-0.016 738 891 601 562 494 78 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 494 78(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
-0.016 738 891 601 562 494 78(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:

|-0.016 738 891 601 562 494 78| = 0.016 738 891 601 562 494 78


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: 0.016 738 891 601 562 494 78.

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) 0.016 738 891 601 562 494 78 × 2 = 0 + 0.033 477 783 203 124 989 56;
  • 2) 0.033 477 783 203 124 989 56 × 2 = 0 + 0.066 955 566 406 249 979 12;
  • 3) 0.066 955 566 406 249 979 12 × 2 = 0 + 0.133 911 132 812 499 958 24;
  • 4) 0.133 911 132 812 499 958 24 × 2 = 0 + 0.267 822 265 624 999 916 48;
  • 5) 0.267 822 265 624 999 916 48 × 2 = 0 + 0.535 644 531 249 999 832 96;
  • 6) 0.535 644 531 249 999 832 96 × 2 = 1 + 0.071 289 062 499 999 665 92;
  • 7) 0.071 289 062 499 999 665 92 × 2 = 0 + 0.142 578 124 999 999 331 84;
  • 8) 0.142 578 124 999 999 331 84 × 2 = 0 + 0.285 156 249 999 998 663 68;
  • 9) 0.285 156 249 999 998 663 68 × 2 = 0 + 0.570 312 499 999 997 327 36;
  • 10) 0.570 312 499 999 997 327 36 × 2 = 1 + 0.140 624 999 999 994 654 72;
  • 11) 0.140 624 999 999 994 654 72 × 2 = 0 + 0.281 249 999 999 989 309 44;
  • 12) 0.281 249 999 999 989 309 44 × 2 = 0 + 0.562 499 999 999 978 618 88;
  • 13) 0.562 499 999 999 978 618 88 × 2 = 1 + 0.124 999 999 999 957 237 76;
  • 14) 0.124 999 999 999 957 237 76 × 2 = 0 + 0.249 999 999 999 914 475 52;
  • 15) 0.249 999 999 999 914 475 52 × 2 = 0 + 0.499 999 999 999 828 951 04;
  • 16) 0.499 999 999 999 828 951 04 × 2 = 0 + 0.999 999 999 999 657 902 08;
  • 17) 0.999 999 999 999 657 902 08 × 2 = 1 + 0.999 999 999 999 315 804 16;
  • 18) 0.999 999 999 999 315 804 16 × 2 = 1 + 0.999 999 999 998 631 608 32;
  • 19) 0.999 999 999 998 631 608 32 × 2 = 1 + 0.999 999 999 997 263 216 64;
  • 20) 0.999 999 999 997 263 216 64 × 2 = 1 + 0.999 999 999 994 526 433 28;
  • 21) 0.999 999 999 994 526 433 28 × 2 = 1 + 0.999 999 999 989 052 866 56;
  • 22) 0.999 999 999 989 052 866 56 × 2 = 1 + 0.999 999 999 978 105 733 12;
  • 23) 0.999 999 999 978 105 733 12 × 2 = 1 + 0.999 999 999 956 211 466 24;
  • 24) 0.999 999 999 956 211 466 24 × 2 = 1 + 0.999 999 999 912 422 932 48;
  • 25) 0.999 999 999 912 422 932 48 × 2 = 1 + 0.999 999 999 824 845 864 96;
  • 26) 0.999 999 999 824 845 864 96 × 2 = 1 + 0.999 999 999 649 691 729 92;
  • 27) 0.999 999 999 649 691 729 92 × 2 = 1 + 0.999 999 999 299 383 459 84;
  • 28) 0.999 999 999 299 383 459 84 × 2 = 1 + 0.999 999 998 598 766 919 68;
  • 29) 0.999 999 998 598 766 919 68 × 2 = 1 + 0.999 999 997 197 533 839 36;
  • 30) 0.999 999 997 197 533 839 36 × 2 = 1 + 0.999 999 994 395 067 678 72;
  • 31) 0.999 999 994 395 067 678 72 × 2 = 1 + 0.999 999 988 790 135 357 44;
  • 32) 0.999 999 988 790 135 357 44 × 2 = 1 + 0.999 999 977 580 270 714 88;
  • 33) 0.999 999 977 580 270 714 88 × 2 = 1 + 0.999 999 955 160 541 429 76;
  • 34) 0.999 999 955 160 541 429 76 × 2 = 1 + 0.999 999 910 321 082 859 52;
  • 35) 0.999 999 910 321 082 859 52 × 2 = 1 + 0.999 999 820 642 165 719 04;
  • 36) 0.999 999 820 642 165 719 04 × 2 = 1 + 0.999 999 641 284 331 438 08;
  • 37) 0.999 999 641 284 331 438 08 × 2 = 1 + 0.999 999 282 568 662 876 16;
  • 38) 0.999 999 282 568 662 876 16 × 2 = 1 + 0.999 998 565 137 325 752 32;
  • 39) 0.999 998 565 137 325 752 32 × 2 = 1 + 0.999 997 130 274 651 504 64;
  • 40) 0.999 997 130 274 651 504 64 × 2 = 1 + 0.999 994 260 549 303 009 28;
  • 41) 0.999 994 260 549 303 009 28 × 2 = 1 + 0.999 988 521 098 606 018 56;
  • 42) 0.999 988 521 098 606 018 56 × 2 = 1 + 0.999 977 042 197 212 037 12;
  • 43) 0.999 977 042 197 212 037 12 × 2 = 1 + 0.999 954 084 394 424 074 24;
  • 44) 0.999 954 084 394 424 074 24 × 2 = 1 + 0.999 908 168 788 848 148 48;
  • 45) 0.999 908 168 788 848 148 48 × 2 = 1 + 0.999 816 337 577 696 296 96;
  • 46) 0.999 816 337 577 696 296 96 × 2 = 1 + 0.999 632 675 155 392 593 92;
  • 47) 0.999 632 675 155 392 593 92 × 2 = 1 + 0.999 265 350 310 785 187 84;
  • 48) 0.999 265 350 310 785 187 84 × 2 = 1 + 0.998 530 700 621 570 375 68;
  • 49) 0.998 530 700 621 570 375 68 × 2 = 1 + 0.997 061 401 243 140 751 36;
  • 50) 0.997 061 401 243 140 751 36 × 2 = 1 + 0.994 122 802 486 281 502 72;
  • 51) 0.994 122 802 486 281 502 72 × 2 = 1 + 0.988 245 604 972 563 005 44;
  • 52) 0.988 245 604 972 563 005 44 × 2 = 1 + 0.976 491 209 945 126 010 88;
  • 53) 0.976 491 209 945 126 010 88 × 2 = 1 + 0.952 982 419 890 252 021 76;
  • 54) 0.952 982 419 890 252 021 76 × 2 = 1 + 0.905 964 839 780 504 043 52;
  • 55) 0.905 964 839 780 504 043 52 × 2 = 1 + 0.811 929 679 561 008 087 04;
  • 56) 0.811 929 679 561 008 087 04 × 2 = 1 + 0.623 859 359 122 016 174 08;
  • 57) 0.623 859 359 122 016 174 08 × 2 = 1 + 0.247 718 718 244 032 348 16;
  • 58) 0.247 718 718 244 032 348 16 × 2 = 0 + 0.495 437 436 488 064 696 32;

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:


0.016 738 891 601 562 494 78(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 494 78(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:


0.016 738 891 601 562 494 78(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 -0.016 738 891 601 562 494 78 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