-0.000 281 877 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 281 877(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
-0.000 281 877(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:

|-0.000 281 877| = 0.000 281 877


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: 0.000 281 877.

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) 0.000 281 877 × 2 = 0 + 0.000 563 754;
  • 2) 0.000 563 754 × 2 = 0 + 0.001 127 508;
  • 3) 0.001 127 508 × 2 = 0 + 0.002 255 016;
  • 4) 0.002 255 016 × 2 = 0 + 0.004 510 032;
  • 5) 0.004 510 032 × 2 = 0 + 0.009 020 064;
  • 6) 0.009 020 064 × 2 = 0 + 0.018 040 128;
  • 7) 0.018 040 128 × 2 = 0 + 0.036 080 256;
  • 8) 0.036 080 256 × 2 = 0 + 0.072 160 512;
  • 9) 0.072 160 512 × 2 = 0 + 0.144 321 024;
  • 10) 0.144 321 024 × 2 = 0 + 0.288 642 048;
  • 11) 0.288 642 048 × 2 = 0 + 0.577 284 096;
  • 12) 0.577 284 096 × 2 = 1 + 0.154 568 192;
  • 13) 0.154 568 192 × 2 = 0 + 0.309 136 384;
  • 14) 0.309 136 384 × 2 = 0 + 0.618 272 768;
  • 15) 0.618 272 768 × 2 = 1 + 0.236 545 536;
  • 16) 0.236 545 536 × 2 = 0 + 0.473 091 072;
  • 17) 0.473 091 072 × 2 = 0 + 0.946 182 144;
  • 18) 0.946 182 144 × 2 = 1 + 0.892 364 288;
  • 19) 0.892 364 288 × 2 = 1 + 0.784 728 576;
  • 20) 0.784 728 576 × 2 = 1 + 0.569 457 152;
  • 21) 0.569 457 152 × 2 = 1 + 0.138 914 304;
  • 22) 0.138 914 304 × 2 = 0 + 0.277 828 608;
  • 23) 0.277 828 608 × 2 = 0 + 0.555 657 216;
  • 24) 0.555 657 216 × 2 = 1 + 0.111 314 432;
  • 25) 0.111 314 432 × 2 = 0 + 0.222 628 864;
  • 26) 0.222 628 864 × 2 = 0 + 0.445 257 728;
  • 27) 0.445 257 728 × 2 = 0 + 0.890 515 456;
  • 28) 0.890 515 456 × 2 = 1 + 0.781 030 912;
  • 29) 0.781 030 912 × 2 = 1 + 0.562 061 824;
  • 30) 0.562 061 824 × 2 = 1 + 0.124 123 648;
  • 31) 0.124 123 648 × 2 = 0 + 0.248 247 296;
  • 32) 0.248 247 296 × 2 = 0 + 0.496 494 592;
  • 33) 0.496 494 592 × 2 = 0 + 0.992 989 184;
  • 34) 0.992 989 184 × 2 = 1 + 0.985 978 368;
  • 35) 0.985 978 368 × 2 = 1 + 0.971 956 736;
  • 36) 0.971 956 736 × 2 = 1 + 0.943 913 472;
  • 37) 0.943 913 472 × 2 = 1 + 0.887 826 944;
  • 38) 0.887 826 944 × 2 = 1 + 0.775 653 888;
  • 39) 0.775 653 888 × 2 = 1 + 0.551 307 776;
  • 40) 0.551 307 776 × 2 = 1 + 0.102 615 552;
  • 41) 0.102 615 552 × 2 = 0 + 0.205 231 104;
  • 42) 0.205 231 104 × 2 = 0 + 0.410 462 208;
  • 43) 0.410 462 208 × 2 = 0 + 0.820 924 416;
  • 44) 0.820 924 416 × 2 = 1 + 0.641 848 832;
  • 45) 0.641 848 832 × 2 = 1 + 0.283 697 664;
  • 46) 0.283 697 664 × 2 = 0 + 0.567 395 328;
  • 47) 0.567 395 328 × 2 = 1 + 0.134 790 656;
  • 48) 0.134 790 656 × 2 = 0 + 0.269 581 312;
  • 49) 0.269 581 312 × 2 = 0 + 0.539 162 624;
  • 50) 0.539 162 624 × 2 = 1 + 0.078 325 248;
  • 51) 0.078 325 248 × 2 = 0 + 0.156 650 496;
  • 52) 0.156 650 496 × 2 = 0 + 0.313 300 992;
  • 53) 0.313 300 992 × 2 = 0 + 0.626 601 984;
  • 54) 0.626 601 984 × 2 = 1 + 0.253 203 968;
  • 55) 0.253 203 968 × 2 = 0 + 0.506 407 936;
  • 56) 0.506 407 936 × 2 = 1 + 0.012 815 872;
  • 57) 0.012 815 872 × 2 = 0 + 0.025 631 744;
  • 58) 0.025 631 744 × 2 = 0 + 0.051 263 488;
  • 59) 0.051 263 488 × 2 = 0 + 0.102 526 976;
  • 60) 0.102 526 976 × 2 = 0 + 0.205 053 952;
  • 61) 0.205 053 952 × 2 = 0 + 0.410 107 904;
  • 62) 0.410 107 904 × 2 = 0 + 0.820 215 808;
  • 63) 0.820 215 808 × 2 = 1 + 0.640 431 616;
  • 64) 0.640 431 616 × 2 = 1 + 0.280 863 232;

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:


0.000 281 877(10) =

0.0000 0000 0001 0010 0111 1001 0001 1100 0111 1111 0001 1010 0100 0101 0000 0011(2)

6. Positive number before normalization:

0.000 281 877(10) =

0.0000 0000 0001 0010 0111 1001 0001 1100 0111 1111 0001 1010 0100 0101 0000 0011(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 12 positions to the right, so that only one non zero digit remains to the left of it:


0.000 281 877(10) =

0.0000 0000 0001 0010 0111 1001 0001 1100 0111 1111 0001 1010 0100 0101 0000 0011(2) =

0.0000 0000 0001 0010 0111 1001 0001 1100 0111 1111 0001 1010 0100 0101 0000 0011(2) × 20 =

1.0010 0111 1001 0001 1100 0111 1111 0001 1010 0100 0101 0000 0011(2) × 2-12


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): -12

Mantissa (not normalized):
1.0010 0111 1001 0001 1100 0111 1111 0001 1010 0100 0101 0000 0011


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-12 + 2(11-1) - 1 =

(-12 + 1 023)(10) =

1 011(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;
  • 1 011 ÷ 2 = 505 + 1;
  • 505 ÷ 2 = 252 + 1;
  • 252 ÷ 2 = 126 + 0;
  • 126 ÷ 2 = 63 + 0;
  • 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) =

1011(10) =

011 1111 0011(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. 0010 0111 1001 0001 1100 0111 1111 0001 1010 0100 0101 0000 0011 =

0010 0111 1001 0001 1100 0111 1111 0001 1010 0100 0101 0000 0011


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 0011

Mantissa (52 bits) =
0010 0111 1001 0001 1100 0111 1111 0001 1010 0100 0101 0000 0011


Decimal number -0.000 281 877 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 0011 - 0010 0111 1001 0001 1100 0111 1111 0001 1010 0100 0101 0000 0011

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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