-0.016 738 891 601 562 496 530 239 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 239(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
-0.016 738 891 601 562 496 530 239(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:

|-0.016 738 891 601 562 496 530 239| = 0.016 738 891 601 562 496 530 239


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: 0.016 738 891 601 562 496 530 239.

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) 0.016 738 891 601 562 496 530 239 × 2 = 0 + 0.033 477 783 203 124 993 060 478;
  • 2) 0.033 477 783 203 124 993 060 478 × 2 = 0 + 0.066 955 566 406 249 986 120 956;
  • 3) 0.066 955 566 406 249 986 120 956 × 2 = 0 + 0.133 911 132 812 499 972 241 912;
  • 4) 0.133 911 132 812 499 972 241 912 × 2 = 0 + 0.267 822 265 624 999 944 483 824;
  • 5) 0.267 822 265 624 999 944 483 824 × 2 = 0 + 0.535 644 531 249 999 888 967 648;
  • 6) 0.535 644 531 249 999 888 967 648 × 2 = 1 + 0.071 289 062 499 999 777 935 296;
  • 7) 0.071 289 062 499 999 777 935 296 × 2 = 0 + 0.142 578 124 999 999 555 870 592;
  • 8) 0.142 578 124 999 999 555 870 592 × 2 = 0 + 0.285 156 249 999 999 111 741 184;
  • 9) 0.285 156 249 999 999 111 741 184 × 2 = 0 + 0.570 312 499 999 998 223 482 368;
  • 10) 0.570 312 499 999 998 223 482 368 × 2 = 1 + 0.140 624 999 999 996 446 964 736;
  • 11) 0.140 624 999 999 996 446 964 736 × 2 = 0 + 0.281 249 999 999 992 893 929 472;
  • 12) 0.281 249 999 999 992 893 929 472 × 2 = 0 + 0.562 499 999 999 985 787 858 944;
  • 13) 0.562 499 999 999 985 787 858 944 × 2 = 1 + 0.124 999 999 999 971 575 717 888;
  • 14) 0.124 999 999 999 971 575 717 888 × 2 = 0 + 0.249 999 999 999 943 151 435 776;
  • 15) 0.249 999 999 999 943 151 435 776 × 2 = 0 + 0.499 999 999 999 886 302 871 552;
  • 16) 0.499 999 999 999 886 302 871 552 × 2 = 0 + 0.999 999 999 999 772 605 743 104;
  • 17) 0.999 999 999 999 772 605 743 104 × 2 = 1 + 0.999 999 999 999 545 211 486 208;
  • 18) 0.999 999 999 999 545 211 486 208 × 2 = 1 + 0.999 999 999 999 090 422 972 416;
  • 19) 0.999 999 999 999 090 422 972 416 × 2 = 1 + 0.999 999 999 998 180 845 944 832;
  • 20) 0.999 999 999 998 180 845 944 832 × 2 = 1 + 0.999 999 999 996 361 691 889 664;
  • 21) 0.999 999 999 996 361 691 889 664 × 2 = 1 + 0.999 999 999 992 723 383 779 328;
  • 22) 0.999 999 999 992 723 383 779 328 × 2 = 1 + 0.999 999 999 985 446 767 558 656;
  • 23) 0.999 999 999 985 446 767 558 656 × 2 = 1 + 0.999 999 999 970 893 535 117 312;
  • 24) 0.999 999 999 970 893 535 117 312 × 2 = 1 + 0.999 999 999 941 787 070 234 624;
  • 25) 0.999 999 999 941 787 070 234 624 × 2 = 1 + 0.999 999 999 883 574 140 469 248;
  • 26) 0.999 999 999 883 574 140 469 248 × 2 = 1 + 0.999 999 999 767 148 280 938 496;
  • 27) 0.999 999 999 767 148 280 938 496 × 2 = 1 + 0.999 999 999 534 296 561 876 992;
  • 28) 0.999 999 999 534 296 561 876 992 × 2 = 1 + 0.999 999 999 068 593 123 753 984;
  • 29) 0.999 999 999 068 593 123 753 984 × 2 = 1 + 0.999 999 998 137 186 247 507 968;
