-0.016 738 891 601 562 496 561 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.016 738 891 601 562 496 561 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
-0.016 738 891 601 562 496 561 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:

|-0.016 738 891 601 562 496 561 9| = 0.016 738 891 601 562 496 561 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: 0.016 738 891 601 562 496 561 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) 0.016 738 891 601 562 496 561 9 × 2 = 0 + 0.033 477 783 203 124 993 123 8;
  • 2) 0.033 477 783 203 124 993 123 8 × 2 = 0 + 0.066 955 566 406 249 986 247 6;
  • 3) 0.066 955 566 406 249 986 247 6 × 2 = 0 + 0.133 911 132 812 499 972 495 2;
  • 4) 0.133 911 132 812 499 972 495 2 × 2 = 0 + 0.267 822 265 624 999 944 990 4;
  • 5) 0.267 822 265 624 999 944 990 4 × 2 = 0 + 0.535 644 531 249 999 889 980 8;
  • 6) 0.535 644 531 249 999 889 980 8 × 2 = 1 + 0.071 289 062 499 999 779 961 6;
  • 7) 0.071 289 062 499 999 779 961 6 × 2 = 0 + 0.142 578 124 999 999 559 923 2;
  • 8) 0.142 578 124 999 999 559 923 2 × 2 = 0 + 0.285 156 249 999 999 119 846 4;
  • 9) 0.285 156 249 999 999 119 846 4 × 2 = 0 + 0.570 312 499 999 998 239 692 8;
  • 10) 0.570 312 499 999 998 239 692 8 × 2 = 1 + 0.140 624 999 999 996 479 385 6;
  • 11) 0.140 624 999 999 996 479 385 6 × 2 = 0 + 0.281 249 999 999 992 958 771 2;
  • 12) 0.281 249 999 999 992 958 771 2 × 2 = 0 + 0.562 499 999 999 985 917 542 4;
  • 13) 0.562 499 999 999 985 917 542 4 × 2 = 1 + 0.124 999 999 999 971 835 084 8;
  • 14) 0.124 999 999 999 971 835 084 8 × 2 = 0 + 0.249 999 999 999 943 670 169 6;
  • 15) 0.249 999 999 999 943 670 169 6 × 2 = 0 + 0.499 999 999 999 887 340 339 2;
  • 16) 0.499 999 999 999 887 340 339 2 × 2 = 0 + 0.999 999 999 999 774 680 678 4;
  • 17) 0.999 999 999 999 774 680 678 4 × 2 = 1 + 0.999 999 999 999 549 361 356 8;
  • 18) 0.999 999 999 999 549 361 356 8 × 2 = 1 + 0.999 999 999 999 098 722 713 6;
  • 19) 0.999 999 999 999 098 722 713 6 × 2 = 1 + 0.999 999 999 998 197 445 427 2;
  • 20) 0.999 999 999 998 197 445 427 2 × 2 = 1 + 0.999 999 999 996 394 890 854 4;
  • 21) 0.999 999 999 996 394 890 854 4 × 2 = 1 + 0.999 999 999 992 789 781 708 8;
  • 22) 0.999 999 999 992 789 781 708 8 × 2 = 1 + 0.999 999 999 985 579 563 417 6;
  • 23) 0.999 999 999 985 579 563 417 6 × 2 = 1 + 0.999 999 999 971 159 126 835 2;
  • 24) 0.999 999 999 971 159 126 835 2 × 2 = 1 + 0.999 999 999 942 318 253 670 4;
  • 25) 0.999 999 999 942 318 253 670 4 × 2 = 1 + 0.999 999 999 884 636 507 340 8;
  • 26) 0.999 999 999 884 636 507 340 8 × 2 = 1 + 0.999 999 999 769 273 014 681 6;
  • 27) 0.999 999 999 769 273 014 681 6 × 2 = 1 + 0.999 999 999 538 546 029 363 2;
  • 28) 0.999 999 999 538 546 029 363 2 × 2 = 1 + 0.999 999 999 077 092 058 726 4;
  • 29) 0.999 999 999 077 092 058 726 4 × 2 = 1 + 0.999 999 998 154 184 117 452 8;
