-0.016 738 891 601 562 496 530 45 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 45(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 45(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 45| = 0.016 738 891 601 562 496 530 45


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 45.

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 45 × 2 = 0 + 0.033 477 783 203 124 993 060 9;
  • 2) 0.033 477 783 203 124 993 060 9 × 2 = 0 + 0.066 955 566 406 249 986 121 8;
  • 3) 0.066 955 566 406 249 986 121 8 × 2 = 0 + 0.133 911 132 812 499 972 243 6;
  • 4) 0.133 911 132 812 499 972 243 6 × 2 = 0 + 0.267 822 265 624 999 944 487 2;
  • 5) 0.267 822 265 624 999 944 487 2 × 2 = 0 + 0.535 644 531 249 999 888 974 4;
  • 6) 0.535 644 531 249 999 888 974 4 × 2 = 1 + 0.071 289 062 499 999 777 948 8;
  • 7) 0.071 289 062 499 999 777 948 8 × 2 = 0 + 0.142 578 124 999 999 555 897 6;
  • 8) 0.142 578 124 999 999 555 897 6 × 2 = 0 + 0.285 156 249 999 999 111 795 2;
  • 9) 0.285 156 249 999 999 111 795 2 × 2 = 0 + 0.570 312 499 999 998 223 590 4;
  • 10) 0.570 312 499 999 998 223 590 4 × 2 = 1 + 0.140 624 999 999 996 447 180 8;
  • 11) 0.140 624 999 999 996 447 180 8 × 2 = 0 + 0.281 249 999 999 992 894 361 6;
  • 12) 0.281 249 999 999 992 894 361 6 × 2 = 0 + 0.562 499 999 999 985 788 723 2;
  • 13) 0.562 499 999 999 985 788 723 2 × 2 = 1 + 0.124 999 999 999 971 577 446 4;
  • 14) 0.124 999 999 999 971 577 446 4 × 2 = 0 + 0.249 999 999 999 943 154 892 8;
  • 15) 0.249 999 999 999 943 154 892 8 × 2 = 0 + 0.499 999 999 999 886 309 785 6;
  • 16) 0.499 999 999 999 886 309 785 6 × 2 = 0 + 0.999 999 999 999 772 619 571 2;
  • 17) 0.999 999 999 999 772 619 571 2 × 2 = 1 + 0.999 999 999 999 545 239 142 4;
  • 18) 0.999 999 999 999 545 239 142 4 × 2 = 1 + 0.999 999 999 999 090 478 284 8;
  • 19) 0.999 999 999 999 090 478 284 8 × 2 = 1 + 0.999 999 999 998 180 956 569 6;
  • 20) 0.999 999 999 998 180 956 569 6 × 2 = 1 + 0.999 999 999 996 361 913 139 2;
  • 21) 0.999 999 999 996 361 913 139 2 × 2 = 1 + 0.999 999 999 992 723 826 278 4;
  • 22) 0.999 999 999 992 723 826 278 4 × 2 = 1 + 0.999 999 999 985 447 652 556 8;
  • 23) 0.999 999 999 985 447 652 556 8 × 2 = 1 + 0.999 999 999 970 895 305 113 6;
  • 24) 0.999 999 999 970 895 305 113 6 × 2 = 1 + 0.999 999 999 941 790 610 227 2;
  • 25) 0.999 999 999 941 790 610 227 2 × 2 = 1 + 0.999 999 999 883 581 220 454 4;
  • 26) 0.999 999 999 883 581 220 454 4 × 2 = 1 + 0.999 999 999 767 162 440 908 8;
  • 27) 0.999 999 999 767 162 440 908 8 × 2 = 1 + 0.999 999 999 534 324 881 817 6;
  • 28) 0.999 999 999 534 324 881 817 6 × 2 = 1 + 0.999 999 999 068 649 763 635 2;
  • 29) 0.999 999 999 068 649 763 635 2 × 2 = 1 + 0.999 999 998 137 299 527 270 4;
