-0.016 738 891 601 562 496 539 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 539 3(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
-0.016 738 891 601 562 496 539 3(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:

|-0.016 738 891 601 562 496 539 3| = 0.016 738 891 601 562 496 539 3


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: 0.016 738 891 601 562 496 539 3.

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) 0.016 738 891 601 562 496 539 3 × 2 = 0 + 0.033 477 783 203 124 993 078 6;
  • 2) 0.033 477 783 203 124 993 078 6 × 2 = 0 + 0.066 955 566 406 249 986 157 2;
  • 3) 0.066 955 566 406 249 986 157 2 × 2 = 0 + 0.133 911 132 812 499 972 314 4;
  • 4) 0.133 911 132 812 499 972 314 4 × 2 = 0 + 0.267 822 265 624 999 944 628 8;
  • 5) 0.267 822 265 624 999 944 628 8 × 2 = 0 + 0.535 644 531 249 999 889 257 6;
  • 6) 0.535 644 531 249 999 889 257 6 × 2 = 1 + 0.071 289 062 499 999 778 515 2;
  • 7) 0.071 289 062 499 999 778 515 2 × 2 = 0 + 0.142 578 124 999 999 557 030 4;
  • 8) 0.142 578 124 999 999 557 030 4 × 2 = 0 + 0.285 156 249 999 999 114 060 8;
  • 9) 0.285 156 249 999 999 114 060 8 × 2 = 0 + 0.570 312 499 999 998 228 121 6;
  • 10) 0.570 312 499 999 998 228 121 6 × 2 = 1 + 0.140 624 999 999 996 456 243 2;
  • 11) 0.140 624 999 999 996 456 243 2 × 2 = 0 + 0.281 249 999 999 992 912 486 4;
  • 12) 0.281 249 999 999 992 912 486 4 × 2 = 0 + 0.562 499 999 999 985 824 972 8;
  • 13) 0.562 499 999 999 985 824 972 8 × 2 = 1 + 0.124 999 999 999 971 649 945 6;
  • 14) 0.124 999 999 999 971 649 945 6 × 2 = 0 + 0.249 999 999 999 943 299 891 2;
  • 15) 0.249 999 999 999 943 299 891 2 × 2 = 0 + 0.499 999 999 999 886 599 782 4;
  • 16) 0.499 999 999 999 886 599 782 4 × 2 = 0 + 0.999 999 999 999 773 199 564 8;
  • 17) 0.999 999 999 999 773 199 564 8 × 2 = 1 + 0.999 999 999 999 546 399 129 6;
  • 18) 0.999 999 999 999 546 399 129 6 × 2 = 1 + 0.999 999 999 999 092 798 259 2;
  • 19) 0.999 999 999 999 092 798 259 2 × 2 = 1 + 0.999 999 999 998 185 596 518 4;
  • 20) 0.999 999 999 998 185 596 518 4 × 2 = 1 + 0.999 999 999 996 371 193 036 8;
  • 21) 0.999 999 999 996 371 193 036 8 × 2 = 1 + 0.999 999 999 992 742 386 073 6;
  • 22) 0.999 999 999 992 742 386 073 6 × 2 = 1 + 0.999 999 999 985 484 772 147 2;
  • 23) 0.999 999 999 985 484 772 147 2 × 2 = 1 + 0.999 999 999 970 969 544 294 4;
  • 24) 0.999 999 999 970 969 544 294 4 × 2 = 1 + 0.999 999 999 941 939 088 588 8;
  • 25) 0.999 999 999 941 939 088 588 8 × 2 = 1 + 0.999 999 999 883 878 177 177 6;
  • 26) 0.999 999 999 883 878 177 177 6 × 2 = 1 + 0.999 999 999 767 756 354 355 2;
  • 27) 0.999 999 999 767 756 354 355 2 × 2 = 1 + 0.999 999 999 535 512 708 710 4;
  • 28) 0.999 999 999 535 512 708 710 4 × 2 = 1 + 0.999 999 999 071 025 417 420 8;
  • 29) 0.999 999 999 071 025 417 420 8 × 2 = 1 + 0.999 999 998 142 050 834 841 6;
