-0.016 738 891 601 562 496 548 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 548 9(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
-0.016 738 891 601 562 496 548 9(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:

|-0.016 738 891 601 562 496 548 9| = 0.016 738 891 601 562 496 548 9


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: 0.016 738 891 601 562 496 548 9.

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) 0.016 738 891 601 562 496 548 9 × 2 = 0 + 0.033 477 783 203 124 993 097 8;
  • 2) 0.033 477 783 203 124 993 097 8 × 2 = 0 + 0.066 955 566 406 249 986 195 6;
  • 3) 0.066 955 566 406 249 986 195 6 × 2 = 0 + 0.133 911 132 812 499 972 391 2;
  • 4) 0.133 911 132 812 499 972 391 2 × 2 = 0 + 0.267 822 265 624 999 944 782 4;
  • 5) 0.267 822 265 624 999 944 782 4 × 2 = 0 + 0.535 644 531 249 999 889 564 8;
  • 6) 0.535 644 531 249 999 889 564 8 × 2 = 1 + 0.071 289 062 499 999 779 129 6;
  • 7) 0.071 289 062 499 999 779 129 6 × 2 = 0 + 0.142 578 124 999 999 558 259 2;
  • 8) 0.142 578 124 999 999 558 259 2 × 2 = 0 + 0.285 156 249 999 999 116 518 4;
  • 9) 0.285 156 249 999 999 116 518 4 × 2 = 0 + 0.570 312 499 999 998 233 036 8;
  • 10) 0.570 312 499 999 998 233 036 8 × 2 = 1 + 0.140 624 999 999 996 466 073 6;
  • 11) 0.140 624 999 999 996 466 073 6 × 2 = 0 + 0.281 249 999 999 992 932 147 2;
  • 12) 0.281 249 999 999 992 932 147 2 × 2 = 0 + 0.562 499 999 999 985 864 294 4;
  • 13) 0.562 499 999 999 985 864 294 4 × 2 = 1 + 0.124 999 999 999 971 728 588 8;
  • 14) 0.124 999 999 999 971 728 588 8 × 2 = 0 + 0.249 999 999 999 943 457 177 6;
  • 15) 0.249 999 999 999 943 457 177 6 × 2 = 0 + 0.499 999 999 999 886 914 355 2;
  • 16) 0.499 999 999 999 886 914 355 2 × 2 = 0 + 0.999 999 999 999 773 828 710 4;
  • 17) 0.999 999 999 999 773 828 710 4 × 2 = 1 + 0.999 999 999 999 547 657 420 8;
  • 18) 0.999 999 999 999 547 657 420 8 × 2 = 1 + 0.999 999 999 999 095 314 841 6;
  • 19) 0.999 999 999 999 095 314 841 6 × 2 = 1 + 0.999 999 999 998 190 629 683 2;
  • 20) 0.999 999 999 998 190 629 683 2 × 2 = 1 + 0.999 999 999 996 381 259 366 4;
  • 21) 0.999 999 999 996 381 259 366 4 × 2 = 1 + 0.999 999 999 992 762 518 732 8;
  • 22) 0.999 999 999 992 762 518 732 8 × 2 = 1 + 0.999 999 999 985 525 037 465 6;
  • 23) 0.999 999 999 985 525 037 465 6 × 2 = 1 + 0.999 999 999 971 050 074 931 2;
  • 24) 0.999 999 999 971 050 074 931 2 × 2 = 1 + 0.999 999 999 942 100 149 862 4;
  • 25) 0.999 999 999 942 100 149 862 4 × 2 = 1 + 0.999 999 999 884 200 299 724 8;
  • 26) 0.999 999 999 884 200 299 724 8 × 2 = 1 + 0.999 999 999 768 400 599 449 6;
  • 27) 0.999 999 999 768 400 599 449 6 × 2 = 1 + 0.999 999 999 536 801 198 899 2;
  • 28) 0.999 999 999 536 801 198 899 2 × 2 = 1 + 0.999 999 999 073 602 397 798 4;
  • 29) 0.999 999 999 073 602 397 798 4 × 2 = 1 + 0.999 999 998 147 204 795 596 8;
