-0.016 738 891 601 562 496 540 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 540 2(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
-0.016 738 891 601 562 496 540 2(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:

|-0.016 738 891 601 562 496 540 2| = 0.016 738 891 601 562 496 540 2


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: 0.016 738 891 601 562 496 540 2.

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) 0.016 738 891 601 562 496 540 2 × 2 = 0 + 0.033 477 783 203 124 993 080 4;
  • 2) 0.033 477 783 203 124 993 080 4 × 2 = 0 + 0.066 955 566 406 249 986 160 8;
  • 3) 0.066 955 566 406 249 986 160 8 × 2 = 0 + 0.133 911 132 812 499 972 321 6;
  • 4) 0.133 911 132 812 499 972 321 6 × 2 = 0 + 0.267 822 265 624 999 944 643 2;
  • 5) 0.267 822 265 624 999 944 643 2 × 2 = 0 + 0.535 644 531 249 999 889 286 4;
  • 6) 0.535 644 531 249 999 889 286 4 × 2 = 1 + 0.071 289 062 499 999 778 572 8;
  • 7) 0.071 289 062 499 999 778 572 8 × 2 = 0 + 0.142 578 124 999 999 557 145 6;
  • 8) 0.142 578 124 999 999 557 145 6 × 2 = 0 + 0.285 156 249 999 999 114 291 2;
  • 9) 0.285 156 249 999 999 114 291 2 × 2 = 0 + 0.570 312 499 999 998 228 582 4;
  • 10) 0.570 312 499 999 998 228 582 4 × 2 = 1 + 0.140 624 999 999 996 457 164 8;
  • 11) 0.140 624 999 999 996 457 164 8 × 2 = 0 + 0.281 249 999 999 992 914 329 6;
  • 12) 0.281 249 999 999 992 914 329 6 × 2 = 0 + 0.562 499 999 999 985 828 659 2;
  • 13) 0.562 499 999 999 985 828 659 2 × 2 = 1 + 0.124 999 999 999 971 657 318 4;
  • 14) 0.124 999 999 999 971 657 318 4 × 2 = 0 + 0.249 999 999 999 943 314 636 8;
  • 15) 0.249 999 999 999 943 314 636 8 × 2 = 0 + 0.499 999 999 999 886 629 273 6;
  • 16) 0.499 999 999 999 886 629 273 6 × 2 = 0 + 0.999 999 999 999 773 258 547 2;
  • 17) 0.999 999 999 999 773 258 547 2 × 2 = 1 + 0.999 999 999 999 546 517 094 4;
  • 18) 0.999 999 999 999 546 517 094 4 × 2 = 1 + 0.999 999 999 999 093 034 188 8;
  • 19) 0.999 999 999 999 093 034 188 8 × 2 = 1 + 0.999 999 999 998 186 068 377 6;
  • 20) 0.999 999 999 998 186 068 377 6 × 2 = 1 + 0.999 999 999 996 372 136 755 2;
  • 21) 0.999 999 999 996 372 136 755 2 × 2 = 1 + 0.999 999 999 992 744 273 510 4;
  • 22) 0.999 999 999 992 744 273 510 4 × 2 = 1 + 0.999 999 999 985 488 547 020 8;
  • 23) 0.999 999 999 985 488 547 020 8 × 2 = 1 + 0.999 999 999 970 977 094 041 6;
  • 24) 0.999 999 999 970 977 094 041 6 × 2 = 1 + 0.999 999 999 941 954 188 083 2;
  • 25) 0.999 999 999 941 954 188 083 2 × 2 = 1 + 0.999 999 999 883 908 376 166 4;
  • 26) 0.999 999 999 883 908 376 166 4 × 2 = 1 + 0.999 999 999 767 816 752 332 8;
  • 27) 0.999 999 999 767 816 752 332 8 × 2 = 1 + 0.999 999 999 535 633 504 665 6;
  • 28) 0.999 999 999 535 633 504 665 6 × 2 = 1 + 0.999 999 999 071 267 009 331 2;
  • 29) 0.999 999 999 071 267 009 331 2 × 2 = 1 + 0.999 999 998 142 534 018 662 4;