  • 30) 0.999 999 998 142 050 834 841 6 × 2 = 1 + 0.999 999 996 284 101 669 683 2;
  • 31) 0.999 999 996 284 101 669 683 2 × 2 = 1 + 0.999 999 992 568 203 339 366 4;
  • 32) 0.999 999 992 568 203 339 366 4 × 2 = 1 + 0.999 999 985 136 406 678 732 8;
  • 33) 0.999 999 985 136 406 678 732 8 × 2 = 1 + 0.999 999 970 272 813 357 465 6;
  • 34) 0.999 999 970 272 813 357 465 6 × 2 = 1 + 0.999 999 940 545 626 714 931 2;
  • 35) 0.999 999 940 545 626 714 931 2 × 2 = 1 + 0.999 999 881 091 253 429 862 4;
  • 36) 0.999 999 881 091 253 429 862 4 × 2 = 1 + 0.999 999 762 182 506 859 724 8;
  • 37) 0.999 999 762 182 506 859 724 8 × 2 = 1 + 0.999 999 524 365 013 719 449 6;
  • 38) 0.999 999 524 365 013 719 449 6 × 2 = 1 + 0.999 999 048 730 027 438 899 2;
  • 39) 0.999 999 048 730 027 438 899 2 × 2 = 1 + 0.999 998 097 460 054 877 798 4;
  • 40) 0.999 998 097 460 054 877 798 4 × 2 = 1 + 0.999 996 194 920 109 755 596 8;
  • 41) 0.999 996 194 920 109 755 596 8 × 2 = 1 + 0.999 992 389 840 219 511 193 6;
  • 42) 0.999 992 389 840 219 511 193 6 × 2 = 1 + 0.999 984 779 680 439 022 387 2;
  • 43) 0.999 984 779 680 439 022 387 2 × 2 = 1 + 0.999 969 559 360 878 044 774 4;
  • 44) 0.999 969 559 360 878 044 774 4 × 2 = 1 + 0.999 939 118 721 756 089 548 8;
  • 45) 0.999 939 118 721 756 089 548 8 × 2 = 1 + 0.999 878 237 443 512 179 097 6;
  • 46) 0.999 878 237 443 512 179 097 6 × 2 = 1 + 0.999 756 474 887 024 358 195 2;
  • 47) 0.999 756 474 887 024 358 195 2 × 2 = 1 + 0.999 512 949 774 048 716 390 4;
  • 48) 0.999 512 949 774 048 716 390 4 × 2 = 1 + 0.999 025 899 548 097 432 780 8;
  • 49) 0.999 025 899 548 097 432 780 8 × 2 = 1 + 0.998 051 799 096 194 865 561 6;
  • 50) 0.998 051 799 096 194 865 561 6 × 2 = 1 + 0.996 103 598 192 389 731 123 2;
  • 51) 0.996 103 598 192 389 731 123 2 × 2 = 1 + 0.992 207 196 384 779 462 246 4;
  • 52) 0.992 207 196 384 779 462 246 4 × 2 = 1 + 0.984 414 392 769 558 924 492 8;
  • 53) 0.984 414 392 769 558 924 492 8 × 2 = 1 + 0.968 828 785 539 117 848 985 6;
  • 54) 0.968 828 785 539 117 848 985 6 × 2 = 1 + 0.937 657 571 078 235 697 971 2;
  • 55) 0.937 657 571 078 235 697 971 2 × 2 = 1 + 0.875 315 142 156 471 395 942 4;
  • 56) 0.875 315 142 156 471 395 942 4 × 2 = 1 + 0.750 630 284 312 942 791 884 8;
  • 57) 0.750 630 284 312 942 791 884 8 × 2 = 1 + 0.501 260 568 625 885 583 769 6;
  • 58) 0.501 260 568 625 885 583 769 6 × 2 = 1 + 0.002 521 137 251 771 167 539 2;

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:


0.016 738 891 601 562 496 539 3(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 539 3(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:


0.016 738 891 601 562 496 539 3(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 -0.016 738 891 601 562 496 539 3 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