-0.016 738 891 601 562 496 624 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 624(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
-0.016 738 891 601 562 496 624(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:

|-0.016 738 891 601 562 496 624| = 0.016 738 891 601 562 496 624


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: 0.016 738 891 601 562 496 624.

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) 0.016 738 891 601 562 496 624 × 2 = 0 + 0.033 477 783 203 124 993 248;
  • 2) 0.033 477 783 203 124 993 248 × 2 = 0 + 0.066 955 566 406 249 986 496;
  • 3) 0.066 955 566 406 249 986 496 × 2 = 0 + 0.133 911 132 812 499 972 992;
  • 4) 0.133 911 132 812 499 972 992 × 2 = 0 + 0.267 822 265 624 999 945 984;
  • 5) 0.267 822 265 624 999 945 984 × 2 = 0 + 0.535 644 531 249 999 891 968;
  • 6) 0.535 644 531 249 999 891 968 × 2 = 1 + 0.071 289 062 499 999 783 936;
  • 7) 0.071 289 062 499 999 783 936 × 2 = 0 + 0.142 578 124 999 999 567 872;
  • 8) 0.142 578 124 999 999 567 872 × 2 = 0 + 0.285 156 249 999 999 135 744;
  • 9) 0.285 156 249 999 999 135 744 × 2 = 0 + 0.570 312 499 999 998 271 488;
  • 10) 0.570 312 499 999 998 271 488 × 2 = 1 + 0.140 624 999 999 996 542 976;
  • 11) 0.140 624 999 999 996 542 976 × 2 = 0 + 0.281 249 999 999 993 085 952;
  • 12) 0.281 249 999 999 993 085 952 × 2 = 0 + 0.562 499 999 999 986 171 904;
  • 13) 0.562 499 999 999 986 171 904 × 2 = 1 + 0.124 999 999 999 972 343 808;
  • 14) 0.124 999 999 999 972 343 808 × 2 = 0 + 0.249 999 999 999 944 687 616;
  • 15) 0.249 999 999 999 944 687 616 × 2 = 0 + 0.499 999 999 999 889 375 232;
  • 16) 0.499 999 999 999 889 375 232 × 2 = 0 + 0.999 999 999 999 778 750 464;
  • 17) 0.999 999 999 999 778 750 464 × 2 = 1 + 0.999 999 999 999 557 500 928;
  • 18) 0.999 999 999 999 557 500 928 × 2 = 1 + 0.999 999 999 999 115 001 856;
  • 19) 0.999 999 999 999 115 001 856 × 2 = 1 + 0.999 999 999 998 230 003 712;
  • 20) 0.999 999 999 998 230 003 712 × 2 = 1 + 0.999 999 999 996 460 007 424;
  • 21) 0.999 999 999 996 460 007 424 × 2 = 1 + 0.999 999 999 992 920 014 848;
  • 22) 0.999 999 999 992 920 014 848 × 2 = 1 + 0.999 999 999 985 840 029 696;
  • 23) 0.999 999 999 985 840 029 696 × 2 = 1 + 0.999 999 999 971 680 059 392;
  • 24) 0.999 999 999 971 680 059 392 × 2 = 1 + 0.999 999 999 943 360 118 784;
  • 25) 0.999 999 999 943 360 118 784 × 2 = 1 + 0.999 999 999 886 720 237 568;
  • 26) 0.999 999 999 886 720 237 568 × 2 = 1 + 0.999 999 999 773 440 475 136;
  • 27) 0.999 999 999 773 440 475 136 × 2 = 1 + 0.999 999 999 546 880 950 272;
  • 28) 0.999 999 999 546 880 950 272 × 2 = 1 + 0.999 999 999 093 761 900 544;
  • 29) 0.999 999 999 093 761 900 544 × 2 = 1 + 0.999 999 998 187 523 801 088;
  • 30) 0.999 999 998 187 523 801 088 × 2 = 1 + 0.999 999 996 375 047 602 176;
  • 31) 0.999 999 996 375 047 602 176 × 2 = 1 + 0.999 999 992 750 095 204 352;
  • 32) 0.999 999 992 750 095 204 352 × 2 = 1 + 0.999 999 985 500 190 408 704;
  • 33) 0.999 999 985 500 190 408 704 × 2 = 1 + 0.999 999 971 000 380 817 408;
  • 34) 0.999 999 971 000 380 817 408 × 2 = 1 + 0.999 999 942 000 761 634 816;
  • 35) 0.999 999 942 000 761 634 816 × 2 = 1 + 0.999 999 884 001 523 269 632;
  • 36) 0.999 999 884 001 523 269 632 × 2 = 1 + 0.999 999 768 003 046 539 264;
  • 37) 0.999 999 768 003 046 539 264 × 2 = 1 + 0.999 999 536 006 093 078 528;
  • 38) 0.999 999 536 006 093 078 528 × 2 = 1 + 0.999 999 072 012 186 157 056;
  • 39) 0.999 999 072 012 186 157 056 × 2 = 1 + 0.999 998 144 024 372 314 112;
  • 40) 0.999 998 144 024 372 314 112 × 2 = 1 + 0.999 996 288 048 744 628 224;
  • 41) 0.999 996 288 048 744 628 224 × 2 = 1 + 0.999 992 576 097 489 256 448;
  • 42) 0.999 992 576 097 489 256 448 × 2 = 1 + 0.999 985 152 194 978 512 896;
  • 43) 0.999 985 152 194 978 512 896 × 2 = 1 + 0.999 970 304 389 957 025 792;
  • 44) 0.999 970 304 389 957 025 792 × 2 = 1 + 0.999 940 608 779 914 051 584;
  • 45) 0.999 940 608 779 914 051 584 × 2 = 1 + 0.999 881 217 559 828 103 168;
  • 46) 0.999 881 217 559 828 103 168 × 2 = 1 + 0.999 762 435 119 656 206 336;
  • 47) 0.999 762 435 119 656 206 336 × 2 = 1 + 0.999 524 870 239 312 412 672;
  • 48) 0.999 524 870 239 312 412 672 × 2 = 1 + 0.999 049 740 478 624 825 344;
  • 49) 0.999 049 740 478 624 825 344 × 2 = 1 + 0.998 099 480 957 249 650 688;
  • 50) 0.998 099 480 957 249 650 688 × 2 = 1 + 0.996 198 961 914 499 301 376;
  • 51) 0.996 198 961 914 499 301 376 × 2 = 1 + 0.992 397 923 828 998 602 752;
  • 52) 0.992 397 923 828 998 602 752 × 2 = 1 + 0.984 795 847 657 997 205 504;
  • 53) 0.984 795 847 657 997 205 504 × 2 = 1 + 0.969 591 695 315 994 411 008;
  • 54) 0.969 591 695 315 994 411 008 × 2 = 1 + 0.939 183 390 631 988 822 016;
  • 55) 0.939 183 390 631 988 822 016 × 2 = 1 + 0.878 366 781 263 977 644 032;
  • 56) 0.878 366 781 263 977 644 032 × 2 = 1 + 0.756 733 562 527 955 288 064;
  • 57) 0.756 733 562 527 955 288 064 × 2 = 1 + 0.513 467 125 055 910 576 128;
  • 58) 0.513 467 125 055 910 576 128 × 2 = 1 + 0.026 934 250 111 821 152 256;

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:


0.016 738 891 601 562 496 624(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 624(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:


0.016 738 891 601 562 496 624(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 -0.016 738 891 601 562 496 624 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