  • 30) 0.999 999 998 142 534 018 662 4 × 2 = 1 + 0.999 999 996 285 068 037 324 8;
  • 31) 0.999 999 996 285 068 037 324 8 × 2 = 1 + 0.999 999 992 570 136 074 649 6;
  • 32) 0.999 999 992 570 136 074 649 6 × 2 = 1 + 0.999 999 985 140 272 149 299 2;
  • 33) 0.999 999 985 140 272 149 299 2 × 2 = 1 + 0.999 999 970 280 544 298 598 4;
  • 34) 0.999 999 970 280 544 298 598 4 × 2 = 1 + 0.999 999 940 561 088 597 196 8;
  • 35) 0.999 999 940 561 088 597 196 8 × 2 = 1 + 0.999 999 881 122 177 194 393 6;
  • 36) 0.999 999 881 122 177 194 393 6 × 2 = 1 + 0.999 999 762 244 354 388 787 2;
  • 37) 0.999 999 762 244 354 388 787 2 × 2 = 1 + 0.999 999 524 488 708 777 574 4;
  • 38) 0.999 999 524 488 708 777 574 4 × 2 = 1 + 0.999 999 048 977 417 555 148 8;
  • 39) 0.999 999 048 977 417 555 148 8 × 2 = 1 + 0.999 998 097 954 835 110 297 6;
  • 40) 0.999 998 097 954 835 110 297 6 × 2 = 1 + 0.999 996 195 909 670 220 595 2;
  • 41) 0.999 996 195 909 670 220 595 2 × 2 = 1 + 0.999 992 391 819 340 441 190 4;
  • 42) 0.999 992 391 819 340 441 190 4 × 2 = 1 + 0.999 984 783 638 680 882 380 8;
  • 43) 0.999 984 783 638 680 882 380 8 × 2 = 1 + 0.999 969 567 277 361 764 761 6;
  • 44) 0.999 969 567 277 361 764 761 6 × 2 = 1 + 0.999 939 134 554 723 529 523 2;
  • 45) 0.999 939 134 554 723 529 523 2 × 2 = 1 + 0.999 878 269 109 447 059 046 4;
  • 46) 0.999 878 269 109 447 059 046 4 × 2 = 1 + 0.999 756 538 218 894 118 092 8;
  • 47) 0.999 756 538 218 894 118 092 8 × 2 = 1 + 0.999 513 076 437 788 236 185 6;
  • 48) 0.999 513 076 437 788 236 185 6 × 2 = 1 + 0.999 026 152 875 576 472 371 2;
  • 49) 0.999 026 152 875 576 472 371 2 × 2 = 1 + 0.998 052 305 751 152 944 742 4;
  • 50) 0.998 052 305 751 152 944 742 4 × 2 = 1 + 0.996 104 611 502 305 889 484 8;
  • 51) 0.996 104 611 502 305 889 484 8 × 2 = 1 + 0.992 209 223 004 611 778 969 6;
  • 52) 0.992 209 223 004 611 778 969 6 × 2 = 1 + 0.984 418 446 009 223 557 939 2;
  • 53) 0.984 418 446 009 223 557 939 2 × 2 = 1 + 0.968 836 892 018 447 115 878 4;
  • 54) 0.968 836 892 018 447 115 878 4 × 2 = 1 + 0.937 673 784 036 894 231 756 8;
  • 55) 0.937 673 784 036 894 231 756 8 × 2 = 1 + 0.875 347 568 073 788 463 513 6;
  • 56) 0.875 347 568 073 788 463 513 6 × 2 = 1 + 0.750 695 136 147 576 927 027 2;
  • 57) 0.750 695 136 147 576 927 027 2 × 2 = 1 + 0.501 390 272 295 153 854 054 4;
  • 58) 0.501 390 272 295 153 854 054 4 × 2 = 1 + 0.002 780 544 590 307 708 108 8;

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:


0.016 738 891 601 562 496 540 2(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 540 2(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:


0.016 738 891 601 562 496 540 2(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 -0.016 738 891 601 562 496 540 2 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