-0.016 738 891 601 562 496 587 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 587(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
-0.016 738 891 601 562 496 587(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:

|-0.016 738 891 601 562 496 587| = 0.016 738 891 601 562 496 587


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: 0.016 738 891 601 562 496 587.

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) 0.016 738 891 601 562 496 587 × 2 = 0 + 0.033 477 783 203 124 993 174;
  • 2) 0.033 477 783 203 124 993 174 × 2 = 0 + 0.066 955 566 406 249 986 348;
  • 3) 0.066 955 566 406 249 986 348 × 2 = 0 + 0.133 911 132 812 499 972 696;
  • 4) 0.133 911 132 812 499 972 696 × 2 = 0 + 0.267 822 265 624 999 945 392;
  • 5) 0.267 822 265 624 999 945 392 × 2 = 0 + 0.535 644 531 249 999 890 784;
  • 6) 0.535 644 531 249 999 890 784 × 2 = 1 + 0.071 289 062 499 999 781 568;
  • 7) 0.071 289 062 499 999 781 568 × 2 = 0 + 0.142 578 124 999 999 563 136;
  • 8) 0.142 578 124 999 999 563 136 × 2 = 0 + 0.285 156 249 999 999 126 272;
  • 9) 0.285 156 249 999 999 126 272 × 2 = 0 + 0.570 312 499 999 998 252 544;
  • 10) 0.570 312 499 999 998 252 544 × 2 = 1 + 0.140 624 999 999 996 505 088;
  • 11) 0.140 624 999 999 996 505 088 × 2 = 0 + 0.281 249 999 999 993 010 176;
  • 12) 0.281 249 999 999 993 010 176 × 2 = 0 + 0.562 499 999 999 986 020 352;
  • 13) 0.562 499 999 999 986 020 352 × 2 = 1 + 0.124 999 999 999 972 040 704;
  • 14) 0.124 999 999 999 972 040 704 × 2 = 0 + 0.249 999 999 999 944 081 408;
  • 15) 0.249 999 999 999 944 081 408 × 2 = 0 + 0.499 999 999 999 888 162 816;
  • 16) 0.499 999 999 999 888 162 816 × 2 = 0 + 0.999 999 999 999 776 325 632;
  • 17) 0.999 999 999 999 776 325 632 × 2 = 1 + 0.999 999 999 999 552 651 264;
  • 18) 0.999 999 999 999 552 651 264 × 2 = 1 + 0.999 999 999 999 105 302 528;
  • 19) 0.999 999 999 999 105 302 528 × 2 = 1 + 0.999 999 999 998 210 605 056;
  • 20) 0.999 999 999 998 210 605 056 × 2 = 1 + 0.999 999 999 996 421 210 112;
  • 21) 0.999 999 999 996 421 210 112 × 2 = 1 + 0.999 999 999 992 842 420 224;
  • 22) 0.999 999 999 992 842 420 224 × 2 = 1 + 0.999 999 999 985 684 840 448;
  • 23) 0.999 999 999 985 684 840 448 × 2 = 1 + 0.999 999 999 971 369 680 896;
  • 24) 0.999 999 999 971 369 680 896 × 2 = 1 + 0.999 999 999 942 739 361 792;
  • 25) 0.999 999 999 942 739 361 792 × 2 = 1 + 0.999 999 999 885 478 723 584;
  • 26) 0.999 999 999 885 478 723 584 × 2 = 1 + 0.999 999 999 770 957 447 168;
  • 27) 0.999 999 999 770 957 447 168 × 2 = 1 + 0.999 999 999 541 914 894 336;
  • 28) 0.999 999 999 541 914 894 336 × 2 = 1 + 0.999 999 999 083 829 788 672;
  • 29) 0.999 999 999 083 829 788 672 × 2 = 1 + 0.999 999 998 167 659 577 344;
  • 30) 0.999 999 998 167 659 577 344 × 2 = 1 + 0.999 999 996 335 319 154 688;
  • 31) 0.999 999 996 335 319 154 688 × 2 = 1 + 0.999 999 992 670 638 309 376;
  • 32) 0.999 999 992 670 638 309 376 × 2 = 1 + 0.999 999 985 341 276 618 752;
  • 33) 0.999 999 985 341 276 618 752 × 2 = 1 + 0.999 999 970 682 553 237 504;
  • 34) 0.999 999 970 682 553 237 504 × 2 = 1 + 0.999 999 941 365 106 475 008;
  • 35) 0.999 999 941 365 106 475 008 × 2 = 1 + 0.999 999 882 730 212 950 016;
  • 36) 0.999 999 882 730 212 950 016 × 2 = 1 + 0.999 999 765 460 425 900 032;
  • 37) 0.999 999 765 460 425 900 032 × 2 = 1 + 0.999 999 530 920 851 800 064;
  • 38) 0.999 999 530 920 851 800 064 × 2 = 1 + 0.999 999 061 841 703 600 128;
  • 39) 0.999 999 061 841 703 600 128 × 2 = 1 + 0.999 998 123 683 407 200 256;
  • 40) 0.999 998 123 683 407 200 256 × 2 = 1 + 0.999 996 247 366 814 400 512;
  • 41) 0.999 996 247 366 814 400 512 × 2 = 1 + 0.999 992 494 733 628 801 024;
  • 42) 0.999 992 494 733 628 801 024 × 2 = 1 + 0.999 984 989 467 257 602 048;
  • 43) 0.999 984 989 467 257 602 048 × 2 = 1 + 0.999 969 978 934 515 204 096;
  • 44) 0.999 969 978 934 515 204 096 × 2 = 1 + 0.999 939 957 869 030 408 192;
  • 45) 0.999 939 957 869 030 408 192 × 2 = 1 + 0.999 879 915 738 060 816 384;
  • 46) 0.999 879 915 738 060 816 384 × 2 = 1 + 0.999 759 831 476 121 632 768;
  • 47) 0.999 759 831 476 121 632 768 × 2 = 1 + 0.999 519 662 952 243 265 536;
  • 48) 0.999 519 662 952 243 265 536 × 2 = 1 + 0.999 039 325 904 486 531 072;
  • 49) 0.999 039 325 904 486 531 072 × 2 = 1 + 0.998 078 651 808 973 062 144;
  • 50) 0.998 078 651 808 973 062 144 × 2 = 1 + 0.996 157 303 617 946 124 288;
  • 51) 0.996 157 303 617 946 124 288 × 2 = 1 + 0.992 314 607 235 892 248 576;
  • 52) 0.992 314 607 235 892 248 576 × 2 = 1 + 0.984 629 214 471 784 497 152;
  • 53) 0.984 629 214 471 784 497 152 × 2 = 1 + 0.969 258 428 943 568 994 304;
  • 54) 0.969 258 428 943 568 994 304 × 2 = 1 + 0.938 516 857 887 137 988 608;
  • 55) 0.938 516 857 887 137 988 608 × 2 = 1 + 0.877 033 715 774 275 977 216;
  • 56) 0.877 033 715 774 275 977 216 × 2 = 1 + 0.754 067 431 548 551 954 432;
  • 57) 0.754 067 431 548 551 954 432 × 2 = 1 + 0.508 134 863 097 103 908 864;
  • 58) 0.508 134 863 097 103 908 864 × 2 = 1 + 0.016 269 726 194 207 817 728;

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:


0.016 738 891 601 562 496 587(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 587(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:


0.016 738 891 601 562 496 587(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 -0.016 738 891 601 562 496 587 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