-0.016 738 891 601 562 496 152 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 152(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
-0.016 738 891 601 562 496 152(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:

|-0.016 738 891 601 562 496 152| = 0.016 738 891 601 562 496 152


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: 0.016 738 891 601 562 496 152.

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) 0.016 738 891 601 562 496 152 × 2 = 0 + 0.033 477 783 203 124 992 304;
  • 2) 0.033 477 783 203 124 992 304 × 2 = 0 + 0.066 955 566 406 249 984 608;
  • 3) 0.066 955 566 406 249 984 608 × 2 = 0 + 0.133 911 132 812 499 969 216;
  • 4) 0.133 911 132 812 499 969 216 × 2 = 0 + 0.267 822 265 624 999 938 432;
  • 5) 0.267 822 265 624 999 938 432 × 2 = 0 + 0.535 644 531 249 999 876 864;
  • 6) 0.535 644 531 249 999 876 864 × 2 = 1 + 0.071 289 062 499 999 753 728;
  • 7) 0.071 289 062 499 999 753 728 × 2 = 0 + 0.142 578 124 999 999 507 456;
  • 8) 0.142 578 124 999 999 507 456 × 2 = 0 + 0.285 156 249 999 999 014 912;
  • 9) 0.285 156 249 999 999 014 912 × 2 = 0 + 0.570 312 499 999 998 029 824;
  • 10) 0.570 312 499 999 998 029 824 × 2 = 1 + 0.140 624 999 999 996 059 648;
  • 11) 0.140 624 999 999 996 059 648 × 2 = 0 + 0.281 249 999 999 992 119 296;
  • 12) 0.281 249 999 999 992 119 296 × 2 = 0 + 0.562 499 999 999 984 238 592;
  • 13) 0.562 499 999 999 984 238 592 × 2 = 1 + 0.124 999 999 999 968 477 184;
  • 14) 0.124 999 999 999 968 477 184 × 2 = 0 + 0.249 999 999 999 936 954 368;
  • 15) 0.249 999 999 999 936 954 368 × 2 = 0 + 0.499 999 999 999 873 908 736;
  • 16) 0.499 999 999 999 873 908 736 × 2 = 0 + 0.999 999 999 999 747 817 472;
  • 17) 0.999 999 999 999 747 817 472 × 2 = 1 + 0.999 999 999 999 495 634 944;
  • 18) 0.999 999 999 999 495 634 944 × 2 = 1 + 0.999 999 999 998 991 269 888;
  • 19) 0.999 999 999 998 991 269 888 × 2 = 1 + 0.999 999 999 997 982 539 776;
  • 20) 0.999 999 999 997 982 539 776 × 2 = 1 + 0.999 999 999 995 965 079 552;
  • 21) 0.999 999 999 995 965 079 552 × 2 = 1 + 0.999 999 999 991 930 159 104;
  • 22) 0.999 999 999 991 930 159 104 × 2 = 1 + 0.999 999 999 983 860 318 208;
  • 23) 0.999 999 999 983 860 318 208 × 2 = 1 + 0.999 999 999 967 720 636 416;
  • 24) 0.999 999 999 967 720 636 416 × 2 = 1 + 0.999 999 999 935 441 272 832;
  • 25) 0.999 999 999 935 441 272 832 × 2 = 1 + 0.999 999 999 870 882 545 664;
  • 26) 0.999 999 999 870 882 545 664 × 2 = 1 + 0.999 999 999 741 765 091 328;
  • 27) 0.999 999 999 741 765 091 328 × 2 = 1 + 0.999 999 999 483 530 182 656;
  • 28) 0.999 999 999 483 530 182 656 × 2 = 1 + 0.999 999 998 967 060 365 312;
  • 29) 0.999 999 998 967 060 365 312 × 2 = 1 + 0.999 999 997 934 120 730 624;
  • 30) 0.999 999 997 934 120 730 624 × 2 = 1 + 0.999 999 995 868 241 461 248;
  • 31) 0.999 999 995 868 241 461 248 × 2 = 1 + 0.999 999 991 736 482 922 496;
  • 32) 0.999 999 991 736 482 922 496 × 2 = 1 + 0.999 999 983 472 965 844 992;
  • 33) 0.999 999 983 472 965 844 992 × 2 = 1 + 0.999 999 966 945 931 689 984;
  • 34) 0.999 999 966 945 931 689 984 × 2 = 1 + 0.999 999 933 891 863 379 968;
  • 35) 0.999 999 933 891 863 379 968 × 2 = 1 + 0.999 999 867 783 726 759 936;
  • 36) 0.999 999 867 783 726 759 936 × 2 = 1 + 0.999 999 735 567 453 519 872;
  • 37) 0.999 999 735 567 453 519 872 × 2 = 1 + 0.999 999 471 134 907 039 744;
  • 38) 0.999 999 471 134 907 039 744 × 2 = 1 + 0.999 998 942 269 814 079 488;
  • 39) 0.999 998 942 269 814 079 488 × 2 = 1 + 0.999 997 884 539 628 158 976;
  • 40) 0.999 997 884 539 628 158 976 × 2 = 1 + 0.999 995 769 079 256 317 952;
  • 41) 0.999 995 769 079 256 317 952 × 2 = 1 + 0.999 991 538 158 512 635 904;
  • 42) 0.999 991 538 158 512 635 904 × 2 = 1 + 0.999 983 076 317 025 271 808;
  • 43) 0.999 983 076 317 025 271 808 × 2 = 1 + 0.999 966 152 634 050 543 616;
  • 44) 0.999 966 152 634 050 543 616 × 2 = 1 + 0.999 932 305 268 101 087 232;
  • 45) 0.999 932 305 268 101 087 232 × 2 = 1 + 0.999 864 610 536 202 174 464;
  • 46) 0.999 864 610 536 202 174 464 × 2 = 1 + 0.999 729 221 072 404 348 928;
  • 47) 0.999 729 221 072 404 348 928 × 2 = 1 + 0.999 458 442 144 808 697 856;
  • 48) 0.999 458 442 144 808 697 856 × 2 = 1 + 0.998 916 884 289 617 395 712;
  • 49) 0.998 916 884 289 617 395 712 × 2 = 1 + 0.997 833 768 579 234 791 424;
  • 50) 0.997 833 768 579 234 791 424 × 2 = 1 + 0.995 667 537 158 469 582 848;
  • 51) 0.995 667 537 158 469 582 848 × 2 = 1 + 0.991 335 074 316 939 165 696;
  • 52) 0.991 335 074 316 939 165 696 × 2 = 1 + 0.982 670 148 633 878 331 392;
  • 53) 0.982 670 148 633 878 331 392 × 2 = 1 + 0.965 340 297 267 756 662 784;
  • 54) 0.965 340 297 267 756 662 784 × 2 = 1 + 0.930 680 594 535 513 325 568;
  • 55) 0.930 680 594 535 513 325 568 × 2 = 1 + 0.861 361 189 071 026 651 136;
  • 56) 0.861 361 189 071 026 651 136 × 2 = 1 + 0.722 722 378 142 053 302 272;
  • 57) 0.722 722 378 142 053 302 272 × 2 = 1 + 0.445 444 756 284 106 604 544;
  • 58) 0.445 444 756 284 106 604 544 × 2 = 0 + 0.890 889 512 568 213 209 088;

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:


0.016 738 891 601 562 496 152(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 152(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:


0.016 738 891 601 562 496 152(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 -0.016 738 891 601 562 496 152 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