-0.016 738 891 601 562 496 408 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 408(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
-0.016 738 891 601 562 496 408(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:

|-0.016 738 891 601 562 496 408| = 0.016 738 891 601 562 496 408


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: 0.016 738 891 601 562 496 408.

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) 0.016 738 891 601 562 496 408 × 2 = 0 + 0.033 477 783 203 124 992 816;
  • 2) 0.033 477 783 203 124 992 816 × 2 = 0 + 0.066 955 566 406 249 985 632;
  • 3) 0.066 955 566 406 249 985 632 × 2 = 0 + 0.133 911 132 812 499 971 264;
  • 4) 0.133 911 132 812 499 971 264 × 2 = 0 + 0.267 822 265 624 999 942 528;
  • 5) 0.267 822 265 624 999 942 528 × 2 = 0 + 0.535 644 531 249 999 885 056;
  • 6) 0.535 644 531 249 999 885 056 × 2 = 1 + 0.071 289 062 499 999 770 112;
  • 7) 0.071 289 062 499 999 770 112 × 2 = 0 + 0.142 578 124 999 999 540 224;
  • 8) 0.142 578 124 999 999 540 224 × 2 = 0 + 0.285 156 249 999 999 080 448;
  • 9) 0.285 156 249 999 999 080 448 × 2 = 0 + 0.570 312 499 999 998 160 896;
  • 10) 0.570 312 499 999 998 160 896 × 2 = 1 + 0.140 624 999 999 996 321 792;
  • 11) 0.140 624 999 999 996 321 792 × 2 = 0 + 0.281 249 999 999 992 643 584;
  • 12) 0.281 249 999 999 992 643 584 × 2 = 0 + 0.562 499 999 999 985 287 168;
  • 13) 0.562 499 999 999 985 287 168 × 2 = 1 + 0.124 999 999 999 970 574 336;
  • 14) 0.124 999 999 999 970 574 336 × 2 = 0 + 0.249 999 999 999 941 148 672;
  • 15) 0.249 999 999 999 941 148 672 × 2 = 0 + 0.499 999 999 999 882 297 344;
  • 16) 0.499 999 999 999 882 297 344 × 2 = 0 + 0.999 999 999 999 764 594 688;
  • 17) 0.999 999 999 999 764 594 688 × 2 = 1 + 0.999 999 999 999 529 189 376;
  • 18) 0.999 999 999 999 529 189 376 × 2 = 1 + 0.999 999 999 999 058 378 752;
  • 19) 0.999 999 999 999 058 378 752 × 2 = 1 + 0.999 999 999 998 116 757 504;
  • 20) 0.999 999 999 998 116 757 504 × 2 = 1 + 0.999 999 999 996 233 515 008;
  • 21) 0.999 999 999 996 233 515 008 × 2 = 1 + 0.999 999 999 992 467 030 016;
  • 22) 0.999 999 999 992 467 030 016 × 2 = 1 + 0.999 999 999 984 934 060 032;
  • 23) 0.999 999 999 984 934 060 032 × 2 = 1 + 0.999 999 999 969 868 120 064;
  • 24) 0.999 999 999 969 868 120 064 × 2 = 1 + 0.999 999 999 939 736 240 128;
  • 25) 0.999 999 999 939 736 240 128 × 2 = 1 + 0.999 999 999 879 472 480 256;
  • 26) 0.999 999 999 879 472 480 256 × 2 = 1 + 0.999 999 999 758 944 960 512;
  • 27) 0.999 999 999 758 944 960 512 × 2 = 1 + 0.999 999 999 517 889 921 024;
  • 28) 0.999 999 999 517 889 921 024 × 2 = 1 + 0.999 999 999 035 779 842 048;
  • 29) 0.999 999 999 035 779 842 048 × 2 = 1 + 0.999 999 998 071 559 684 096;
  • 30) 0.999 999 998 071 559 684 096 × 2 = 1 + 0.999 999 996 143 119 368 192;
  • 31) 0.999 999 996 143 119 368 192 × 2 = 1 + 0.999 999 992 286 238 736 384;
  • 32) 0.999 999 992 286 238 736 384 × 2 = 1 + 0.999 999 984 572 477 472 768;
  • 33) 0.999 999 984 572 477 472 768 × 2 = 1 + 0.999 999 969 144 954 945 536;
  • 34) 0.999 999 969 144 954 945 536 × 2 = 1 + 0.999 999 938 289 909 891 072;
  • 35) 0.999 999 938 289 909 891 072 × 2 = 1 + 0.999 999 876 579 819 782 144;
  • 36) 0.999 999 876 579 819 782 144 × 2 = 1 + 0.999 999 753 159 639 564 288;
  • 37) 0.999 999 753 159 639 564 288 × 2 = 1 + 0.999 999 506 319 279 128 576;
  • 38) 0.999 999 506 319 279 128 576 × 2 = 1 + 0.999 999 012 638 558 257 152;
  • 39) 0.999 999 012 638 558 257 152 × 2 = 1 + 0.999 998 025 277 116 514 304;
  • 40) 0.999 998 025 277 116 514 304 × 2 = 1 + 0.999 996 050 554 233 028 608;
  • 41) 0.999 996 050 554 233 028 608 × 2 = 1 + 0.999 992 101 108 466 057 216;
  • 42) 0.999 992 101 108 466 057 216 × 2 = 1 + 0.999 984 202 216 932 114 432;
  • 43) 0.999 984 202 216 932 114 432 × 2 = 1 + 0.999 968 404 433 864 228 864;
  • 44) 0.999 968 404 433 864 228 864 × 2 = 1 + 0.999 936 808 867 728 457 728;
  • 45) 0.999 936 808 867 728 457 728 × 2 = 1 + 0.999 873 617 735 456 915 456;
  • 46) 0.999 873 617 735 456 915 456 × 2 = 1 + 0.999 747 235 470 913 830 912;
  • 47) 0.999 747 235 470 913 830 912 × 2 = 1 + 0.999 494 470 941 827 661 824;
  • 48) 0.999 494 470 941 827 661 824 × 2 = 1 + 0.998 988 941 883 655 323 648;
  • 49) 0.998 988 941 883 655 323 648 × 2 = 1 + 0.997 977 883 767 310 647 296;
  • 50) 0.997 977 883 767 310 647 296 × 2 = 1 + 0.995 955 767 534 621 294 592;
  • 51) 0.995 955 767 534 621 294 592 × 2 = 1 + 0.991 911 535 069 242 589 184;
  • 52) 0.991 911 535 069 242 589 184 × 2 = 1 + 0.983 823 070 138 485 178 368;
  • 53) 0.983 823 070 138 485 178 368 × 2 = 1 + 0.967 646 140 276 970 356 736;
  • 54) 0.967 646 140 276 970 356 736 × 2 = 1 + 0.935 292 280 553 940 713 472;
  • 55) 0.935 292 280 553 940 713 472 × 2 = 1 + 0.870 584 561 107 881 426 944;
  • 56) 0.870 584 561 107 881 426 944 × 2 = 1 + 0.741 169 122 215 762 853 888;
  • 57) 0.741 169 122 215 762 853 888 × 2 = 1 + 0.482 338 244 431 525 707 776;
  • 58) 0.482 338 244 431 525 707 776 × 2 = 0 + 0.964 676 488 863 051 415 552;

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:


0.016 738 891 601 562 496 408(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 408(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:


0.016 738 891 601 562 496 408(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 -0.016 738 891 601 562 496 408 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