-0.016 738 891 601 562 496 536 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 536 6(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
-0.016 738 891 601 562 496 536 6(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:

|-0.016 738 891 601 562 496 536 6| = 0.016 738 891 601 562 496 536 6


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: 0.016 738 891 601 562 496 536 6.

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) 0.016 738 891 601 562 496 536 6 × 2 = 0 + 0.033 477 783 203 124 993 073 2;
  • 2) 0.033 477 783 203 124 993 073 2 × 2 = 0 + 0.066 955 566 406 249 986 146 4;
  • 3) 0.066 955 566 406 249 986 146 4 × 2 = 0 + 0.133 911 132 812 499 972 292 8;
  • 4) 0.133 911 132 812 499 972 292 8 × 2 = 0 + 0.267 822 265 624 999 944 585 6;
  • 5) 0.267 822 265 624 999 944 585 6 × 2 = 0 + 0.535 644 531 249 999 889 171 2;
  • 6) 0.535 644 531 249 999 889 171 2 × 2 = 1 + 0.071 289 062 499 999 778 342 4;
  • 7) 0.071 289 062 499 999 778 342 4 × 2 = 0 + 0.142 578 124 999 999 556 684 8;
  • 8) 0.142 578 124 999 999 556 684 8 × 2 = 0 + 0.285 156 249 999 999 113 369 6;
  • 9) 0.285 156 249 999 999 113 369 6 × 2 = 0 + 0.570 312 499 999 998 226 739 2;
  • 10) 0.570 312 499 999 998 226 739 2 × 2 = 1 + 0.140 624 999 999 996 453 478 4;
  • 11) 0.140 624 999 999 996 453 478 4 × 2 = 0 + 0.281 249 999 999 992 906 956 8;
  • 12) 0.281 249 999 999 992 906 956 8 × 2 = 0 + 0.562 499 999 999 985 813 913 6;
  • 13) 0.562 499 999 999 985 813 913 6 × 2 = 1 + 0.124 999 999 999 971 627 827 2;
  • 14) 0.124 999 999 999 971 627 827 2 × 2 = 0 + 0.249 999 999 999 943 255 654 4;
  • 15) 0.249 999 999 999 943 255 654 4 × 2 = 0 + 0.499 999 999 999 886 511 308 8;
  • 16) 0.499 999 999 999 886 511 308 8 × 2 = 0 + 0.999 999 999 999 773 022 617 6;
  • 17) 0.999 999 999 999 773 022 617 6 × 2 = 1 + 0.999 999 999 999 546 045 235 2;
  • 18) 0.999 999 999 999 546 045 235 2 × 2 = 1 + 0.999 999 999 999 092 090 470 4;
  • 19) 0.999 999 999 999 092 090 470 4 × 2 = 1 + 0.999 999 999 998 184 180 940 8;
  • 20) 0.999 999 999 998 184 180 940 8 × 2 = 1 + 0.999 999 999 996 368 361 881 6;
  • 21) 0.999 999 999 996 368 361 881 6 × 2 = 1 + 0.999 999 999 992 736 723 763 2;
  • 22) 0.999 999 999 992 736 723 763 2 × 2 = 1 + 0.999 999 999 985 473 447 526 4;
  • 23) 0.999 999 999 985 473 447 526 4 × 2 = 1 + 0.999 999 999 970 946 895 052 8;
  • 24) 0.999 999 999 970 946 895 052 8 × 2 = 1 + 0.999 999 999 941 893 790 105 6;
  • 25) 0.999 999 999 941 893 790 105 6 × 2 = 1 + 0.999 999 999 883 787 580 211 2;
  • 26) 0.999 999 999 883 787 580 211 2 × 2 = 1 + 0.999 999 999 767 575 160 422 4;
  • 27) 0.999 999 999 767 575 160 422 4 × 2 = 1 + 0.999 999 999 535 150 320 844 8;
  • 28) 0.999 999 999 535 150 320 844 8 × 2 = 1 + 0.999 999 999 070 300 641 689 6;
  • 29) 0.999 999 999 070 300 641 689 6 × 2 = 1 + 0.999 999 998 140 601 283 379 2;
