-0.016 738 891 601 562 496 63 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 63(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 63(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 63| = 0.016 738 891 601 562 496 63


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 63.

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 63 × 2 = 0 + 0.033 477 783 203 124 993 26;
  • 2) 0.033 477 783 203 124 993 26 × 2 = 0 + 0.066 955 566 406 249 986 52;
  • 3) 0.066 955 566 406 249 986 52 × 2 = 0 + 0.133 911 132 812 499 973 04;
  • 4) 0.133 911 132 812 499 973 04 × 2 = 0 + 0.267 822 265 624 999 946 08;
  • 5) 0.267 822 265 624 999 946 08 × 2 = 0 + 0.535 644 531 249 999 892 16;
  • 6) 0.535 644 531 249 999 892 16 × 2 = 1 + 0.071 289 062 499 999 784 32;
  • 7) 0.071 289 062 499 999 784 32 × 2 = 0 + 0.142 578 124 999 999 568 64;
  • 8) 0.142 578 124 999 999 568 64 × 2 = 0 + 0.285 156 249 999 999 137 28;
  • 9) 0.285 156 249 999 999 137 28 × 2 = 0 + 0.570 312 499 999 998 274 56;
  • 10) 0.570 312 499 999 998 274 56 × 2 = 1 + 0.140 624 999 999 996 549 12;
  • 11) 0.140 624 999 999 996 549 12 × 2 = 0 + 0.281 249 999 999 993 098 24;
  • 12) 0.281 249 999 999 993 098 24 × 2 = 0 + 0.562 499 999 999 986 196 48;
  • 13) 0.562 499 999 999 986 196 48 × 2 = 1 + 0.124 999 999 999 972 392 96;
  • 14) 0.124 999 999 999 972 392 96 × 2 = 0 + 0.249 999 999 999 944 785 92;
  • 15) 0.249 999 999 999 944 785 92 × 2 = 0 + 0.499 999 999 999 889 571 84;
  • 16) 0.499 999 999 999 889 571 84 × 2 = 0 + 0.999 999 999 999 779 143 68;
  • 17) 0.999 999 999 999 779 143 68 × 2 = 1 + 0.999 999 999 999 558 287 36;
  • 18) 0.999 999 999 999 558 287 36 × 2 = 1 + 0.999 999 999 999 116 574 72;
  • 19) 0.999 999 999 999 116 574 72 × 2 = 1 + 0.999 999 999 998 233 149 44;
  • 20) 0.999 999 999 998 233 149 44 × 2 = 1 + 0.999 999 999 996 466 298 88;
  • 21) 0.999 999 999 996 466 298 88 × 2 = 1 + 0.999 999 999 992 932 597 76;
  • 22) 0.999 999 999 992 932 597 76 × 2 = 1 + 0.999 999 999 985 865 195 52;
  • 23) 0.999 999 999 985 865 195 52 × 2 = 1 + 0.999 999 999 971 730 391 04;
  • 24) 0.999 999 999 971 730 391 04 × 2 = 1 + 0.999 999 999 943 460 782 08;
  • 25) 0.999 999 999 943 460 782 08 × 2 = 1 + 0.999 999 999 886 921 564 16;
  • 26) 0.999 999 999 886 921 564 16 × 2 = 1 + 0.999 999 999 773 843 128 32;
  • 27) 0.999 999 999 773 843 128 32 × 2 = 1 + 0.999 999 999 547 686 256 64;
  • 28) 0.999 999 999 547 686 256 64 × 2 = 1 + 0.999 999 999 095 372 513 28;
  • 29) 0.999 999 999 095 372 513 28 × 2 = 1 + 0.999 999 998 190 745 026 56;
  • 30) 0.999 999 998 190 745 026 56 × 2 = 1 + 0.999 999 996 381 490 053 12;
  • 31) 0.999 999 996 381 490 053 12 × 2 = 1 + 0.999 999 992 762 980 106 24;
  • 32) 0.999 999 992 762 980 106 24 × 2 = 1 + 0.999 999 985 525 960 212 48;
  • 33) 0.999 999 985 525 960 212 48 × 2 = 1 + 0.999 999 971 051 920 424 96;
  • 34) 0.999 999 971 051 920 424 96 × 2 = 1 + 0.999 999 942 103 840 849 92;
  • 35) 0.999 999 942 103 840 849 92 × 2 = 1 + 0.999 999 884 207 681 699 84;
  • 36) 0.999 999 884 207 681 699 84 × 2 = 1 + 0.999 999 768 415 363 399 68;
  • 37) 0.999 999 768 415 363 399 68 × 2 = 1 + 0.999 999 536 830 726 799 36;
  • 38) 0.999 999 536 830 726 799 36 × 2 = 1 + 0.999 999 073 661 453 598 72;
  • 39) 0.999 999 073 661 453 598 72 × 2 = 1 + 0.999 998 147 322 907 197 44;
  • 40) 0.999 998 147 322 907 197 44 × 2 = 1 + 0.999 996 294 645 814 394 88;
  • 41) 0.999 996 294 645 814 394 88 × 2 = 1 + 0.999 992 589 291 628 789 76;
  • 42) 0.999 992 589 291 628 789 76 × 2 = 1 + 0.999 985 178 583 257 579 52;
  • 43) 0.999 985 178 583 257 579 52 × 2 = 1 + 0.999 970 357 166 515 159 04;
  • 44) 0.999 970 357 166 515 159 04 × 2 = 1 + 0.999 940 714 333 030 318 08;
  • 45) 0.999 940 714 333 030 318 08 × 2 = 1 + 0.999 881 428 666 060 636 16;
  • 46) 0.999 881 428 666 060 636 16 × 2 = 1 + 0.999 762 857 332 121 272 32;
  • 47) 0.999 762 857 332 121 272 32 × 2 = 1 + 0.999 525 714 664 242 544 64;
  • 48) 0.999 525 714 664 242 544 64 × 2 = 1 + 0.999 051 429 328 485 089 28;
  • 49) 0.999 051 429 328 485 089 28 × 2 = 1 + 0.998 102 858 656 970 178 56;
  • 50) 0.998 102 858 656 970 178 56 × 2 = 1 + 0.996 205 717 313 940 357 12;
  • 51) 0.996 205 717 313 940 357 12 × 2 = 1 + 0.992 411 434 627 880 714 24;
  • 52) 0.992 411 434 627 880 714 24 × 2 = 1 + 0.984 822 869 255 761 428 48;
  • 53) 0.984 822 869 255 761 428 48 × 2 = 1 + 0.969 645 738 511 522 856 96;
  • 54) 0.969 645 738 511 522 856 96 × 2 = 1 + 0.939 291 477 023 045 713 92;
  • 55) 0.939 291 477 023 045 713 92 × 2 = 1 + 0.878 582 954 046 091 427 84;
  • 56) 0.878 582 954 046 091 427 84 × 2 = 1 + 0.757 165 908 092 182 855 68;
  • 57) 0.757 165 908 092 182 855 68 × 2 = 1 + 0.514 331 816 184 365 711 36;
  • 58) 0.514 331 816 184 365 711 36 × 2 = 1 + 0.028 663 632 368 731 422 72;

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 63(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 63(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 63(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 63 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