  • 30) 0.999 999 998 154 184 117 452 8 × 2 = 1 + 0.999 999 996 308 368 234 905 6;
  • 31) 0.999 999 996 308 368 234 905 6 × 2 = 1 + 0.999 999 992 616 736 469 811 2;
  • 32) 0.999 999 992 616 736 469 811 2 × 2 = 1 + 0.999 999 985 233 472 939 622 4;
  • 33) 0.999 999 985 233 472 939 622 4 × 2 = 1 + 0.999 999 970 466 945 879 244 8;
  • 34) 0.999 999 970 466 945 879 244 8 × 2 = 1 + 0.999 999 940 933 891 758 489 6;
  • 35) 0.999 999 940 933 891 758 489 6 × 2 = 1 + 0.999 999 881 867 783 516 979 2;
  • 36) 0.999 999 881 867 783 516 979 2 × 2 = 1 + 0.999 999 763 735 567 033 958 4;
  • 37) 0.999 999 763 735 567 033 958 4 × 2 = 1 + 0.999 999 527 471 134 067 916 8;
  • 38) 0.999 999 527 471 134 067 916 8 × 2 = 1 + 0.999 999 054 942 268 135 833 6;
  • 39) 0.999 999 054 942 268 135 833 6 × 2 = 1 + 0.999 998 109 884 536 271 667 2;
  • 40) 0.999 998 109 884 536 271 667 2 × 2 = 1 + 0.999 996 219 769 072 543 334 4;
  • 41) 0.999 996 219 769 072 543 334 4 × 2 = 1 + 0.999 992 439 538 145 086 668 8;
  • 42) 0.999 992 439 538 145 086 668 8 × 2 = 1 + 0.999 984 879 076 290 173 337 6;
  • 43) 0.999 984 879 076 290 173 337 6 × 2 = 1 + 0.999 969 758 152 580 346 675 2;
  • 44) 0.999 969 758 152 580 346 675 2 × 2 = 1 + 0.999 939 516 305 160 693 350 4;
  • 45) 0.999 939 516 305 160 693 350 4 × 2 = 1 + 0.999 879 032 610 321 386 700 8;
  • 46) 0.999 879 032 610 321 386 700 8 × 2 = 1 + 0.999 758 065 220 642 773 401 6;
  • 47) 0.999 758 065 220 642 773 401 6 × 2 = 1 + 0.999 516 130 441 285 546 803 2;
  • 48) 0.999 516 130 441 285 546 803 2 × 2 = 1 + 0.999 032 260 882 571 093 606 4;
  • 49) 0.999 032 260 882 571 093 606 4 × 2 = 1 + 0.998 064 521 765 142 187 212 8;
  • 50) 0.998 064 521 765 142 187 212 8 × 2 = 1 + 0.996 129 043 530 284 374 425 6;
  • 51) 0.996 129 043 530 284 374 425 6 × 2 = 1 + 0.992 258 087 060 568 748 851 2;
  • 52) 0.992 258 087 060 568 748 851 2 × 2 = 1 + 0.984 516 174 121 137 497 702 4;
  • 53) 0.984 516 174 121 137 497 702 4 × 2 = 1 + 0.969 032 348 242 274 995 404 8;
  • 54) 0.969 032 348 242 274 995 404 8 × 2 = 1 + 0.938 064 696 484 549 990 809 6;
  • 55) 0.938 064 696 484 549 990 809 6 × 2 = 1 + 0.876 129 392 969 099 981 619 2;
  • 56) 0.876 129 392 969 099 981 619 2 × 2 = 1 + 0.752 258 785 938 199 963 238 4;
  • 57) 0.752 258 785 938 199 963 238 4 × 2 = 1 + 0.504 517 571 876 399 926 476 8;
  • 58) 0.504 517 571 876 399 926 476 8 × 2 = 1 + 0.009 035 143 752 799 852 953 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:


0.016 738 891 601 562 496 561 9(10) =

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

6. Positive number before normalization:

0.016 738 891 601 562 496 561 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:


0.016 738 891 601 562 496 561 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 -0.016 738 891 601 562 496 561 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