  • 30) 0.999 999 998 140 601 283 379 2 × 2 = 1 + 0.999 999 996 281 202 566 758 4;
  • 31) 0.999 999 996 281 202 566 758 4 × 2 = 1 + 0.999 999 992 562 405 133 516 8;
  • 32) 0.999 999 992 562 405 133 516 8 × 2 = 1 + 0.999 999 985 124 810 267 033 6;
  • 33) 0.999 999 985 124 810 267 033 6 × 2 = 1 + 0.999 999 970 249 620 534 067 2;
  • 34) 0.999 999 970 249 620 534 067 2 × 2 = 1 + 0.999 999 940 499 241 068 134 4;
  • 35) 0.999 999 940 499 241 068 134 4 × 2 = 1 + 0.999 999 880 998 482 136 268 8;
  • 36) 0.999 999 880 998 482 136 268 8 × 2 = 1 + 0.999 999 761 996 964 272 537 6;
  • 37) 0.999 999 761 996 964 272 537 6 × 2 = 1 + 0.999 999 523 993 928 545 075 2;
  • 38) 0.999 999 523 993 928 545 075 2 × 2 = 1 + 0.999 999 047 987 857 090 150 4;
  • 39) 0.999 999 047 987 857 090 150 4 × 2 = 1 + 0.999 998 095 975 714 180 300 8;
  • 40) 0.999 998 095 975 714 180 300 8 × 2 = 1 + 0.999 996 191 951 428 360 601 6;
  • 41) 0.999 996 191 951 428 360 601 6 × 2 = 1 + 0.999 992 383 902 856 721 203 2;
  • 42) 0.999 992 383 902 856 721 203 2 × 2 = 1 + 0.999 984 767 805 713 442 406 4;
  • 43) 0.999 984 767 805 713 442 406 4 × 2 = 1 + 0.999 969 535 611 426 884 812 8;
  • 44) 0.999 969 535 611 426 884 812 8 × 2 = 1 + 0.999 939 071 222 853 769 625 6;
  • 45) 0.999 939 071 222 853 769 625 6 × 2 = 1 + 0.999 878 142 445 707 539 251 2;
  • 46) 0.999 878 142 445 707 539 251 2 × 2 = 1 + 0.999 756 284 891 415 078 502 4;
  • 47) 0.999 756 284 891 415 078 502 4 × 2 = 1 + 0.999 512 569 782 830 157 004 8;
  • 48) 0.999 512 569 782 830 157 004 8 × 2 = 1 + 0.999 025 139 565 660 314 009 6;
  • 49) 0.999 025 139 565 660 314 009 6 × 2 = 1 + 0.998 050 279 131 320 628 019 2;
  • 50) 0.998 050 279 131 320 628 019 2 × 2 = 1 + 0.996 100 558 262 641 256 038 4;
  • 51) 0.996 100 558 262 641 256 038 4 × 2 = 1 + 0.992 201 116 525 282 512 076 8;
  • 52) 0.992 201 116 525 282 512 076 8 × 2 = 1 + 0.984 402 233 050 565 024 153 6;
  • 53) 0.984 402 233 050 565 024 153 6 × 2 = 1 + 0.968 804 466 101 130 048 307 2;
  • 54) 0.968 804 466 101 130 048 307 2 × 2 = 1 + 0.937 608 932 202 260 096 614 4;
  • 55) 0.937 608 932 202 260 096 614 4 × 2 = 1 + 0.875 217 864 404 520 193 228 8;
  • 56) 0.875 217 864 404 520 193 228 8 × 2 = 1 + 0.750 435 728 809 040 386 457 6;
  • 57) 0.750 435 728 809 040 386 457 6 × 2 = 1 + 0.500 871 457 618 080 772 915 2;
  • 58) 0.500 871 457 618 080 772 915 2 × 2 = 1 + 0.001 742 915 236 161 545 830 4;

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:


0.016 738 891 601 562 496 536 6(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 536 6(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:


0.016 738 891 601 562 496 536 6(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 -0.016 738 891 601 562 496 536 6 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