-0.016 738 891 601 562 496 530 499 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 499(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
-0.016 738 891 601 562 496 530 499(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:

|-0.016 738 891 601 562 496 530 499| = 0.016 738 891 601 562 496 530 499


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: 0.016 738 891 601 562 496 530 499.

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) 0.016 738 891 601 562 496 530 499 × 2 = 0 + 0.033 477 783 203 124 993 060 998;
  • 2) 0.033 477 783 203 124 993 060 998 × 2 = 0 + 0.066 955 566 406 249 986 121 996;
  • 3) 0.066 955 566 406 249 986 121 996 × 2 = 0 + 0.133 911 132 812 499 972 243 992;
  • 4) 0.133 911 132 812 499 972 243 992 × 2 = 0 + 0.267 822 265 624 999 944 487 984;
  • 5) 0.267 822 265 624 999 944 487 984 × 2 = 0 + 0.535 644 531 249 999 888 975 968;
  • 6) 0.535 644 531 249 999 888 975 968 × 2 = 1 + 0.071 289 062 499 999 777 951 936;
  • 7) 0.071 289 062 499 999 777 951 936 × 2 = 0 + 0.142 578 124 999 999 555 903 872;
  • 8) 0.142 578 124 999 999 555 903 872 × 2 = 0 + 0.285 156 249 999 999 111 807 744;
  • 9) 0.285 156 249 999 999 111 807 744 × 2 = 0 + 0.570 312 499 999 998 223 615 488;
  • 10) 0.570 312 499 999 998 223 615 488 × 2 = 1 + 0.140 624 999 999 996 447 230 976;
  • 11) 0.140 624 999 999 996 447 230 976 × 2 = 0 + 0.281 249 999 999 992 894 461 952;
  • 12) 0.281 249 999 999 992 894 461 952 × 2 = 0 + 0.562 499 999 999 985 788 923 904;
  • 13) 0.562 499 999 999 985 788 923 904 × 2 = 1 + 0.124 999 999 999 971 577 847 808;
  • 14) 0.124 999 999 999 971 577 847 808 × 2 = 0 + 0.249 999 999 999 943 155 695 616;
  • 15) 0.249 999 999 999 943 155 695 616 × 2 = 0 + 0.499 999 999 999 886 311 391 232;
  • 16) 0.499 999 999 999 886 311 391 232 × 2 = 0 + 0.999 999 999 999 772 622 782 464;
  • 17) 0.999 999 999 999 772 622 782 464 × 2 = 1 + 0.999 999 999 999 545 245 564 928;
  • 18) 0.999 999 999 999 545 245 564 928 × 2 = 1 + 0.999 999 999 999 090 491 129 856;
  • 19) 0.999 999 999 999 090 491 129 856 × 2 = 1 + 0.999 999 999 998 180 982 259 712;
  • 20) 0.999 999 999 998 180 982 259 712 × 2 = 1 + 0.999 999 999 996 361 964 519 424;
  • 21) 0.999 999 999 996 361 964 519 424 × 2 = 1 + 0.999 999 999 992 723 929 038 848;
  • 22) 0.999 999 999 992 723 929 038 848 × 2 = 1 + 0.999 999 999 985 447 858 077 696;
  • 23) 0.999 999 999 985 447 858 077 696 × 2 = 1 + 0.999 999 999 970 895 716 155 392;
  • 24) 0.999 999 999 970 895 716 155 392 × 2 = 1 + 0.999 999 999 941 791 432 310 784;
  • 25) 0.999 999 999 941 791 432 310 784 × 2 = 1 + 0.999 999 999 883 582 864 621 568;
  • 26) 0.999 999 999 883 582 864 621 568 × 2 = 1 + 0.999 999 999 767 165 729 243 136;
  • 27) 0.999 999 999 767 165 729 243 136 × 2 = 1 + 0.999 999 999 534 331 458 486 272;
  • 28) 0.999 999 999 534 331 458 486 272 × 2 = 1 + 0.999 999 999 068 662 916 972 544;
  • 29) 0.999 999 999 068 662 916 972 544 × 2 = 1 + 0.999 999 998 137 325 833 945 088;
