-0.016 738 891 601 562 495 91 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 495 91(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
-0.016 738 891 601 562 495 91(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:

|-0.016 738 891 601 562 495 91| = 0.016 738 891 601 562 495 91


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: 0.016 738 891 601 562 495 91.

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) 0.016 738 891 601 562 495 91 × 2 = 0 + 0.033 477 783 203 124 991 82;
  • 2) 0.033 477 783 203 124 991 82 × 2 = 0 + 0.066 955 566 406 249 983 64;
  • 3) 0.066 955 566 406 249 983 64 × 2 = 0 + 0.133 911 132 812 499 967 28;
  • 4) 0.133 911 132 812 499 967 28 × 2 = 0 + 0.267 822 265 624 999 934 56;
  • 5) 0.267 822 265 624 999 934 56 × 2 = 0 + 0.535 644 531 249 999 869 12;
  • 6) 0.535 644 531 249 999 869 12 × 2 = 1 + 0.071 289 062 499 999 738 24;
  • 7) 0.071 289 062 499 999 738 24 × 2 = 0 + 0.142 578 124 999 999 476 48;
  • 8) 0.142 578 124 999 999 476 48 × 2 = 0 + 0.285 156 249 999 998 952 96;
  • 9) 0.285 156 249 999 998 952 96 × 2 = 0 + 0.570 312 499 999 997 905 92;
  • 10) 0.570 312 499 999 997 905 92 × 2 = 1 + 0.140 624 999 999 995 811 84;
  • 11) 0.140 624 999 999 995 811 84 × 2 = 0 + 0.281 249 999 999 991 623 68;
  • 12) 0.281 249 999 999 991 623 68 × 2 = 0 + 0.562 499 999 999 983 247 36;
  • 13) 0.562 499 999 999 983 247 36 × 2 = 1 + 0.124 999 999 999 966 494 72;
  • 14) 0.124 999 999 999 966 494 72 × 2 = 0 + 0.249 999 999 999 932 989 44;
  • 15) 0.249 999 999 999 932 989 44 × 2 = 0 + 0.499 999 999 999 865 978 88;
  • 16) 0.499 999 999 999 865 978 88 × 2 = 0 + 0.999 999 999 999 731 957 76;
  • 17) 0.999 999 999 999 731 957 76 × 2 = 1 + 0.999 999 999 999 463 915 52;
  • 18) 0.999 999 999 999 463 915 52 × 2 = 1 + 0.999 999 999 998 927 831 04;
  • 19) 0.999 999 999 998 927 831 04 × 2 = 1 + 0.999 999 999 997 855 662 08;
  • 20) 0.999 999 999 997 855 662 08 × 2 = 1 + 0.999 999 999 995 711 324 16;
  • 21) 0.999 999 999 995 711 324 16 × 2 = 1 + 0.999 999 999 991 422 648 32;
  • 22) 0.999 999 999 991 422 648 32 × 2 = 1 + 0.999 999 999 982 845 296 64;
  • 23) 0.999 999 999 982 845 296 64 × 2 = 1 + 0.999 999 999 965 690 593 28;
  • 24) 0.999 999 999 965 690 593 28 × 2 = 1 + 0.999 999 999 931 381 186 56;
  • 25) 0.999 999 999 931 381 186 56 × 2 = 1 + 0.999 999 999 862 762 373 12;
  • 26) 0.999 999 999 862 762 373 12 × 2 = 1 + 0.999 999 999 725 524 746 24;
  • 27) 0.999 999 999 725 524 746 24 × 2 = 1 + 0.999 999 999 451 049 492 48;
  • 28) 0.999 999 999 451 049 492 48 × 2 = 1 + 0.999 999 998 902 098 984 96;
  • 29) 0.999 999 998 902 098 984 96 × 2 = 1 + 0.999 999 997 804 197 969 92;
  • 30) 0.999 999 997 804 197 969 92 × 2 = 1 + 0.999 999 995 608 395 939 84;
  • 31) 0.999 999 995 608 395 939 84 × 2 = 1 + 0.999 999 991 216 791 879 68;
  • 32) 0.999 999 991 216 791 879 68 × 2 = 1 + 0.999 999 982 433 583 759 36;
  • 33) 0.999 999 982 433 583 759 36 × 2 = 1 + 0.999 999 964 867 167 518 72;
  • 34) 0.999 999 964 867 167 518 72 × 2 = 1 + 0.999 999 929 734 335 037 44;
  • 35) 0.999 999 929 734 335 037 44 × 2 = 1 + 0.999 999 859 468 670 074 88;
  • 36) 0.999 999 859 468 670 074 88 × 2 = 1 + 0.999 999 718 937 340 149 76;
  • 37) 0.999 999 718 937 340 149 76 × 2 = 1 + 0.999 999 437 874 680 299 52;
  • 38) 0.999 999 437 874 680 299 52 × 2 = 1 + 0.999 998 875 749 360 599 04;
  • 39) 0.999 998 875 749 360 599 04 × 2 = 1 + 0.999 997 751 498 721 198 08;
  • 40) 0.999 997 751 498 721 198 08 × 2 = 1 + 0.999 995 502 997 442 396 16;
  • 41) 0.999 995 502 997 442 396 16 × 2 = 1 + 0.999 991 005 994 884 792 32;
  • 42) 0.999 991 005 994 884 792 32 × 2 = 1 + 0.999 982 011 989 769 584 64;
  • 43) 0.999 982 011 989 769 584 64 × 2 = 1 + 0.999 964 023 979 539 169 28;
  • 44) 0.999 964 023 979 539 169 28 × 2 = 1 + 0.999 928 047 959 078 338 56;
  • 45) 0.999 928 047 959 078 338 56 × 2 = 1 + 0.999 856 095 918 156 677 12;
  • 46) 0.999 856 095 918 156 677 12 × 2 = 1 + 0.999 712 191 836 313 354 24;
  • 47) 0.999 712 191 836 313 354 24 × 2 = 1 + 0.999 424 383 672 626 708 48;
  • 48) 0.999 424 383 672 626 708 48 × 2 = 1 + 0.998 848 767 345 253 416 96;
  • 49) 0.998 848 767 345 253 416 96 × 2 = 1 + 0.997 697 534 690 506 833 92;
  • 50) 0.997 697 534 690 506 833 92 × 2 = 1 + 0.995 395 069 381 013 667 84;
  • 51) 0.995 395 069 381 013 667 84 × 2 = 1 + 0.990 790 138 762 027 335 68;
  • 52) 0.990 790 138 762 027 335 68 × 2 = 1 + 0.981 580 277 524 054 671 36;
  • 53) 0.981 580 277 524 054 671 36 × 2 = 1 + 0.963 160 555 048 109 342 72;
  • 54) 0.963 160 555 048 109 342 72 × 2 = 1 + 0.926 321 110 096 218 685 44;
  • 55) 0.926 321 110 096 218 685 44 × 2 = 1 + 0.852 642 220 192 437 370 88;
  • 56) 0.852 642 220 192 437 370 88 × 2 = 1 + 0.705 284 440 384 874 741 76;
  • 57) 0.705 284 440 384 874 741 76 × 2 = 1 + 0.410 568 880 769 749 483 52;
  • 58) 0.410 568 880 769 749 483 52 × 2 = 0 + 0.821 137 761 539 498 967 04;

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:


0.016 738 891 601 562 495 91(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 495 91(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:


0.016 738 891 601 562 495 91(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 -0.016 738 891 601 562 495 91 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