-0.016 738 891 601 562 496 809 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 809(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
-0.016 738 891 601 562 496 809(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:

|-0.016 738 891 601 562 496 809| = 0.016 738 891 601 562 496 809


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: 0.016 738 891 601 562 496 809.

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) 0.016 738 891 601 562 496 809 × 2 = 0 + 0.033 477 783 203 124 993 618;
  • 2) 0.033 477 783 203 124 993 618 × 2 = 0 + 0.066 955 566 406 249 987 236;
  • 3) 0.066 955 566 406 249 987 236 × 2 = 0 + 0.133 911 132 812 499 974 472;
  • 4) 0.133 911 132 812 499 974 472 × 2 = 0 + 0.267 822 265 624 999 948 944;
  • 5) 0.267 822 265 624 999 948 944 × 2 = 0 + 0.535 644 531 249 999 897 888;
  • 6) 0.535 644 531 249 999 897 888 × 2 = 1 + 0.071 289 062 499 999 795 776;
  • 7) 0.071 289 062 499 999 795 776 × 2 = 0 + 0.142 578 124 999 999 591 552;
  • 8) 0.142 578 124 999 999 591 552 × 2 = 0 + 0.285 156 249 999 999 183 104;
  • 9) 0.285 156 249 999 999 183 104 × 2 = 0 + 0.570 312 499 999 998 366 208;
  • 10) 0.570 312 499 999 998 366 208 × 2 = 1 + 0.140 624 999 999 996 732 416;
  • 11) 0.140 624 999 999 996 732 416 × 2 = 0 + 0.281 249 999 999 993 464 832;
  • 12) 0.281 249 999 999 993 464 832 × 2 = 0 + 0.562 499 999 999 986 929 664;
  • 13) 0.562 499 999 999 986 929 664 × 2 = 1 + 0.124 999 999 999 973 859 328;
  • 14) 0.124 999 999 999 973 859 328 × 2 = 0 + 0.249 999 999 999 947 718 656;
  • 15) 0.249 999 999 999 947 718 656 × 2 = 0 + 0.499 999 999 999 895 437 312;
  • 16) 0.499 999 999 999 895 437 312 × 2 = 0 + 0.999 999 999 999 790 874 624;
  • 17) 0.999 999 999 999 790 874 624 × 2 = 1 + 0.999 999 999 999 581 749 248;
  • 18) 0.999 999 999 999 581 749 248 × 2 = 1 + 0.999 999 999 999 163 498 496;
  • 19) 0.999 999 999 999 163 498 496 × 2 = 1 + 0.999 999 999 998 326 996 992;
  • 20) 0.999 999 999 998 326 996 992 × 2 = 1 + 0.999 999 999 996 653 993 984;
  • 21) 0.999 999 999 996 653 993 984 × 2 = 1 + 0.999 999 999 993 307 987 968;
  • 22) 0.999 999 999 993 307 987 968 × 2 = 1 + 0.999 999 999 986 615 975 936;
  • 23) 0.999 999 999 986 615 975 936 × 2 = 1 + 0.999 999 999 973 231 951 872;
  • 24) 0.999 999 999 973 231 951 872 × 2 = 1 + 0.999 999 999 946 463 903 744;
  • 25) 0.999 999 999 946 463 903 744 × 2 = 1 + 0.999 999 999 892 927 807 488;
  • 26) 0.999 999 999 892 927 807 488 × 2 = 1 + 0.999 999 999 785 855 614 976;
  • 27) 0.999 999 999 785 855 614 976 × 2 = 1 + 0.999 999 999 571 711 229 952;
  • 28) 0.999 999 999 571 711 229 952 × 2 = 1 + 0.999 999 999 143 422 459 904;
  • 29) 0.999 999 999 143 422 459 904 × 2 = 1 + 0.999 999 998 286 844 919 808;
  • 30) 0.999 999 998 286 844 919 808 × 2 = 1 + 0.999 999 996 573 689 839 616;
  • 31) 0.999 999 996 573 689 839 616 × 2 = 1 + 0.999 999 993 147 379 679 232;
  • 32) 0.999 999 993 147 379 679 232 × 2 = 1 + 0.999 999 986 294 759 358 464;
  • 33) 0.999 999 986 294 759 358 464 × 2 = 1 + 0.999 999 972 589 518 716 928;
  • 34) 0.999 999 972 589 518 716 928 × 2 = 1 + 0.999 999 945 179 037 433 856;
  • 35) 0.999 999 945 179 037 433 856 × 2 = 1 + 0.999 999 890 358 074 867 712;
  • 36) 0.999 999 890 358 074 867 712 × 2 = 1 + 0.999 999 780 716 149 735 424;
  • 37) 0.999 999 780 716 149 735 424 × 2 = 1 + 0.999 999 561 432 299 470 848;
  • 38) 0.999 999 561 432 299 470 848 × 2 = 1 + 0.999 999 122 864 598 941 696;
  • 39) 0.999 999 122 864 598 941 696 × 2 = 1 + 0.999 998 245 729 197 883 392;
  • 40) 0.999 998 245 729 197 883 392 × 2 = 1 + 0.999 996 491 458 395 766 784;
  • 41) 0.999 996 491 458 395 766 784 × 2 = 1 + 0.999 992 982 916 791 533 568;
  • 42) 0.999 992 982 916 791 533 568 × 2 = 1 + 0.999 985 965 833 583 067 136;
  • 43) 0.999 985 965 833 583 067 136 × 2 = 1 + 0.999 971 931 667 166 134 272;
  • 44) 0.999 971 931 667 166 134 272 × 2 = 1 + 0.999 943 863 334 332 268 544;
  • 45) 0.999 943 863 334 332 268 544 × 2 = 1 + 0.999 887 726 668 664 537 088;
  • 46) 0.999 887 726 668 664 537 088 × 2 = 1 + 0.999 775 453 337 329 074 176;
  • 47) 0.999 775 453 337 329 074 176 × 2 = 1 + 0.999 550 906 674 658 148 352;
  • 48) 0.999 550 906 674 658 148 352 × 2 = 1 + 0.999 101 813 349 316 296 704;
  • 49) 0.999 101 813 349 316 296 704 × 2 = 1 + 0.998 203 626 698 632 593 408;
  • 50) 0.998 203 626 698 632 593 408 × 2 = 1 + 0.996 407 253 397 265 186 816;
  • 51) 0.996 407 253 397 265 186 816 × 2 = 1 + 0.992 814 506 794 530 373 632;
  • 52) 0.992 814 506 794 530 373 632 × 2 = 1 + 0.985 629 013 589 060 747 264;
  • 53) 0.985 629 013 589 060 747 264 × 2 = 1 + 0.971 258 027 178 121 494 528;
  • 54) 0.971 258 027 178 121 494 528 × 2 = 1 + 0.942 516 054 356 242 989 056;
  • 55) 0.942 516 054 356 242 989 056 × 2 = 1 + 0.885 032 108 712 485 978 112;
  • 56) 0.885 032 108 712 485 978 112 × 2 = 1 + 0.770 064 217 424 971 956 224;
  • 57) 0.770 064 217 424 971 956 224 × 2 = 1 + 0.540 128 434 849 943 912 448;
  • 58) 0.540 128 434 849 943 912 448 × 2 = 1 + 0.080 256 869 699 887 824 896;

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:


0.016 738 891 601 562 496 809(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 809(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:


0.016 738 891 601 562 496 809(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 -0.016 738 891 601 562 496 809 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