-0.016 738 891 601 562 496 530 409 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 409(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
-0.016 738 891 601 562 496 530 409(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:

|-0.016 738 891 601 562 496 530 409| = 0.016 738 891 601 562 496 530 409


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: 0.016 738 891 601 562 496 530 409.

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) 0.016 738 891 601 562 496 530 409 × 2 = 0 + 0.033 477 783 203 124 993 060 818;
  • 2) 0.033 477 783 203 124 993 060 818 × 2 = 0 + 0.066 955 566 406 249 986 121 636;
  • 3) 0.066 955 566 406 249 986 121 636 × 2 = 0 + 0.133 911 132 812 499 972 243 272;
  • 4) 0.133 911 132 812 499 972 243 272 × 2 = 0 + 0.267 822 265 624 999 944 486 544;
  • 5) 0.267 822 265 624 999 944 486 544 × 2 = 0 + 0.535 644 531 249 999 888 973 088;
  • 6) 0.535 644 531 249 999 888 973 088 × 2 = 1 + 0.071 289 062 499 999 777 946 176;
  • 7) 0.071 289 062 499 999 777 946 176 × 2 = 0 + 0.142 578 124 999 999 555 892 352;
  • 8) 0.142 578 124 999 999 555 892 352 × 2 = 0 + 0.285 156 249 999 999 111 784 704;
  • 9) 0.285 156 249 999 999 111 784 704 × 2 = 0 + 0.570 312 499 999 998 223 569 408;
  • 10) 0.570 312 499 999 998 223 569 408 × 2 = 1 + 0.140 624 999 999 996 447 138 816;
  • 11) 0.140 624 999 999 996 447 138 816 × 2 = 0 + 0.281 249 999 999 992 894 277 632;
  • 12) 0.281 249 999 999 992 894 277 632 × 2 = 0 + 0.562 499 999 999 985 788 555 264;
  • 13) 0.562 499 999 999 985 788 555 264 × 2 = 1 + 0.124 999 999 999 971 577 110 528;
  • 14) 0.124 999 999 999 971 577 110 528 × 2 = 0 + 0.249 999 999 999 943 154 221 056;
  • 15) 0.249 999 999 999 943 154 221 056 × 2 = 0 + 0.499 999 999 999 886 308 442 112;
  • 16) 0.499 999 999 999 886 308 442 112 × 2 = 0 + 0.999 999 999 999 772 616 884 224;
  • 17) 0.999 999 999 999 772 616 884 224 × 2 = 1 + 0.999 999 999 999 545 233 768 448;
  • 18) 0.999 999 999 999 545 233 768 448 × 2 = 1 + 0.999 999 999 999 090 467 536 896;
  • 19) 0.999 999 999 999 090 467 536 896 × 2 = 1 + 0.999 999 999 998 180 935 073 792;
  • 20) 0.999 999 999 998 180 935 073 792 × 2 = 1 + 0.999 999 999 996 361 870 147 584;
  • 21) 0.999 999 999 996 361 870 147 584 × 2 = 1 + 0.999 999 999 992 723 740 295 168;
  • 22) 0.999 999 999 992 723 740 295 168 × 2 = 1 + 0.999 999 999 985 447 480 590 336;
  • 23) 0.999 999 999 985 447 480 590 336 × 2 = 1 + 0.999 999 999 970 894 961 180 672;
  • 24) 0.999 999 999 970 894 961 180 672 × 2 = 1 + 0.999 999 999 941 789 922 361 344;
  • 25) 0.999 999 999 941 789 922 361 344 × 2 = 1 + 0.999 999 999 883 579 844 722 688;
  • 26) 0.999 999 999 883 579 844 722 688 × 2 = 1 + 0.999 999 999 767 159 689 445 376;
  • 27) 0.999 999 999 767 159 689 445 376 × 2 = 1 + 0.999 999 999 534 319 378 890 752;
  • 28) 0.999 999 999 534 319 378 890 752 × 2 = 1 + 0.999 999 999 068 638 757 781 504;
  • 29) 0.999 999 999 068 638 757 781 504 × 2 = 1 + 0.999 999 998 137 277 515 563 008;
