-0.016 738 891 601 562 496 39 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 39(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 39(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 39| = 0.016 738 891 601 562 496 39


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 39.

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 39 × 2 = 0 + 0.033 477 783 203 124 992 78;
  • 2) 0.033 477 783 203 124 992 78 × 2 = 0 + 0.066 955 566 406 249 985 56;
  • 3) 0.066 955 566 406 249 985 56 × 2 = 0 + 0.133 911 132 812 499 971 12;
  • 4) 0.133 911 132 812 499 971 12 × 2 = 0 + 0.267 822 265 624 999 942 24;
  • 5) 0.267 822 265 624 999 942 24 × 2 = 0 + 0.535 644 531 249 999 884 48;
  • 6) 0.535 644 531 249 999 884 48 × 2 = 1 + 0.071 289 062 499 999 768 96;
  • 7) 0.071 289 062 499 999 768 96 × 2 = 0 + 0.142 578 124 999 999 537 92;
  • 8) 0.142 578 124 999 999 537 92 × 2 = 0 + 0.285 156 249 999 999 075 84;
  • 9) 0.285 156 249 999 999 075 84 × 2 = 0 + 0.570 312 499 999 998 151 68;
  • 10) 0.570 312 499 999 998 151 68 × 2 = 1 + 0.140 624 999 999 996 303 36;
  • 11) 0.140 624 999 999 996 303 36 × 2 = 0 + 0.281 249 999 999 992 606 72;
  • 12) 0.281 249 999 999 992 606 72 × 2 = 0 + 0.562 499 999 999 985 213 44;
  • 13) 0.562 499 999 999 985 213 44 × 2 = 1 + 0.124 999 999 999 970 426 88;
  • 14) 0.124 999 999 999 970 426 88 × 2 = 0 + 0.249 999 999 999 940 853 76;
  • 15) 0.249 999 999 999 940 853 76 × 2 = 0 + 0.499 999 999 999 881 707 52;
  • 16) 0.499 999 999 999 881 707 52 × 2 = 0 + 0.999 999 999 999 763 415 04;
  • 17) 0.999 999 999 999 763 415 04 × 2 = 1 + 0.999 999 999 999 526 830 08;
  • 18) 0.999 999 999 999 526 830 08 × 2 = 1 + 0.999 999 999 999 053 660 16;
  • 19) 0.999 999 999 999 053 660 16 × 2 = 1 + 0.999 999 999 998 107 320 32;
  • 20) 0.999 999 999 998 107 320 32 × 2 = 1 + 0.999 999 999 996 214 640 64;
  • 21) 0.999 999 999 996 214 640 64 × 2 = 1 + 0.999 999 999 992 429 281 28;
  • 22) 0.999 999 999 992 429 281 28 × 2 = 1 + 0.999 999 999 984 858 562 56;
  • 23) 0.999 999 999 984 858 562 56 × 2 = 1 + 0.999 999 999 969 717 125 12;
  • 24) 0.999 999 999 969 717 125 12 × 2 = 1 + 0.999 999 999 939 434 250 24;
  • 25) 0.999 999 999 939 434 250 24 × 2 = 1 + 0.999 999 999 878 868 500 48;
  • 26) 0.999 999 999 878 868 500 48 × 2 = 1 + 0.999 999 999 757 737 000 96;
  • 27) 0.999 999 999 757 737 000 96 × 2 = 1 + 0.999 999 999 515 474 001 92;
  • 28) 0.999 999 999 515 474 001 92 × 2 = 1 + 0.999 999 999 030 948 003 84;
  • 29) 0.999 999 999 030 948 003 84 × 2 = 1 + 0.999 999 998 061 896 007 68;
  • 30) 0.999 999 998 061 896 007 68 × 2 = 1 + 0.999 999 996 123 792 015 36;
  • 31) 0.999 999 996 123 792 015 36 × 2 = 1 + 0.999 999 992 247 584 030 72;
  • 32) 0.999 999 992 247 584 030 72 × 2 = 1 + 0.999 999 984 495 168 061 44;
  • 33) 0.999 999 984 495 168 061 44 × 2 = 1 + 0.999 999 968 990 336 122 88;
  • 34) 0.999 999 968 990 336 122 88 × 2 = 1 + 0.999 999 937 980 672 245 76;
  • 35) 0.999 999 937 980 672 245 76 × 2 = 1 + 0.999 999 875 961 344 491 52;
  • 36) 0.999 999 875 961 344 491 52 × 2 = 1 + 0.999 999 751 922 688 983 04;
  • 37) 0.999 999 751 922 688 983 04 × 2 = 1 + 0.999 999 503 845 377 966 08;
  • 38) 0.999 999 503 845 377 966 08 × 2 = 1 + 0.999 999 007 690 755 932 16;
  • 39) 0.999 999 007 690 755 932 16 × 2 = 1 + 0.999 998 015 381 511 864 32;
  • 40) 0.999 998 015 381 511 864 32 × 2 = 1 + 0.999 996 030 763 023 728 64;
  • 41) 0.999 996 030 763 023 728 64 × 2 = 1 + 0.999 992 061 526 047 457 28;
  • 42) 0.999 992 061 526 047 457 28 × 2 = 1 + 0.999 984 123 052 094 914 56;
  • 43) 0.999 984 123 052 094 914 56 × 2 = 1 + 0.999 968 246 104 189 829 12;
  • 44) 0.999 968 246 104 189 829 12 × 2 = 1 + 0.999 936 492 208 379 658 24;
  • 45) 0.999 936 492 208 379 658 24 × 2 = 1 + 0.999 872 984 416 759 316 48;
  • 46) 0.999 872 984 416 759 316 48 × 2 = 1 + 0.999 745 968 833 518 632 96;
  • 47) 0.999 745 968 833 518 632 96 × 2 = 1 + 0.999 491 937 667 037 265 92;
  • 48) 0.999 491 937 667 037 265 92 × 2 = 1 + 0.998 983 875 334 074 531 84;
  • 49) 0.998 983 875 334 074 531 84 × 2 = 1 + 0.997 967 750 668 149 063 68;
  • 50) 0.997 967 750 668 149 063 68 × 2 = 1 + 0.995 935 501 336 298 127 36;
  • 51) 0.995 935 501 336 298 127 36 × 2 = 1 + 0.991 871 002 672 596 254 72;
  • 52) 0.991 871 002 672 596 254 72 × 2 = 1 + 0.983 742 005 345 192 509 44;
  • 53) 0.983 742 005 345 192 509 44 × 2 = 1 + 0.967 484 010 690 385 018 88;
  • 54) 0.967 484 010 690 385 018 88 × 2 = 1 + 0.934 968 021 380 770 037 76;
  • 55) 0.934 968 021 380 770 037 76 × 2 = 1 + 0.869 936 042 761 540 075 52;
  • 56) 0.869 936 042 761 540 075 52 × 2 = 1 + 0.739 872 085 523 080 151 04;
  • 57) 0.739 872 085 523 080 151 04 × 2 = 1 + 0.479 744 171 046 160 302 08;
  • 58) 0.479 744 171 046 160 302 08 × 2 = 0 + 0.959 488 342 092 320 604 16;

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 39(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(2)

6. Positive number before normalization:

0.016 738 891 601 562 496 39(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(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 39(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(2) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 10(2) × 20 =

1.0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110(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 1110


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 1110 =

0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110


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 1110


Decimal number -0.016 738 891 601 562 496 39 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 1110

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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