  • 30) 0.999 999 998 147 204 795 596 8 × 2 = 1 + 0.999 999 996 294 409 591 193 6;
  • 31) 0.999 999 996 294 409 591 193 6 × 2 = 1 + 0.999 999 992 588 819 182 387 2;
  • 32) 0.999 999 992 588 819 182 387 2 × 2 = 1 + 0.999 999 985 177 638 364 774 4;
  • 33) 0.999 999 985 177 638 364 774 4 × 2 = 1 + 0.999 999 970 355 276 729 548 8;
  • 34) 0.999 999 970 355 276 729 548 8 × 2 = 1 + 0.999 999 940 710 553 459 097 6;
  • 35) 0.999 999 940 710 553 459 097 6 × 2 = 1 + 0.999 999 881 421 106 918 195 2;
  • 36) 0.999 999 881 421 106 918 195 2 × 2 = 1 + 0.999 999 762 842 213 836 390 4;
  • 37) 0.999 999 762 842 213 836 390 4 × 2 = 1 + 0.999 999 525 684 427 672 780 8;
  • 38) 0.999 999 525 684 427 672 780 8 × 2 = 1 + 0.999 999 051 368 855 345 561 6;
  • 39) 0.999 999 051 368 855 345 561 6 × 2 = 1 + 0.999 998 102 737 710 691 123 2;
  • 40) 0.999 998 102 737 710 691 123 2 × 2 = 1 + 0.999 996 205 475 421 382 246 4;
  • 41) 0.999 996 205 475 421 382 246 4 × 2 = 1 + 0.999 992 410 950 842 764 492 8;
  • 42) 0.999 992 410 950 842 764 492 8 × 2 = 1 + 0.999 984 821 901 685 528 985 6;
  • 43) 0.999 984 821 901 685 528 985 6 × 2 = 1 + 0.999 969 643 803 371 057 971 2;
  • 44) 0.999 969 643 803 371 057 971 2 × 2 = 1 + 0.999 939 287 606 742 115 942 4;
  • 45) 0.999 939 287 606 742 115 942 4 × 2 = 1 + 0.999 878 575 213 484 231 884 8;
  • 46) 0.999 878 575 213 484 231 884 8 × 2 = 1 + 0.999 757 150 426 968 463 769 6;
  • 47) 0.999 757 150 426 968 463 769 6 × 2 = 1 + 0.999 514 300 853 936 927 539 2;
  • 48) 0.999 514 300 853 936 927 539 2 × 2 = 1 + 0.999 028 601 707 873 855 078 4;
  • 49) 0.999 028 601 707 873 855 078 4 × 2 = 1 + 0.998 057 203 415 747 710 156 8;
  • 50) 0.998 057 203 415 747 710 156 8 × 2 = 1 + 0.996 114 406 831 495 420 313 6;
  • 51) 0.996 114 406 831 495 420 313 6 × 2 = 1 + 0.992 228 813 662 990 840 627 2;
  • 52) 0.992 228 813 662 990 840 627 2 × 2 = 1 + 0.984 457 627 325 981 681 254 4;
  • 53) 0.984 457 627 325 981 681 254 4 × 2 = 1 + 0.968 915 254 651 963 362 508 8;
  • 54) 0.968 915 254 651 963 362 508 8 × 2 = 1 + 0.937 830 509 303 926 725 017 6;
  • 55) 0.937 830 509 303 926 725 017 6 × 2 = 1 + 0.875 661 018 607 853 450 035 2;
  • 56) 0.875 661 018 607 853 450 035 2 × 2 = 1 + 0.751 322 037 215 706 900 070 4;
  • 57) 0.751 322 037 215 706 900 070 4 × 2 = 1 + 0.502 644 074 431 413 800 140 8;
  • 58) 0.502 644 074 431 413 800 140 8 × 2 = 1 + 0.005 288 148 862 827 600 281 6;

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:


0.016 738 891 601 562 496 548 9(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 548 9(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:


0.016 738 891 601 562 496 548 9(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 -0.016 738 891 601 562 496 548 9 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