2.718 281 828 469 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 2.718 281 828 469 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
2.718 281 828 469 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. First, convert to binary (in base 2) the integer part: 2.
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;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. 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.

2(10) =

10(2)


3. Convert to binary (base 2) the fractional part: 0.718 281 828 469 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) 0.718 281 828 469 3 × 2 = 1 + 0.436 563 656 938 6;
  • 2) 0.436 563 656 938 6 × 2 = 0 + 0.873 127 313 877 2;
  • 3) 0.873 127 313 877 2 × 2 = 1 + 0.746 254 627 754 4;
  • 4) 0.746 254 627 754 4 × 2 = 1 + 0.492 509 255 508 8;
  • 5) 0.492 509 255 508 8 × 2 = 0 + 0.985 018 511 017 6;
  • 6) 0.985 018 511 017 6 × 2 = 1 + 0.970 037 022 035 2;
  • 7) 0.970 037 022 035 2 × 2 = 1 + 0.940 074 044 070 4;
  • 8) 0.940 074 044 070 4 × 2 = 1 + 0.880 148 088 140 8;
  • 9) 0.880 148 088 140 8 × 2 = 1 + 0.760 296 176 281 6;
  • 10) 0.760 296 176 281 6 × 2 = 1 + 0.520 592 352 563 2;
  • 11) 0.520 592 352 563 2 × 2 = 1 + 0.041 184 705 126 4;
  • 12) 0.041 184 705 126 4 × 2 = 0 + 0.082 369 410 252 8;
  • 13) 0.082 369 410 252 8 × 2 = 0 + 0.164 738 820 505 6;
  • 14) 0.164 738 820 505 6 × 2 = 0 + 0.329 477 641 011 2;
  • 15) 0.329 477 641 011 2 × 2 = 0 + 0.658 955 282 022 4;
  • 16) 0.658 955 282 022 4 × 2 = 1 + 0.317 910 564 044 8;
  • 17) 0.317 910 564 044 8 × 2 = 0 + 0.635 821 128 089 6;
  • 18) 0.635 821 128 089 6 × 2 = 1 + 0.271 642 256 179 2;
  • 19) 0.271 642 256 179 2 × 2 = 0 + 0.543 284 512 358 4;
  • 20) 0.543 284 512 358 4 × 2 = 1 + 0.086 569 024 716 8;
  • 21) 0.086 569 024 716 8 × 2 = 0 + 0.173 138 049 433 6;
  • 22) 0.173 138 049 433 6 × 2 = 0 + 0.346 276 098 867 2;
  • 23) 0.346 276 098 867 2 × 2 = 0 + 0.692 552 197 734 4;
  • 24) 0.692 552 197 734 4 × 2 = 1 + 0.385 104 395 468 8;
  • 25) 0.385 104 395 468 8 × 2 = 0 + 0.770 208 790 937 6;
  • 26) 0.770 208 790 937 6 × 2 = 1 + 0.540 417 581 875 2;
  • 27) 0.540 417 581 875 2 × 2 = 1 + 0.080 835 163 750 4;
  • 28) 0.080 835 163 750 4 × 2 = 0 + 0.161 670 327 500 8;
  • 29) 0.161 670 327 500 8 × 2 = 0 + 0.323 340 655 001 6;
  • 30) 0.323 340 655 001 6 × 2 = 0 + 0.646 681 310 003 2;
  • 31) 0.646 681 310 003 2 × 2 = 1 + 0.293 362 620 006 4;
  • 32) 0.293 362 620 006 4 × 2 = 0 + 0.586 725 240 012 8;
  • 33) 0.586 725 240 012 8 × 2 = 1 + 0.173 450 480 025 6;
  • 34) 0.173 450 480 025 6 × 2 = 0 + 0.346 900 960 051 2;
  • 35) 0.346 900 960 051 2 × 2 = 0 + 0.693 801 920 102 4;
  • 36) 0.693 801 920 102 4 × 2 = 1 + 0.387 603 840 204 8;
  • 37) 0.387 603 840 204 8 × 2 = 0 + 0.775 207 680 409 6;
  • 38) 0.775 207 680 409 6 × 2 = 1 + 0.550 415 360 819 2;
  • 39) 0.550 415 360 819 2 × 2 = 1 + 0.100 830 721 638 4;
  • 40) 0.100 830 721 638 4 × 2 = 0 + 0.201 661 443 276 8;
  • 41) 0.201 661 443 276 8 × 2 = 0 + 0.403 322 886 553 6;
  • 42) 0.403 322 886 553 6 × 2 = 0 + 0.806 645 773 107 2;
  • 43) 0.806 645 773 107 2 × 2 = 1 + 0.613 291 546 214 4;
  • 44) 0.613 291 546 214 4 × 2 = 1 + 0.226 583 092 428 8;
  • 45) 0.226 583 092 428 8 × 2 = 0 + 0.453 166 184 857 6;
  • 46) 0.453 166 184 857 6 × 2 = 0 + 0.906 332 369 715 2;
  • 47) 0.906 332 369 715 2 × 2 = 1 + 0.812 664 739 430 4;
  • 48) 0.812 664 739 430 4 × 2 = 1 + 0.625 329 478 860 8;
  • 49) 0.625 329 478 860 8 × 2 = 1 + 0.250 658 957 721 6;
  • 50) 0.250 658 957 721 6 × 2 = 0 + 0.501 317 915 443 2;
  • 51) 0.501 317 915 443 2 × 2 = 1 + 0.002 635 830 886 4;
  • 52) 0.002 635 830 886 4 × 2 = 0 + 0.005 271 661 772 8;
  • 53) 0.005 271 661 772 8 × 2 = 0 + 0.010 543 323 545 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).


4. 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:


0.718 281 828 469 3(10) =

0.1011 0111 1110 0001 0101 0001 0110 0010 1001 0110 0011 0011 1010 0(2)

5. Positive number before normalization:

2.718 281 828 469 3(10) =

10.1011 0111 1110 0001 0101 0001 0110 0010 1001 0110 0011 0011 1010 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the left, so that only one non zero digit remains to the left of it:


2.718 281 828 469 3(10) =

10.1011 0111 1110 0001 0101 0001 0110 0010 1001 0110 0011 0011 1010 0(2) =

10.1011 0111 1110 0001 0101 0001 0110 0010 1001 0110 0011 0011 1010 0(2) × 20 =

1.0101 1011 1111 0000 1010 1000 1011 0001 0100 1011 0001 1001 1101 00(2) × 21


7. 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): 1

Mantissa (not normalized):
1.0101 1011 1111 0000 1010 1000 1011 0001 0100 1011 0001 1001 1101 00


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

1 + 2(11-1) - 1 =

(1 + 1 023)(10) =

1 024(10)


9. 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;
  • 1 024 ÷ 2 = 512 + 0;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =

1024(10) =

100 0000 0000(2)


11. 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. 0101 1011 1111 0000 1010 1000 1011 0001 0100 1011 0001 1001 1101 00 =

0101 1011 1111 0000 1010 1000 1011 0001 0100 1011 0001 1001 1101


12. 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 0000 0000

Mantissa (52 bits) =
0101 1011 1111 0000 1010 1000 1011 0001 0100 1011 0001 1001 1101


Decimal number 2.718 281 828 469 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 0101 1011 1111 0000 1010 1000 1011 0001 0100 1011 0001 1001 1101

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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