  • 30) 0.999 999 998 137 299 527 270 4 × 2 = 1 + 0.999 999 996 274 599 054 540 8;
  • 31) 0.999 999 996 274 599 054 540 8 × 2 = 1 + 0.999 999 992 549 198 109 081 6;
  • 32) 0.999 999 992 549 198 109 081 6 × 2 = 1 + 0.999 999 985 098 396 218 163 2;
  • 33) 0.999 999 985 098 396 218 163 2 × 2 = 1 + 0.999 999 970 196 792 436 326 4;
  • 34) 0.999 999 970 196 792 436 326 4 × 2 = 1 + 0.999 999 940 393 584 872 652 8;
  • 35) 0.999 999 940 393 584 872 652 8 × 2 = 1 + 0.999 999 880 787 169 745 305 6;
  • 36) 0.999 999 880 787 169 745 305 6 × 2 = 1 + 0.999 999 761 574 339 490 611 2;
  • 37) 0.999 999 761 574 339 490 611 2 × 2 = 1 + 0.999 999 523 148 678 981 222 4;
  • 38) 0.999 999 523 148 678 981 222 4 × 2 = 1 + 0.999 999 046 297 357 962 444 8;
  • 39) 0.999 999 046 297 357 962 444 8 × 2 = 1 + 0.999 998 092 594 715 924 889 6;
  • 40) 0.999 998 092 594 715 924 889 6 × 2 = 1 + 0.999 996 185 189 431 849 779 2;
  • 41) 0.999 996 185 189 431 849 779 2 × 2 = 1 + 0.999 992 370 378 863 699 558 4;
  • 42) 0.999 992 370 378 863 699 558 4 × 2 = 1 + 0.999 984 740 757 727 399 116 8;
  • 43) 0.999 984 740 757 727 399 116 8 × 2 = 1 + 0.999 969 481 515 454 798 233 6;
  • 44) 0.999 969 481 515 454 798 233 6 × 2 = 1 + 0.999 938 963 030 909 596 467 2;
  • 45) 0.999 938 963 030 909 596 467 2 × 2 = 1 + 0.999 877 926 061 819 192 934 4;
  • 46) 0.999 877 926 061 819 192 934 4 × 2 = 1 + 0.999 755 852 123 638 385 868 8;
  • 47) 0.999 755 852 123 638 385 868 8 × 2 = 1 + 0.999 511 704 247 276 771 737 6;
  • 48) 0.999 511 704 247 276 771 737 6 × 2 = 1 + 0.999 023 408 494 553 543 475 2;
  • 49) 0.999 023 408 494 553 543 475 2 × 2 = 1 + 0.998 046 816 989 107 086 950 4;
  • 50) 0.998 046 816 989 107 086 950 4 × 2 = 1 + 0.996 093 633 978 214 173 900 8;
  • 51) 0.996 093 633 978 214 173 900 8 × 2 = 1 + 0.992 187 267 956 428 347 801 6;
  • 52) 0.992 187 267 956 428 347 801 6 × 2 = 1 + 0.984 374 535 912 856 695 603 2;
  • 53) 0.984 374 535 912 856 695 603 2 × 2 = 1 + 0.968 749 071 825 713 391 206 4;
  • 54) 0.968 749 071 825 713 391 206 4 × 2 = 1 + 0.937 498 143 651 426 782 412 8;
  • 55) 0.937 498 143 651 426 782 412 8 × 2 = 1 + 0.874 996 287 302 853 564 825 6;
  • 56) 0.874 996 287 302 853 564 825 6 × 2 = 1 + 0.749 992 574 605 707 129 651 2;
  • 57) 0.749 992 574 605 707 129 651 2 × 2 = 1 + 0.499 985 149 211 414 259 302 4;
  • 58) 0.499 985 149 211 414 259 302 4 × 2 = 0 + 0.999 970 298 422 828 518 604 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:


0.016 738 891 601 562 496 530 45(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 45(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 45(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 45 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