Decimal to 64 Bit IEEE 754 Binary: Convert Number 18 446 462 598 732 839 871 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number 18 446 462 598 732 839 871(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. 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;
  • 18 446 462 598 732 839 871 ÷ 2 = 9 223 231 299 366 419 935 + 1;
  • 9 223 231 299 366 419 935 ÷ 2 = 4 611 615 649 683 209 967 + 1;
  • 4 611 615 649 683 209 967 ÷ 2 = 2 305 807 824 841 604 983 + 1;
  • 2 305 807 824 841 604 983 ÷ 2 = 1 152 903 912 420 802 491 + 1;
  • 1 152 903 912 420 802 491 ÷ 2 = 576 451 956 210 401 245 + 1;
  • 576 451 956 210 401 245 ÷ 2 = 288 225 978 105 200 622 + 1;
  • 288 225 978 105 200 622 ÷ 2 = 144 112 989 052 600 311 + 0;
  • 144 112 989 052 600 311 ÷ 2 = 72 056 494 526 300 155 + 1;
  • 72 056 494 526 300 155 ÷ 2 = 36 028 247 263 150 077 + 1;
  • 36 028 247 263 150 077 ÷ 2 = 18 014 123 631 575 038 + 1;
  • 18 014 123 631 575 038 ÷ 2 = 9 007 061 815 787 519 + 0;
  • 9 007 061 815 787 519 ÷ 2 = 4 503 530 907 893 759 + 1;
  • 4 503 530 907 893 759 ÷ 2 = 2 251 765 453 946 879 + 1;
  • 2 251 765 453 946 879 ÷ 2 = 1 125 882 726 973 439 + 1;
  • 1 125 882 726 973 439 ÷ 2 = 562 941 363 486 719 + 1;
  • 562 941 363 486 719 ÷ 2 = 281 470 681 743 359 + 1;
  • 281 470 681 743 359 ÷ 2 = 140 735 340 871 679 + 1;
  • 140 735 340 871 679 ÷ 2 = 70 367 670 435 839 + 1;
  • 70 367 670 435 839 ÷ 2 = 35 183 835 217 919 + 1;
  • 35 183 835 217 919 ÷ 2 = 17 591 917 608 959 + 1;
  • 17 591 917 608 959 ÷ 2 = 8 795 958 804 479 + 1;
  • 8 795 958 804 479 ÷ 2 = 4 397 979 402 239 + 1;
  • 4 397 979 402 239 ÷ 2 = 2 198 989 701 119 + 1;
  • 2 198 989 701 119 ÷ 2 = 1 099 494 850 559 + 1;
  • 1 099 494 850 559 ÷ 2 = 549 747 425 279 + 1;
  • 549 747 425 279 ÷ 2 = 274 873 712 639 + 1;
  • 274 873 712 639 ÷ 2 = 137 436 856 319 + 1;
  • 137 436 856 319 ÷ 2 = 68 718 428 159 + 1;
  • 68 718 428 159 ÷ 2 = 34 359 214 079 + 1;
  • 34 359 214 079 ÷ 2 = 17 179 607 039 + 1;
  • 17 179 607 039 ÷ 2 = 8 589 803 519 + 1;
  • 8 589 803 519 ÷ 2 = 4 294 901 759 + 1;
  • 4 294 901 759 ÷ 2 = 2 147 450 879 + 1;
  • 2 147 450 879 ÷ 2 = 1 073 725 439 + 1;
  • 1 073 725 439 ÷ 2 = 536 862 719 + 1;
  • 536 862 719 ÷ 2 = 268 431 359 + 1;
  • 268 431 359 ÷ 2 = 134 215 679 + 1;
  • 134 215 679 ÷ 2 = 67 107 839 + 1;
  • 67 107 839 ÷ 2 = 33 553 919 + 1;
  • 33 553 919 ÷ 2 = 16 776 959 + 1;
  • 16 776 959 ÷ 2 = 8 388 479 + 1;
  • 8 388 479 ÷ 2 = 4 194 239 + 1;
  • 4 194 239 ÷ 2 = 2 097 119 + 1;
  • 2 097 119 ÷ 2 = 1 048 559 + 1;
  • 1 048 559 ÷ 2 = 524 279 + 1;
  • 524 279 ÷ 2 = 262 139 + 1;
  • 262 139 ÷ 2 = 131 069 + 1;
  • 131 069 ÷ 2 = 65 534 + 1;
  • 65 534 ÷ 2 = 32 767 + 0;
  • 32 767 ÷ 2 = 16 383 + 1;
  • 16 383 ÷ 2 = 8 191 + 1;
  • 8 191 ÷ 2 = 4 095 + 1;
  • 4 095 ÷ 2 = 2 047 + 1;
  • 2 047 ÷ 2 = 1 023 + 1;
  • 1 023 ÷ 2 = 511 + 1;
  • 511 ÷ 2 = 255 + 1;
  • 255 ÷ 2 = 127 + 1;
  • 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;

2. Construct the base 2 representation of the positive number.

Take all the remainders starting from the bottom of the list constructed above.


18 446 462 598 732 839 871(10) =


1111 1111 1111 1110 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1011 1111(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 63 positions to the left, so that only one non zero digit remains to the left of it:


18 446 462 598 732 839 871(10) =


1111 1111 1111 1110 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1011 1111(2) =


1111 1111 1111 1110 1111 1111 1111 1111 1111 1111 1111 1111 1111 1011 1011 1111(2) × 20 =


1.1111 1111 1111 1101 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0111 111(2) × 263


4. 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 0 (a positive number)


Exponent (unadjusted): 63


Mantissa (not normalized):
1.1111 1111 1111 1101 1111 1111 1111 1111 1111 1111 1111 1111 1111 0111 0111 111


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


Exponent (unadjusted) + 2(11-1) - 1 =


63 + 2(11-1) - 1 =


(63 + 1 023)(10) =


1 086(10)


6. 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;
  • 1 086 ÷ 2 = 543 + 0;
  • 543 ÷ 2 = 271 + 1;
  • 271 ÷ 2 = 135 + 1;
  • 135 ÷ 2 = 67 + 1;
  • 67 ÷ 2 = 33 + 1;
  • 33 ÷ 2 = 16 + 1;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

7. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


1086(10) =


100 0011 1110(2)


8. 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =


1. 1111 1111 1111 1101 1111 1111 1111 1111 1111 1111 1111 1111 1111 011 1011 1111 =


1111 1111 1111 1101 1111 1111 1111 1111 1111 1111 1111 1111 1111


9. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)


Exponent (11 bits) =
100 0011 1110


Mantissa (52 bits) =
1111 1111 1111 1101 1111 1111 1111 1111 1111 1111 1111 1111 1111


The base ten decimal number 18 446 462 598 732 839 871 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 0011 1110 - 1111 1111 1111 1101 1111 1111 1111 1111 1111 1111 1111 1111 1111

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