1.999 999 999 999 999 778 042 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.999 999 999 999 999 778 042(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
1.999 999 999 999 999 778 042(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: 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;
  • 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.

1(10) =

1(2)


3. Convert to binary (base 2) the fractional part: 0.999 999 999 999 999 778 042.

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) 0.999 999 999 999 999 778 042 × 2 = 1 + 0.999 999 999 999 999 556 084;
  • 2) 0.999 999 999 999 999 556 084 × 2 = 1 + 0.999 999 999 999 999 112 168;
  • 3) 0.999 999 999 999 999 112 168 × 2 = 1 + 0.999 999 999 999 998 224 336;
  • 4) 0.999 999 999 999 998 224 336 × 2 = 1 + 0.999 999 999 999 996 448 672;
  • 5) 0.999 999 999 999 996 448 672 × 2 = 1 + 0.999 999 999 999 992 897 344;
  • 6) 0.999 999 999 999 992 897 344 × 2 = 1 + 0.999 999 999 999 985 794 688;
  • 7) 0.999 999 999 999 985 794 688 × 2 = 1 + 0.999 999 999 999 971 589 376;
  • 8) 0.999 999 999 999 971 589 376 × 2 = 1 + 0.999 999 999 999 943 178 752;
  • 9) 0.999 999 999 999 943 178 752 × 2 = 1 + 0.999 999 999 999 886 357 504;
  • 10) 0.999 999 999 999 886 357 504 × 2 = 1 + 0.999 999 999 999 772 715 008;
  • 11) 0.999 999 999 999 772 715 008 × 2 = 1 + 0.999 999 999 999 545 430 016;
  • 12) 0.999 999 999 999 545 430 016 × 2 = 1 + 0.999 999 999 999 090 860 032;
  • 13) 0.999 999 999 999 090 860 032 × 2 = 1 + 0.999 999 999 998 181 720 064;
  • 14) 0.999 999 999 998 181 720 064 × 2 = 1 + 0.999 999 999 996 363 440 128;
  • 15) 0.999 999 999 996 363 440 128 × 2 = 1 + 0.999 999 999 992 726 880 256;
  • 16) 0.999 999 999 992 726 880 256 × 2 = 1 + 0.999 999 999 985 453 760 512;
  • 17) 0.999 999 999 985 453 760 512 × 2 = 1 + 0.999 999 999 970 907 521 024;
  • 18) 0.999 999 999 970 907 521 024 × 2 = 1 + 0.999 999 999 941 815 042 048;
  • 19) 0.999 999 999 941 815 042 048 × 2 = 1 + 0.999 999 999 883 630 084 096;
  • 20) 0.999 999 999 883 630 084 096 × 2 = 1 + 0.999 999 999 767 260 168 192;
  • 21) 0.999 999 999 767 260 168 192 × 2 = 1 + 0.999 999 999 534 520 336 384;
  • 22) 0.999 999 999 534 520 336 384 × 2 = 1 + 0.999 999 999 069 040 672 768;
  • 23) 0.999 999 999 069 040 672 768 × 2 = 1 + 0.999 999 998 138 081 345 536;
  • 24) 0.999 999 998 138 081 345 536 × 2 = 1 + 0.999 999 996 276 162 691 072;
  • 25) 0.999 999 996 276 162 691 072 × 2 = 1 + 0.999 999 992 552 325 382 144;
  • 26) 0.999 999 992 552 325 382 144 × 2 = 1 + 0.999 999 985 104 650 764 288;
  • 27) 0.999 999 985 104 650 764 288 × 2 = 1 + 0.999 999 970 209 301 528 576;
  • 28) 0.999 999 970 209 301 528 576 × 2 = 1 + 0.999 999 940 418 603 057 152;
  • 29) 0.999 999 940 418 603 057 152 × 2 = 1 + 0.999 999 880 837 206 114 304;
  • 30) 0.999 999 880 837 206 114 304 × 2 = 1 + 0.999 999 761 674 412 228 608;
  • 31) 0.999 999 761 674 412 228 608 × 2 = 1 + 0.999 999 523 348 824 457 216;
  • 32) 0.999 999 523 348 824 457 216 × 2 = 1 + 0.999 999 046 697 648 914 432;
  • 33) 0.999 999 046 697 648 914 432 × 2 = 1 + 0.999 998 093 395 297 828 864;
  • 34) 0.999 998 093 395 297 828 864 × 2 = 1 + 0.999 996 186 790 595 657 728;
  • 35) 0.999 996 186 790 595 657 728 × 2 = 1 + 0.999 992 373 581 191 315 456;
  • 36) 0.999 992 373 581 191 315 456 × 2 = 1 + 0.999 984 747 162 382 630 912;
  • 37) 0.999 984 747 162 382 630 912 × 2 = 1 + 0.999 969 494 324 765 261 824;
  • 38) 0.999 969 494 324 765 261 824 × 2 = 1 + 0.999 938 988 649 530 523 648;
  • 39) 0.999 938 988 649 530 523 648 × 2 = 1 + 0.999 877 977 299 061 047 296;
  • 40) 0.999 877 977 299 061 047 296 × 2 = 1 + 0.999 755 954 598 122 094 592;
  • 41) 0.999 755 954 598 122 094 592 × 2 = 1 + 0.999 511 909 196 244 189 184;
  • 42) 0.999 511 909 196 244 189 184 × 2 = 1 + 0.999 023 818 392 488 378 368;
  • 43) 0.999 023 818 392 488 378 368 × 2 = 1 + 0.998 047 636 784 976 756 736;
  • 44) 0.998 047 636 784 976 756 736 × 2 = 1 + 0.996 095 273 569 953 513 472;
  • 45) 0.996 095 273 569 953 513 472 × 2 = 1 + 0.992 190 547 139 907 026 944;
  • 46) 0.992 190 547 139 907 026 944 × 2 = 1 + 0.984 381 094 279 814 053 888;
  • 47) 0.984 381 094 279 814 053 888 × 2 = 1 + 0.968 762 188 559 628 107 776;
  • 48) 0.968 762 188 559 628 107 776 × 2 = 1 + 0.937 524 377 119 256 215 552;
  • 49) 0.937 524 377 119 256 215 552 × 2 = 1 + 0.875 048 754 238 512 431 104;
  • 50) 0.875 048 754 238 512 431 104 × 2 = 1 + 0.750 097 508 477 024 862 208;
  • 51) 0.750 097 508 477 024 862 208 × 2 = 1 + 0.500 195 016 954 049 724 416;
  • 52) 0.500 195 016 954 049 724 416 × 2 = 1 + 0.000 390 033 908 099 448 832;
  • 53) 0.000 390 033 908 099 448 832 × 2 = 0 + 0.000 780 067 816 198 897 664;

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:


0.999 999 999 999 999 778 042(10) =

0.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0(2)

5. Positive number before normalization:

1.999 999 999 999 999 778 042(10) =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 0 positions to the left, so that only one non zero digit remains to the left of it:


1.999 999 999 999 999 778 042(10) =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0(2) =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0(2) × 20


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): 0

Mantissa (not normalized):
1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

0 + 2(11-1) - 1 =

(0 + 1 023)(10) =

1 023(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;
  • 1 023 ÷ 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;

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) =

1023(10) =

011 1111 1111(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. 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0 =

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


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) =
011 1111 1111

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


Decimal number 1.999 999 999 999 999 778 042 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 1111 1111 1111 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