-0.016 738 891 601 562 496 530 553 67 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 530 553 67(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
-0.016 738 891 601 562 496 530 553 67(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:

|-0.016 738 891 601 562 496 530 553 67| = 0.016 738 891 601 562 496 530 553 67


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: 0.016 738 891 601 562 496 530 553 67.

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) 0.016 738 891 601 562 496 530 553 67 × 2 = 0 + 0.033 477 783 203 124 993 061 107 34;
  • 2) 0.033 477 783 203 124 993 061 107 34 × 2 = 0 + 0.066 955 566 406 249 986 122 214 68;
  • 3) 0.066 955 566 406 249 986 122 214 68 × 2 = 0 + 0.133 911 132 812 499 972 244 429 36;
  • 4) 0.133 911 132 812 499 972 244 429 36 × 2 = 0 + 0.267 822 265 624 999 944 488 858 72;
  • 5) 0.267 822 265 624 999 944 488 858 72 × 2 = 0 + 0.535 644 531 249 999 888 977 717 44;
  • 6) 0.535 644 531 249 999 888 977 717 44 × 2 = 1 + 0.071 289 062 499 999 777 955 434 88;
  • 7) 0.071 289 062 499 999 777 955 434 88 × 2 = 0 + 0.142 578 124 999 999 555 910 869 76;
  • 8) 0.142 578 124 999 999 555 910 869 76 × 2 = 0 + 0.285 156 249 999 999 111 821 739 52;
  • 9) 0.285 156 249 999 999 111 821 739 52 × 2 = 0 + 0.570 312 499 999 998 223 643 479 04;
  • 10) 0.570 312 499 999 998 223 643 479 04 × 2 = 1 + 0.140 624 999 999 996 447 286 958 08;
  • 11) 0.140 624 999 999 996 447 286 958 08 × 2 = 0 + 0.281 249 999 999 992 894 573 916 16;
  • 12) 0.281 249 999 999 992 894 573 916 16 × 2 = 0 + 0.562 499 999 999 985 789 147 832 32;
  • 13) 0.562 499 999 999 985 789 147 832 32 × 2 = 1 + 0.124 999 999 999 971 578 295 664 64;
  • 14) 0.124 999 999 999 971 578 295 664 64 × 2 = 0 + 0.249 999 999 999 943 156 591 329 28;
  • 15) 0.249 999 999 999 943 156 591 329 28 × 2 = 0 + 0.499 999 999 999 886 313 182 658 56;
  • 16) 0.499 999 999 999 886 313 182 658 56 × 2 = 0 + 0.999 999 999 999 772 626 365 317 12;
  • 17) 0.999 999 999 999 772 626 365 317 12 × 2 = 1 + 0.999 999 999 999 545 252 730 634 24;
  • 18) 0.999 999 999 999 545 252 730 634 24 × 2 = 1 + 0.999 999 999 999 090 505 461 268 48;
  • 19) 0.999 999 999 999 090 505 461 268 48 × 2 = 1 + 0.999 999 999 998 181 010 922 536 96;
  • 20) 0.999 999 999 998 181 010 922 536 96 × 2 = 1 + 0.999 999 999 996 362 021 845 073 92;
  • 21) 0.999 999 999 996 362 021 845 073 92 × 2 = 1 + 0.999 999 999 992 724 043 690 147 84;
  • 22) 0.999 999 999 992 724 043 690 147 84 × 2 = 1 + 0.999 999 999 985 448 087 380 295 68;
  • 23) 0.999 999 999 985 448 087 380 295 68 × 2 = 1 + 0.999 999 999 970 896 174 760 591 36;
  • 24) 0.999 999 999 970 896 174 760 591 36 × 2 = 1 + 0.999 999 999 941 792 349 521 182 72;
  • 25) 0.999 999 999 941 792 349 521 182 72 × 2 = 1 + 0.999 999 999 883 584 699 042 365 44;
  • 26) 0.999 999 999 883 584 699 042 365 44 × 2 = 1 + 0.999 999 999 767 169 398 084 730 88;
  • 27) 0.999 999 999 767 169 398 084 730 88 × 2 = 1 + 0.999 999 999 534 338 796 169 461 76;
  • 28) 0.999 999 999 534 338 796 169 461 76 × 2 = 1 + 0.999 999 999 068 677 592 338 923 52;
  • 29) 0.999 999 999 068 677 592 338 923 52 × 2 = 1 + 0.999 999 998 137 355 184 677 847 04;
