-0.016 738 891 601 562 496 530 687 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 687(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
-0.016 738 891 601 562 496 530 687(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:

|-0.016 738 891 601 562 496 530 687| = 0.016 738 891 601 562 496 530 687


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: 0.016 738 891 601 562 496 530 687.

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) 0.016 738 891 601 562 496 530 687 × 2 = 0 + 0.033 477 783 203 124 993 061 374;
  • 2) 0.033 477 783 203 124 993 061 374 × 2 = 0 + 0.066 955 566 406 249 986 122 748;
  • 3) 0.066 955 566 406 249 986 122 748 × 2 = 0 + 0.133 911 132 812 499 972 245 496;
  • 4) 0.133 911 132 812 499 972 245 496 × 2 = 0 + 0.267 822 265 624 999 944 490 992;
  • 5) 0.267 822 265 624 999 944 490 992 × 2 = 0 + 0.535 644 531 249 999 888 981 984;
  • 6) 0.535 644 531 249 999 888 981 984 × 2 = 1 + 0.071 289 062 499 999 777 963 968;
  • 7) 0.071 289 062 499 999 777 963 968 × 2 = 0 + 0.142 578 124 999 999 555 927 936;
  • 8) 0.142 578 124 999 999 555 927 936 × 2 = 0 + 0.285 156 249 999 999 111 855 872;
  • 9) 0.285 156 249 999 999 111 855 872 × 2 = 0 + 0.570 312 499 999 998 223 711 744;
  • 10) 0.570 312 499 999 998 223 711 744 × 2 = 1 + 0.140 624 999 999 996 447 423 488;
  • 11) 0.140 624 999 999 996 447 423 488 × 2 = 0 + 0.281 249 999 999 992 894 846 976;
  • 12) 0.281 249 999 999 992 894 846 976 × 2 = 0 + 0.562 499 999 999 985 789 693 952;
  • 13) 0.562 499 999 999 985 789 693 952 × 2 = 1 + 0.124 999 999 999 971 579 387 904;
  • 14) 0.124 999 999 999 971 579 387 904 × 2 = 0 + 0.249 999 999 999 943 158 775 808;
  • 15) 0.249 999 999 999 943 158 775 808 × 2 = 0 + 0.499 999 999 999 886 317 551 616;
  • 16) 0.499 999 999 999 886 317 551 616 × 2 = 0 + 0.999 999 999 999 772 635 103 232;
  • 17) 0.999 999 999 999 772 635 103 232 × 2 = 1 + 0.999 999 999 999 545 270 206 464;
  • 18) 0.999 999 999 999 545 270 206 464 × 2 = 1 + 0.999 999 999 999 090 540 412 928;
  • 19) 0.999 999 999 999 090 540 412 928 × 2 = 1 + 0.999 999 999 998 181 080 825 856;
  • 20) 0.999 999 999 998 181 080 825 856 × 2 = 1 + 0.999 999 999 996 362 161 651 712;
  • 21) 0.999 999 999 996 362 161 651 712 × 2 = 1 + 0.999 999 999 992 724 323 303 424;
  • 22) 0.999 999 999 992 724 323 303 424 × 2 = 1 + 0.999 999 999 985 448 646 606 848;
  • 23) 0.999 999 999 985 448 646 606 848 × 2 = 1 + 0.999 999 999 970 897 293 213 696;
  • 24) 0.999 999 999 970 897 293 213 696 × 2 = 1 + 0.999 999 999 941 794 586 427 392;
  • 25) 0.999 999 999 941 794 586 427 392 × 2 = 1 + 0.999 999 999 883 589 172 854 784;
  • 26) 0.999 999 999 883 589 172 854 784 × 2 = 1 + 0.999 999 999 767 178 345 709 568;
  • 27) 0.999 999 999 767 178 345 709 568 × 2 = 1 + 0.999 999 999 534 356 691 419 136;
  • 28) 0.999 999 999 534 356 691 419 136 × 2 = 1 + 0.999 999 999 068 713 382 838 272;
  • 29) 0.999 999 999 068 713 382 838 272 × 2 = 1 + 0.999 999 998 137 426 765 676 544;