  • 30) 0.999 999 998 137 325 833 945 088 × 2 = 1 + 0.999 999 996 274 651 667 890 176;
  • 31) 0.999 999 996 274 651 667 890 176 × 2 = 1 + 0.999 999 992 549 303 335 780 352;
  • 32) 0.999 999 992 549 303 335 780 352 × 2 = 1 + 0.999 999 985 098 606 671 560 704;
  • 33) 0.999 999 985 098 606 671 560 704 × 2 = 1 + 0.999 999 970 197 213 343 121 408;
  • 34) 0.999 999 970 197 213 343 121 408 × 2 = 1 + 0.999 999 940 394 426 686 242 816;
  • 35) 0.999 999 940 394 426 686 242 816 × 2 = 1 + 0.999 999 880 788 853 372 485 632;
  • 36) 0.999 999 880 788 853 372 485 632 × 2 = 1 + 0.999 999 761 577 706 744 971 264;
  • 37) 0.999 999 761 577 706 744 971 264 × 2 = 1 + 0.999 999 523 155 413 489 942 528;
  • 38) 0.999 999 523 155 413 489 942 528 × 2 = 1 + 0.999 999 046 310 826 979 885 056;
  • 39) 0.999 999 046 310 826 979 885 056 × 2 = 1 + 0.999 998 092 621 653 959 770 112;
  • 40) 0.999 998 092 621 653 959 770 112 × 2 = 1 + 0.999 996 185 243 307 919 540 224;
  • 41) 0.999 996 185 243 307 919 540 224 × 2 = 1 + 0.999 992 370 486 615 839 080 448;
  • 42) 0.999 992 370 486 615 839 080 448 × 2 = 1 + 0.999 984 740 973 231 678 160 896;
  • 43) 0.999 984 740 973 231 678 160 896 × 2 = 1 + 0.999 969 481 946 463 356 321 792;
  • 44) 0.999 969 481 946 463 356 321 792 × 2 = 1 + 0.999 938 963 892 926 712 643 584;
  • 45) 0.999 938 963 892 926 712 643 584 × 2 = 1 + 0.999 877 927 785 853 425 287 168;
  • 46) 0.999 877 927 785 853 425 287 168 × 2 = 1 + 0.999 755 855 571 706 850 574 336;
  • 47) 0.999 755 855 571 706 850 574 336 × 2 = 1 + 0.999 511 711 143 413 701 148 672;
  • 48) 0.999 511 711 143 413 701 148 672 × 2 = 1 + 0.999 023 422 286 827 402 297 344;
  • 49) 0.999 023 422 286 827 402 297 344 × 2 = 1 + 0.998 046 844 573 654 804 594 688;
  • 50) 0.998 046 844 573 654 804 594 688 × 2 = 1 + 0.996 093 689 147 309 609 189 376;
  • 51) 0.996 093 689 147 309 609 189 376 × 2 = 1 + 0.992 187 378 294 619 218 378 752;
  • 52) 0.992 187 378 294 619 218 378 752 × 2 = 1 + 0.984 374 756 589 238 436 757 504;
  • 53) 0.984 374 756 589 238 436 757 504 × 2 = 1 + 0.968 749 513 178 476 873 515 008;
  • 54) 0.968 749 513 178 476 873 515 008 × 2 = 1 + 0.937 499 026 356 953 747 030 016;
  • 55) 0.937 499 026 356 953 747 030 016 × 2 = 1 + 0.874 998 052 713 907 494 060 032;
  • 56) 0.874 998 052 713 907 494 060 032 × 2 = 1 + 0.749 996 105 427 814 988 120 064;
  • 57) 0.749 996 105 427 814 988 120 064 × 2 = 1 + 0.499 992 210 855 629 976 240 128;
  • 58) 0.499 992 210 855 629 976 240 128 × 2 = 0 + 0.999 984 421 711 259 952 480 256;

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:


0.016 738 891 601 562 496 530 499(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 499(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:


0.016 738 891 601 562 496 530 499(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 -0.016 738 891 601 562 496 530 499 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