-0.016 738 891 601 562 29 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 29(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 29(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 29| = 0.016 738 891 601 562 29


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 29.

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 29 × 2 = 0 + 0.033 477 783 203 124 58;
  • 2) 0.033 477 783 203 124 58 × 2 = 0 + 0.066 955 566 406 249 16;
  • 3) 0.066 955 566 406 249 16 × 2 = 0 + 0.133 911 132 812 498 32;
  • 4) 0.133 911 132 812 498 32 × 2 = 0 + 0.267 822 265 624 996 64;
  • 5) 0.267 822 265 624 996 64 × 2 = 0 + 0.535 644 531 249 993 28;
  • 6) 0.535 644 531 249 993 28 × 2 = 1 + 0.071 289 062 499 986 56;
  • 7) 0.071 289 062 499 986 56 × 2 = 0 + 0.142 578 124 999 973 12;
  • 8) 0.142 578 124 999 973 12 × 2 = 0 + 0.285 156 249 999 946 24;
  • 9) 0.285 156 249 999 946 24 × 2 = 0 + 0.570 312 499 999 892 48;
  • 10) 0.570 312 499 999 892 48 × 2 = 1 + 0.140 624 999 999 784 96;
  • 11) 0.140 624 999 999 784 96 × 2 = 0 + 0.281 249 999 999 569 92;
  • 12) 0.281 249 999 999 569 92 × 2 = 0 + 0.562 499 999 999 139 84;
  • 13) 0.562 499 999 999 139 84 × 2 = 1 + 0.124 999 999 998 279 68;
  • 14) 0.124 999 999 998 279 68 × 2 = 0 + 0.249 999 999 996 559 36;
  • 15) 0.249 999 999 996 559 36 × 2 = 0 + 0.499 999 999 993 118 72;
  • 16) 0.499 999 999 993 118 72 × 2 = 0 + 0.999 999 999 986 237 44;
  • 17) 0.999 999 999 986 237 44 × 2 = 1 + 0.999 999 999 972 474 88;
  • 18) 0.999 999 999 972 474 88 × 2 = 1 + 0.999 999 999 944 949 76;
  • 19) 0.999 999 999 944 949 76 × 2 = 1 + 0.999 999 999 889 899 52;
  • 20) 0.999 999 999 889 899 52 × 2 = 1 + 0.999 999 999 779 799 04;
  • 21) 0.999 999 999 779 799 04 × 2 = 1 + 0.999 999 999 559 598 08;
  • 22) 0.999 999 999 559 598 08 × 2 = 1 + 0.999 999 999 119 196 16;
  • 23) 0.999 999 999 119 196 16 × 2 = 1 + 0.999 999 998 238 392 32;
  • 24) 0.999 999 998 238 392 32 × 2 = 1 + 0.999 999 996 476 784 64;
  • 25) 0.999 999 996 476 784 64 × 2 = 1 + 0.999 999 992 953 569 28;
  • 26) 0.999 999 992 953 569 28 × 2 = 1 + 0.999 999 985 907 138 56;
  • 27) 0.999 999 985 907 138 56 × 2 = 1 + 0.999 999 971 814 277 12;
  • 28) 0.999 999 971 814 277 12 × 2 = 1 + 0.999 999 943 628 554 24;
  • 29) 0.999 999 943 628 554 24 × 2 = 1 + 0.999 999 887 257 108 48;
  • 30) 0.999 999 887 257 108 48 × 2 = 1 + 0.999 999 774 514 216 96;
  • 31) 0.999 999 774 514 216 96 × 2 = 1 + 0.999 999 549 028 433 92;
  • 32) 0.999 999 549 028 433 92 × 2 = 1 + 0.999 999 098 056 867 84;
  • 33) 0.999 999 098 056 867 84 × 2 = 1 + 0.999 998 196 113 735 68;
  • 34) 0.999 998 196 113 735 68 × 2 = 1 + 0.999 996 392 227 471 36;
  • 35) 0.999 996 392 227 471 36 × 2 = 1 + 0.999 992 784 454 942 72;
  • 36) 0.999 992 784 454 942 72 × 2 = 1 + 0.999 985 568 909 885 44;
  • 37) 0.999 985 568 909 885 44 × 2 = 1 + 0.999 971 137 819 770 88;
  • 38) 0.999 971 137 819 770 88 × 2 = 1 + 0.999 942 275 639 541 76;
  • 39) 0.999 942 275 639 541 76 × 2 = 1 + 0.999 884 551 279 083 52;
  • 40) 0.999 884 551 279 083 52 × 2 = 1 + 0.999 769 102 558 167 04;
  • 41) 0.999 769 102 558 167 04 × 2 = 1 + 0.999 538 205 116 334 08;
  • 42) 0.999 538 205 116 334 08 × 2 = 1 + 0.999 076 410 232 668 16;
  • 43) 0.999 076 410 232 668 16 × 2 = 1 + 0.998 152 820 465 336 32;
  • 44) 0.998 152 820 465 336 32 × 2 = 1 + 0.996 305 640 930 672 64;
  • 45) 0.996 305 640 930 672 64 × 2 = 1 + 0.992 611 281 861 345 28;
  • 46) 0.992 611 281 861 345 28 × 2 = 1 + 0.985 222 563 722 690 56;
  • 47) 0.985 222 563 722 690 56 × 2 = 1 + 0.970 445 127 445 381 12;
  • 48) 0.970 445 127 445 381 12 × 2 = 1 + 0.940 890 254 890 762 24;
  • 49) 0.940 890 254 890 762 24 × 2 = 1 + 0.881 780 509 781 524 48;
  • 50) 0.881 780 509 781 524 48 × 2 = 1 + 0.763 561 019 563 048 96;
  • 51) 0.763 561 019 563 048 96 × 2 = 1 + 0.527 122 039 126 097 92;
  • 52) 0.527 122 039 126 097 92 × 2 = 1 + 0.054 244 078 252 195 84;
  • 53) 0.054 244 078 252 195 84 × 2 = 0 + 0.108 488 156 504 391 68;
  • 54) 0.108 488 156 504 391 68 × 2 = 0 + 0.216 976 313 008 783 36;
  • 55) 0.216 976 313 008 783 36 × 2 = 0 + 0.433 952 626 017 566 72;
  • 56) 0.433 952 626 017 566 72 × 2 = 0 + 0.867 905 252 035 133 44;
  • 57) 0.867 905 252 035 133 44 × 2 = 1 + 0.735 810 504 070 266 88;
  • 58) 0.735 810 504 070 266 88 × 2 = 1 + 0.471 621 008 140 533 76;

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 29(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 11(2)

6. Positive number before normalization:

0.016 738 891 601 562 29(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 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 29(10) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 11(2) =

0.0000 0100 0100 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 11(2) × 20 =

1.0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1100 0011(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 1100 0011


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 1100 0011 =

0001 0010 0011 1111 1111 1111 1111 1111 1111 1111 1111 1100 0011


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 1100 0011


Decimal number -0.016 738 891 601 562 29 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 1100 0011

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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