  • 30) 0.999 999 998 137 426 765 676 544 × 2 = 1 + 0.999 999 996 274 853 531 353 088;
  • 31) 0.999 999 996 274 853 531 353 088 × 2 = 1 + 0.999 999 992 549 707 062 706 176;
  • 32) 0.999 999 992 549 707 062 706 176 × 2 = 1 + 0.999 999 985 099 414 125 412 352;
  • 33) 0.999 999 985 099 414 125 412 352 × 2 = 1 + 0.999 999 970 198 828 250 824 704;
  • 34) 0.999 999 970 198 828 250 824 704 × 2 = 1 + 0.999 999 940 397 656 501 649 408;
  • 35) 0.999 999 940 397 656 501 649 408 × 2 = 1 + 0.999 999 880 795 313 003 298 816;
  • 36) 0.999 999 880 795 313 003 298 816 × 2 = 1 + 0.999 999 761 590 626 006 597 632;
  • 37) 0.999 999 761 590 626 006 597 632 × 2 = 1 + 0.999 999 523 181 252 013 195 264;
  • 38) 0.999 999 523 181 252 013 195 264 × 2 = 1 + 0.999 999 046 362 504 026 390 528;
  • 39) 0.999 999 046 362 504 026 390 528 × 2 = 1 + 0.999 998 092 725 008 052 781 056;
  • 40) 0.999 998 092 725 008 052 781 056 × 2 = 1 + 0.999 996 185 450 016 105 562 112;
  • 41) 0.999 996 185 450 016 105 562 112 × 2 = 1 + 0.999 992 370 900 032 211 124 224;
  • 42) 0.999 992 370 900 032 211 124 224 × 2 = 1 + 0.999 984 741 800 064 422 248 448;
  • 43) 0.999 984 741 800 064 422 248 448 × 2 = 1 + 0.999 969 483 600 128 844 496 896;
  • 44) 0.999 969 483 600 128 844 496 896 × 2 = 1 + 0.999 938 967 200 257 688 993 792;
  • 45) 0.999 938 967 200 257 688 993 792 × 2 = 1 + 0.999 877 934 400 515 377 987 584;
  • 46) 0.999 877 934 400 515 377 987 584 × 2 = 1 + 0.999 755 868 801 030 755 975 168;
  • 47) 0.999 755 868 801 030 755 975 168 × 2 = 1 + 0.999 511 737 602 061 511 950 336;
  • 48) 0.999 511 737 602 061 511 950 336 × 2 = 1 + 0.999 023 475 204 123 023 900 672;
  • 49) 0.999 023 475 204 123 023 900 672 × 2 = 1 + 0.998 046 950 408 246 047 801 344;
  • 50) 0.998 046 950 408 246 047 801 344 × 2 = 1 + 0.996 093 900 816 492 095 602 688;
  • 51) 0.996 093 900 816 492 095 602 688 × 2 = 1 + 0.992 187 801 632 984 191 205 376;
  • 52) 0.992 187 801 632 984 191 205 376 × 2 = 1 + 0.984 375 603 265 968 382 410 752;
  • 53) 0.984 375 603 265 968 382 410 752 × 2 = 1 + 0.968 751 206 531 936 764 821 504;
  • 54) 0.968 751 206 531 936 764 821 504 × 2 = 1 + 0.937 502 413 063 873 529 643 008;
  • 55) 0.937 502 413 063 873 529 643 008 × 2 = 1 + 0.875 004 826 127 747 059 286 016;
  • 56) 0.875 004 826 127 747 059 286 016 × 2 = 1 + 0.750 009 652 255 494 118 572 032;
  • 57) 0.750 009 652 255 494 118 572 032 × 2 = 1 + 0.500 019 304 510 988 237 144 064;
  • 58) 0.500 019 304 510 988 237 144 064 × 2 = 1 + 0.000 038 609 021 976 474 288 128;

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:


0.016 738 891 601 562 496 530 687(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 687(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:


0.016 738 891 601 562 496 530 687(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 -0.016 738 891 601 562 496 530 687 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