  • 30) 0.999 999 998 137 355 184 677 847 04 × 2 = 1 + 0.999 999 996 274 710 369 355 694 08;
  • 31) 0.999 999 996 274 710 369 355 694 08 × 2 = 1 + 0.999 999 992 549 420 738 711 388 16;
  • 32) 0.999 999 992 549 420 738 711 388 16 × 2 = 1 + 0.999 999 985 098 841 477 422 776 32;
  • 33) 0.999 999 985 098 841 477 422 776 32 × 2 = 1 + 0.999 999 970 197 682 954 845 552 64;
  • 34) 0.999 999 970 197 682 954 845 552 64 × 2 = 1 + 0.999 999 940 395 365 909 691 105 28;
  • 35) 0.999 999 940 395 365 909 691 105 28 × 2 = 1 + 0.999 999 880 790 731 819 382 210 56;
  • 36) 0.999 999 880 790 731 819 382 210 56 × 2 = 1 + 0.999 999 761 581 463 638 764 421 12;
  • 37) 0.999 999 761 581 463 638 764 421 12 × 2 = 1 + 0.999 999 523 162 927 277 528 842 24;
  • 38) 0.999 999 523 162 927 277 528 842 24 × 2 = 1 + 0.999 999 046 325 854 555 057 684 48;
  • 39) 0.999 999 046 325 854 555 057 684 48 × 2 = 1 + 0.999 998 092 651 709 110 115 368 96;
  • 40) 0.999 998 092 651 709 110 115 368 96 × 2 = 1 + 0.999 996 185 303 418 220 230 737 92;
  • 41) 0.999 996 185 303 418 220 230 737 92 × 2 = 1 + 0.999 992 370 606 836 440 461 475 84;
  • 42) 0.999 992 370 606 836 440 461 475 84 × 2 = 1 + 0.999 984 741 213 672 880 922 951 68;
  • 43) 0.999 984 741 213 672 880 922 951 68 × 2 = 1 + 0.999 969 482 427 345 761 845 903 36;
  • 44) 0.999 969 482 427 345 761 845 903 36 × 2 = 1 + 0.999 938 964 854 691 523 691 806 72;
  • 45) 0.999 938 964 854 691 523 691 806 72 × 2 = 1 + 0.999 877 929 709 383 047 383 613 44;
  • 46) 0.999 877 929 709 383 047 383 613 44 × 2 = 1 + 0.999 755 859 418 766 094 767 226 88;
  • 47) 0.999 755 859 418 766 094 767 226 88 × 2 = 1 + 0.999 511 718 837 532 189 534 453 76;
  • 48) 0.999 511 718 837 532 189 534 453 76 × 2 = 1 + 0.999 023 437 675 064 379 068 907 52;
  • 49) 0.999 023 437 675 064 379 068 907 52 × 2 = 1 + 0.998 046 875 350 128 758 137 815 04;
  • 50) 0.998 046 875 350 128 758 137 815 04 × 2 = 1 + 0.996 093 750 700 257 516 275 630 08;
  • 51) 0.996 093 750 700 257 516 275 630 08 × 2 = 1 + 0.992 187 501 400 515 032 551 260 16;
  • 52) 0.992 187 501 400 515 032 551 260 16 × 2 = 1 + 0.984 375 002 801 030 065 102 520 32;
  • 53) 0.984 375 002 801 030 065 102 520 32 × 2 = 1 + 0.968 750 005 602 060 130 205 040 64;
  • 54) 0.968 750 005 602 060 130 205 040 64 × 2 = 1 + 0.937 500 011 204 120 260 410 081 28;
  • 55) 0.937 500 011 204 120 260 410 081 28 × 2 = 1 + 0.875 000 022 408 240 520 820 162 56;
  • 56) 0.875 000 022 408 240 520 820 162 56 × 2 = 1 + 0.750 000 044 816 481 041 640 325 12;
  • 57) 0.750 000 044 816 481 041 640 325 12 × 2 = 1 + 0.500 000 089 632 962 083 280 650 24;
  • 58) 0.500 000 089 632 962 083 280 650 24 × 2 = 1 + 0.000 000 179 265 924 166 561 300 48;

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:


0.016 738 891 601 562 496 530 553 67(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 530 553 67(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:


0.016 738 891 601 562 496 530 553 67(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 -0.016 738 891 601 562 496 530 553 67 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