-0.016 738 891 601 562 496 656 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 656(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
-0.016 738 891 601 562 496 656(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. Start with the positive version of the number:

|-0.016 738 891 601 562 496 656| = 0.016 738 891 601 562 496 656


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: 0.016 738 891 601 562 496 656.

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) 0.016 738 891 601 562 496 656 × 2 = 0 + 0.033 477 783 203 124 993 312;
  • 2) 0.033 477 783 203 124 993 312 × 2 = 0 + 0.066 955 566 406 249 986 624;
  • 3) 0.066 955 566 406 249 986 624 × 2 = 0 + 0.133 911 132 812 499 973 248;
  • 4) 0.133 911 132 812 499 973 248 × 2 = 0 + 0.267 822 265 624 999 946 496;
  • 5) 0.267 822 265 624 999 946 496 × 2 = 0 + 0.535 644 531 249 999 892 992;
  • 6) 0.535 644 531 249 999 892 992 × 2 = 1 + 0.071 289 062 499 999 785 984;
  • 7) 0.071 289 062 499 999 785 984 × 2 = 0 + 0.142 578 124 999 999 571 968;
  • 8) 0.142 578 124 999 999 571 968 × 2 = 0 + 0.285 156 249 999 999 143 936;
  • 9) 0.285 156 249 999 999 143 936 × 2 = 0 + 0.570 312 499 999 998 287 872;
  • 10) 0.570 312 499 999 998 287 872 × 2 = 1 + 0.140 624 999 999 996 575 744;
  • 11) 0.140 624 999 999 996 575 744 × 2 = 0 + 0.281 249 999 999 993 151 488;
  • 12) 0.281 249 999 999 993 151 488 × 2 = 0 + 0.562 499 999 999 986 302 976;
  • 13) 0.562 499 999 999 986 302 976 × 2 = 1 + 0.124 999 999 999 972 605 952;
  • 14) 0.124 999 999 999 972 605 952 × 2 = 0 + 0.249 999 999 999 945 211 904;
  • 15) 0.249 999 999 999 945 211 904 × 2 = 0 + 0.499 999 999 999 890 423 808;
  • 16) 0.499 999 999 999 890 423 808 × 2 = 0 + 0.999 999 999 999 780 847 616;
  • 17) 0.999 999 999 999 780 847 616 × 2 = 1 + 0.999 999 999 999 561 695 232;
  • 18) 0.999 999 999 999 561 695 232 × 2 = 1 + 0.999 999 999 999 123 390 464;
  • 19) 0.999 999 999 999 123 390 464 × 2 = 1 + 0.999 999 999 998 246 780 928;
  • 20) 0.999 999 999 998 246 780 928 × 2 = 1 + 0.999 999 999 996 493 561 856;
  • 21) 0.999 999 999 996 493 561 856 × 2 = 1 + 0.999 999 999 992 987 123 712;
  • 22) 0.999 999 999 992 987 123 712 × 2 = 1 + 0.999 999 999 985 974 247 424;
  • 23) 0.999 999 999 985 974 247 424 × 2 = 1 + 0.999 999 999 971 948 494 848;
  • 24) 0.999 999 999 971 948 494 848 × 2 = 1 + 0.999 999 999 943 896 989 696;
  • 25) 0.999 999 999 943 896 989 696 × 2 = 1 + 0.999 999 999 887 793 979 392;
  • 26) 0.999 999 999 887 793 979 392 × 2 = 1 + 0.999 999 999 775 587 958 784;
  • 27) 0.999 999 999 775 587 958 784 × 2 = 1 + 0.999 999 999 551 175 917 568;
  • 28) 0.999 999 999 551 175 917 568 × 2 = 1 + 0.999 999 999 102 351 835 136;
  • 29) 0.999 999 999 102 351 835 136 × 2 = 1 + 0.999 999 998 204 703 670 272;
  • 30) 0.999 999 998 204 703 670 272 × 2 = 1 + 0.999 999 996 409 407 340 544;
  • 31) 0.999 999 996 409 407 340 544 × 2 = 1 + 0.999 999 992 818 814 681 088;
  • 32) 0.999 999 992 818 814 681 088 × 2 = 1 + 0.999 999 985 637 629 362 176;
  • 33) 0.999 999 985 637 629 362 176 × 2 = 1 + 0.999 999 971 275 258 724 352;
  • 34) 0.999 999 971 275 258 724 352 × 2 = 1 + 0.999 999 942 550 517 448 704;
  • 35) 0.999 999 942 550 517 448 704 × 2 = 1 + 0.999 999 885 101 034 897 408;
  • 36) 0.999 999 885 101 034 897 408 × 2 = 1 + 0.999 999 770 202 069 794 816;