  • 30) 0.999 999 998 137 277 515 563 008 × 2 = 1 + 0.999 999 996 274 555 031 126 016;
  • 31) 0.999 999 996 274 555 031 126 016 × 2 = 1 + 0.999 999 992 549 110 062 252 032;
  • 32) 0.999 999 992 549 110 062 252 032 × 2 = 1 + 0.999 999 985 098 220 124 504 064;
  • 33) 0.999 999 985 098 220 124 504 064 × 2 = 1 + 0.999 999 970 196 440 249 008 128;
  • 34) 0.999 999 970 196 440 249 008 128 × 2 = 1 + 0.999 999 940 392 880 498 016 256;
  • 35) 0.999 999 940 392 880 498 016 256 × 2 = 1 + 0.999 999 880 785 760 996 032 512;
  • 36) 0.999 999 880 785 760 996 032 512 × 2 = 1 + 0.999 999 761 571 521 992 065 024;
  • 37) 0.999 999 761 571 521 992 065 024 × 2 = 1 + 0.999 999 523 143 043 984 130 048;
  • 38) 0.999 999 523 143 043 984 130 048 × 2 = 1 + 0.999 999 046 286 087 968 260 096;
  • 39) 0.999 999 046 286 087 968 260 096 × 2 = 1 + 0.999 998 092 572 175 936 520 192;
  • 40) 0.999 998 092 572 175 936 520 192 × 2 = 1 + 0.999 996 185 144 351 873 040 384;
  • 41) 0.999 996 185 144 351 873 040 384 × 2 = 1 + 0.999 992 370 288 703 746 080 768;
  • 42) 0.999 992 370 288 703 746 080 768 × 2 = 1 + 0.999 984 740 577 407 492 161 536;
  • 43) 0.999 984 740 577 407 492 161 536 × 2 = 1 + 0.999 969 481 154 814 984 323 072;
  • 44) 0.999 969 481 154 814 984 323 072 × 2 = 1 + 0.999 938 962 309 629 968 646 144;
  • 45) 0.999 938 962 309 629 968 646 144 × 2 = 1 + 0.999 877 924 619 259 937 292 288;
  • 46) 0.999 877 924 619 259 937 292 288 × 2 = 1 + 0.999 755 849 238 519 874 584 576;
  • 47) 0.999 755 849 238 519 874 584 576 × 2 = 1 + 0.999 511 698 477 039 749 169 152;
  • 48) 0.999 511 698 477 039 749 169 152 × 2 = 1 + 0.999 023 396 954 079 498 338 304;
  • 49) 0.999 023 396 954 079 498 338 304 × 2 = 1 + 0.998 046 793 908 158 996 676 608;
  • 50) 0.998 046 793 908 158 996 676 608 × 2 = 1 + 0.996 093 587 816 317 993 353 216;
  • 51) 0.996 093 587 816 317 993 353 216 × 2 = 1 + 0.992 187 175 632 635 986 706 432;
  • 52) 0.992 187 175 632 635 986 706 432 × 2 = 1 + 0.984 374 351 265 271 973 412 864;
  • 53) 0.984 374 351 265 271 973 412 864 × 2 = 1 + 0.968 748 702 530 543 946 825 728;
  • 54) 0.968 748 702 530 543 946 825 728 × 2 = 1 + 0.937 497 405 061 087 893 651 456;
  • 55) 0.937 497 405 061 087 893 651 456 × 2 = 1 + 0.874 994 810 122 175 787 302 912;
  • 56) 0.874 994 810 122 175 787 302 912 × 2 = 1 + 0.749 989 620 244 351 574 605 824;
  • 57) 0.749 989 620 244 351 574 605 824 × 2 = 1 + 0.499 979 240 488 703 149 211 648;
  • 58) 0.499 979 240 488 703 149 211 648 × 2 = 0 + 0.999 958 480 977 406 298 423 296;

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:


0.016 738 891 601 562 496 530 409(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 409(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:


0.016 738 891 601 562 496 530 409(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 -0.016 738 891 601 562 496 530 409 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