-0.016 738 891 601 562 497 34 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 497 34(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
-0.016 738 891 601 562 497 34(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:

|-0.016 738 891 601 562 497 34| = 0.016 738 891 601 562 497 34


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: 0.016 738 891 601 562 497 34.

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) 0.016 738 891 601 562 497 34 × 2 = 0 + 0.033 477 783 203 124 994 68;
  • 2) 0.033 477 783 203 124 994 68 × 2 = 0 + 0.066 955 566 406 249 989 36;
  • 3) 0.066 955 566 406 249 989 36 × 2 = 0 + 0.133 911 132 812 499 978 72;
  • 4) 0.133 911 132 812 499 978 72 × 2 = 0 + 0.267 822 265 624 999 957 44;
  • 5) 0.267 822 265 624 999 957 44 × 2 = 0 + 0.535 644 531 249 999 914 88;
  • 6) 0.535 644 531 249 999 914 88 × 2 = 1 + 0.071 289 062 499 999 829 76;
  • 7) 0.071 289 062 499 999 829 76 × 2 = 0 + 0.142 578 124 999 999 659 52;
  • 8) 0.142 578 124 999 999 659 52 × 2 = 0 + 0.285 156 249 999 999 319 04;
  • 9) 0.285 156 249 999 999 319 04 × 2 = 0 + 0.570 312 499 999 998 638 08;
  • 10) 0.570 312 499 999 998 638 08 × 2 = 1 + 0.140 624 999 999 997 276 16;
  • 11) 0.140 624 999 999 997 276 16 × 2 = 0 + 0.281 249 999 999 994 552 32;
  • 12) 0.281 249 999 999 994 552 32 × 2 = 0 + 0.562 499 999 999 989 104 64;
  • 13) 0.562 499 999 999 989 104 64 × 2 = 1 + 0.124 999 999 999 978 209 28;
  • 14) 0.124 999 999 999 978 209 28 × 2 = 0 + 0.249 999 999 999 956 418 56;
  • 15) 0.249 999 999 999 956 418 56 × 2 = 0 + 0.499 999 999 999 912 837 12;
  • 16) 0.499 999 999 999 912 837 12 × 2 = 0 + 0.999 999 999 999 825 674 24;
  • 17) 0.999 999 999 999 825 674 24 × 2 = 1 + 0.999 999 999 999 651 348 48;
  • 18) 0.999 999 999 999 651 348 48 × 2 = 1 + 0.999 999 999 999 302 696 96;
  • 19) 0.999 999 999 999 302 696 96 × 2 = 1 + 0.999 999 999 998 605 393 92;
  • 20) 0.999 999 999 998 605 393 92 × 2 = 1 + 0.999 999 999 997 210 787 84;
  • 21) 0.999 999 999 997 210 787 84 × 2 = 1 + 0.999 999 999 994 421 575 68;
  • 22) 0.999 999 999 994 421 575 68 × 2 = 1 + 0.999 999 999 988 843 151 36;
  • 23) 0.999 999 999 988 843 151 36 × 2 = 1 + 0.999 999 999 977 686 302 72;
  • 24) 0.999 999 999 977 686 302 72 × 2 = 1 + 0.999 999 999 955 372 605 44;
  • 25) 0.999 999 999 955 372 605 44 × 2 = 1 + 0.999 999 999 910 745 210 88;
  • 26) 0.999 999 999 910 745 210 88 × 2 = 1 + 0.999 999 999 821 490 421 76;
  • 27) 0.999 999 999 821 490 421 76 × 2 = 1 + 0.999 999 999 642 980 843 52;
  • 28) 0.999 999 999 642 980 843 52 × 2 = 1 + 0.999 999 999 285 961 687 04;
  • 29) 0.999 999 999 285 961 687 04 × 2 = 1 + 0.999 999 998 571 923 374 08;
  • 30) 0.999 999 998 571 923 374 08 × 2 = 1 + 0.999 999 997 143 846 748 16;
  • 31) 0.999 999 997 143 846 748 16 × 2 = 1 + 0.999 999 994 287 693 496 32;
  • 32) 0.999 999 994 287 693 496 32 × 2 = 1 + 0.999 999 988 575 386 992 64;
  • 33) 0.999 999 988 575 386 992 64 × 2 = 1 + 0.999 999 977 150 773 985 28;
  • 34) 0.999 999 977 150 773 985 28 × 2 = 1 + 0.999 999 954 301 547 970 56;
  • 35) 0.999 999 954 301 547 970 56 × 2 = 1 + 0.999 999 908 603 095 941 12;
  • 36) 0.999 999 908 603 095 941 12 × 2 = 1 + 0.999 999 817 206 191 882 24;
  • 37) 0.999 999 817 206 191 882 24 × 2 = 1 + 0.999 999 634 412 383 764 48;
  • 38) 0.999 999 634 412 383 764 48 × 2 = 1 + 0.999 999 268 824 767 528 96;
  • 39) 0.999 999 268 824 767 528 96 × 2 = 1 + 0.999 998 537 649 535 057 92;
  • 40) 0.999 998 537 649 535 057 92 × 2 = 1 + 0.999 997 075 299 070 115 84;
  • 41) 0.999 997 075 299 070 115 84 × 2 = 1 + 0.999 994 150 598 140 231 68;
  • 42) 0.999 994 150 598 140 231 68 × 2 = 1 + 0.999 988 301 196 280 463 36;
  • 43) 0.999 988 301 196 280 463 36 × 2 = 1 + 0.999 976 602 392 560 926 72;
  • 44) 0.999 976 602 392 560 926 72 × 2 = 1 + 0.999 953 204 785 121 853 44;
  • 45) 0.999 953 204 785 121 853 44 × 2 = 1 + 0.999 906 409 570 243 706 88;
  • 46) 0.999 906 409 570 243 706 88 × 2 = 1 + 0.999 812 819 140 487 413 76;
  • 47) 0.999 812 819 140 487 413 76 × 2 = 1 + 0.999 625 638 280 974 827 52;
  • 48) 0.999 625 638 280 974 827 52 × 2 = 1 + 0.999 251 276 561 949 655 04;
  • 49) 0.999 251 276 561 949 655 04 × 2 = 1 + 0.998 502 553 123 899 310 08;
  • 50) 0.998 502 553 123 899 310 08 × 2 = 1 + 0.997 005 106 247 798 620 16;
  • 51) 0.997 005 106 247 798 620 16 × 2 = 1 + 0.994 010 212 495 597 240 32;
  • 52) 0.994 010 212 495 597 240 32 × 2 = 1 + 0.988 020 424 991 194 480 64;
  • 53) 0.988 020 424 991 194 480 64 × 2 = 1 + 0.976 040 849 982 388 961 28;
  • 54) 0.976 040 849 982 388 961 28 × 2 = 1 + 0.952 081 699 964 777 922 56;
  • 55) 0.952 081 699 964 777 922 56 × 2 = 1 + 0.904 163 399 929 555 845 12;
  • 56) 0.904 163 399 929 555 845 12 × 2 = 1 + 0.808 326 799 859 111 690 24;
  • 57) 0.808 326 799 859 111 690 24 × 2 = 1 + 0.616 653 599 718 223 380 48;
  • 58) 0.616 653 599 718 223 380 48 × 2 = 1 + 0.233 307 199 436 446 760 96;

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:


0.016 738 891 601 562 497 34(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 497 34(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:


0.016 738 891 601 562 497 34(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 -0.016 738 891 601 562 497 34 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