  • 37) 0.999 999 770 202 069 794 816 × 2 = 1 + 0.999 999 540 404 139 589 632;
  • 38) 0.999 999 540 404 139 589 632 × 2 = 1 + 0.999 999 080 808 279 179 264;
  • 39) 0.999 999 080 808 279 179 264 × 2 = 1 + 0.999 998 161 616 558 358 528;
  • 40) 0.999 998 161 616 558 358 528 × 2 = 1 + 0.999 996 323 233 116 717 056;
  • 41) 0.999 996 323 233 116 717 056 × 2 = 1 + 0.999 992 646 466 233 434 112;
  • 42) 0.999 992 646 466 233 434 112 × 2 = 1 + 0.999 985 292 932 466 868 224;
  • 43) 0.999 985 292 932 466 868 224 × 2 = 1 + 0.999 970 585 864 933 736 448;
  • 44) 0.999 970 585 864 933 736 448 × 2 = 1 + 0.999 941 171 729 867 472 896;
  • 45) 0.999 941 171 729 867 472 896 × 2 = 1 + 0.999 882 343 459 734 945 792;
  • 46) 0.999 882 343 459 734 945 792 × 2 = 1 + 0.999 764 686 919 469 891 584;
  • 47) 0.999 764 686 919 469 891 584 × 2 = 1 + 0.999 529 373 838 939 783 168;
  • 48) 0.999 529 373 838 939 783 168 × 2 = 1 + 0.999 058 747 677 879 566 336;
  • 49) 0.999 058 747 677 879 566 336 × 2 = 1 + 0.998 117 495 355 759 132 672;
  • 50) 0.998 117 495 355 759 132 672 × 2 = 1 + 0.996 234 990 711 518 265 344;
  • 51) 0.996 234 990 711 518 265 344 × 2 = 1 + 0.992 469 981 423 036 530 688;
  • 52) 0.992 469 981 423 036 530 688 × 2 = 1 + 0.984 939 962 846 073 061 376;
  • 53) 0.984 939 962 846 073 061 376 × 2 = 1 + 0.969 879 925 692 146 122 752;
  • 54) 0.969 879 925 692 146 122 752 × 2 = 1 + 0.939 759 851 384 292 245 504;
  • 55) 0.939 759 851 384 292 245 504 × 2 = 1 + 0.879 519 702 768 584 491 008;
  • 56) 0.879 519 702 768 584 491 008 × 2 = 1 + 0.759 039 405 537 168 982 016;
  • 57) 0.759 039 405 537 168 982 016 × 2 = 1 + 0.518 078 811 074 337 964 032;
  • 58) 0.518 078 811 074 337 964 032 × 2 = 1 + 0.036 157 622 148 675 928 064;

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).


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:


0.016 738 891 601 562 496 656(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2)

6. Positive number before normalization:

0.016 738 891 601 562 496 656(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 6 positions to the right, so that only one non zero digit remains to the left of it:


0.016 738 891 601 562 496 656(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 11(2) × 20 =

1.0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111(2) × 2-6


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): -6

Mantissa (not normalized):
1.0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-6 + 2(11-1) - 1 =

(-6 + 1 023)(10) =

1 017(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;
  • 1 017 ÷ 2 = 508 + 1;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 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) =

1017(10) =

011 1111 1001(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. 0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 =

0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


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 1111 1001

Mantissa (52 bits) =
0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


Decimal number -0.016 738 891 601 562 496 656 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1001 - 0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111

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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