998 724 470 533 521 408 × 2 = 1 + 0.499 999 997 448 941 067 042 816;
  • 46) 0.499 999 997 448 941 067 042 816 × 2 = 0 + 0.999 999 994 897 882 134 085 632;
  • 47) 0.999 999 994 897 882 134 085 632 × 2 = 1 + 0.999 999 989 795 764 268 171 264;
  • 48) 0.999 999 989 795 764 268 171 264 × 2 = 1 + 0.999 999 979 591 528 536 342 528;
  • 49) 0.999 999 979 591 528 536 342 528 × 2 = 1 + 0.999 999 959 183 057 072 685 056;
  • 50) 0.999 999 959 183 057 072 685 056 × 2 = 1 + 0.999 999 918 366 114 145 370 112;
  • 51) 0.999 999 918 366 114 145 370 112 × 2 = 1 + 0.999 999 836 732 228 290 740 224;
  • 52) 0.999 999 836 732 228 290 740 224 × 2 = 1 + 0.999 999 673 464 456 581 480 448;
  • 53) 0.999 999 673 464 456 581 480 448 × 2 = 1 + 0.999 999 346 928 913 162 960 896;

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:


0.035 499 999 999 998 976 818 388(10) =

0.0000 1001 0001 0110 1000 0111 0010 1011 0000 0010 0000 1011 1111 1(2)

5. Positive number before normalization:

17.035 499 999 999 998 976 818 388(10) =

1 0001.0000 1001 0001 0110 1000 0111 0010 1011 0000 0010 0000 1011 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 4 positions to the left, so that only one non zero digit remains to the left of it:


17.035 499 999 999 998 976 818 388(10) =

1 0001.0000 1001 0001 0110 1000 0111 0010 1011 0000 0010 0000 1011 1111 1(2) =

1 0001.0000 1001 0001 0110 1000 0111 0010 1011 0000 0010 0000 1011 1111 1(2) × 20 =

1.0001 0000 1001 0001 0110 1000 0111 0010 1011 0000 0010 0000 1011 1111 1(2) × 24


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): 4

Mantissa (not normalized):
1.0001 0000 1001 0001 0110 1000 0111 0010 1011 0000 0010 0000 1011 1111 1


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

4 + 2(11-1) - 1 =

(4 + 1 023)(10) =

1 027(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;
  • 1 027 ÷ 2 = 513 + 1;
  • 513 ÷ 2 = 256 + 1;
  • 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) =

1027(10) =

100 0000 0011(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. 0001 0000 1001 0001 0110 1000 0111 0010 1011 0000 0010 0000 1011 1 1111 =

0001 0000 1001 0001 0110 1000 0111 0010 1011 0000 0010 0000 1011


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 0011

Mantissa (52 bits) =
0001 0000 1001 0001 0110 1000 0111 0010 1011 0000 0010 0000 1011


Decimal number 17.035 499 999 999 998 976 818 388 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0011 - 0001 0000 1001 0001 0110 1000 0111 0010 1011 0000 0010 0000 1011

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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