  • 30) 0.999 999 998 137 186 247 507 968 × 2 = 1 + 0.999 999 996 274 372 495 015 936;
  • 31) 0.999 999 996 274 372 495 015 936 × 2 = 1 + 0.999 999 992 548 744 990 031 872;
  • 32) 0.999 999 992 548 744 990 031 872 × 2 = 1 + 0.999 999 985 097 489 980 063 744;
  • 33) 0.999 999 985 097 489 980 063 744 × 2 = 1 + 0.999 999 970 194 979 960 127 488;
  • 34) 0.999 999 970 194 979 960 127 488 × 2 = 1 + 0.999 999 940 389 959 920 254 976;
  • 35) 0.999 999 940 389 959 920 254 976 × 2 = 1 + 0.999 999 880 779 919 840 509 952;
  • 36) 0.999 999 880 779 919 840 509 952 × 2 = 1 + 0.999 999 761 559 839 681 019 904;
  • 37) 0.999 999 761 559 839 681 019 904 × 2 = 1 + 0.999 999 523 119 679 362 039 808;
  • 38) 0.999 999 523 119 679 362 039 808 × 2 = 1 + 0.999 999 046 239 358 724 079 616;
  • 39) 0.999 999 046 239 358 724 079 616 × 2 = 1 + 0.999 998 092 478 717 448 159 232;
  • 40) 0.999 998 092 478 717 448 159 232 × 2 = 1 + 0.999 996 184 957 434 896 318 464;
  • 41) 0.999 996 184 957 434 896 318 464 × 2 = 1 + 0.999 992 369 914 869 792 636 928;
  • 42) 0.999 992 369 914 869 792 636 928 × 2 = 1 + 0.999 984 739 829 739 585 273 856;
  • 43) 0.999 984 739 829 739 585 273 856 × 2 = 1 + 0.999 969 479 659 479 170 547 712;
  • 44) 0.999 969 479 659 479 170 547 712 × 2 = 1 + 0.999 938 959 318 958 341 095 424;
  • 45) 0.999 938 959 318 958 341 095 424 × 2 = 1 + 0.999 877 918 637 916 682 190 848;
  • 46) 0.999 877 918 637 916 682 190 848 × 2 = 1 + 0.999 755 837 275 833 364 381 696;
  • 47) 0.999 755 837 275 833 364 381 696 × 2 = 1 + 0.999 511 674 551 666 728 763 392;
  • 48) 0.999 511 674 551 666 728 763 392 × 2 = 1 + 0.999 023 349 103 333 457 526 784;
  • 49) 0.999 023 349 103 333 457 526 784 × 2 = 1 + 0.998 046 698 206 666 915 053 568;
  • 50) 0.998 046 698 206 666 915 053 568 × 2 = 1 + 0.996 093 396 413 333 830 107 136;
  • 51) 0.996 093 396 413 333 830 107 136 × 2 = 1 + 0.992 186 792 826 667 660 214 272;
  • 52) 0.992 186 792 826 667 660 214 272 × 2 = 1 + 0.984 373 585 653 335 320 428 544;
  • 53) 0.984 373 585 653 335 320 428 544 × 2 = 1 + 0.968 747 171 306 670 640 857 088;
  • 54) 0.968 747 171 306 670 640 857 088 × 2 = 1 + 0.937 494 342 613 341 281 714 176;
  • 55) 0.937 494 342 613 341 281 714 176 × 2 = 1 + 0.874 988 685 226 682 563 428 352;
  • 56) 0.874 988 685 226 682 563 428 352 × 2 = 1 + 0.749 977 370 453 365 126 856 704;
  • 57) 0.749 977 370 453 365 126 856 704 × 2 = 1 + 0.499 954 740 906 730 253 713 408;
  • 58) 0.499 954 740 906 730 253 713 408 × 2 = 0 + 0.999 909 481 813 460 507 426 816;

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:


0.016 738 891 601 562 496 530 239(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 239(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:


0.016 738 891 601 562 496 530 239(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 -0.016 738 891 601 562 496 530 239 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