-1.833 880 941 75 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -1.833 880 941 75(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
-1.833 880 941 75(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:

|-1.833 880 941 75| = 1.833 880 941 75


2. 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;

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.

1(10) =

1(2)


4. Convert to binary (base 2) the fractional part: 0.833 880 941 75.

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) 0.833 880 941 75 × 2 = 1 + 0.667 761 883 5;
  • 2) 0.667 761 883 5 × 2 = 1 + 0.335 523 767;
  • 3) 0.335 523 767 × 2 = 0 + 0.671 047 534;
  • 4) 0.671 047 534 × 2 = 1 + 0.342 095 068;
  • 5) 0.342 095 068 × 2 = 0 + 0.684 190 136;
  • 6) 0.684 190 136 × 2 = 1 + 0.368 380 272;
  • 7) 0.368 380 272 × 2 = 0 + 0.736 760 544;
  • 8) 0.736 760 544 × 2 = 1 + 0.473 521 088;
  • 9) 0.473 521 088 × 2 = 0 + 0.947 042 176;
  • 10) 0.947 042 176 × 2 = 1 + 0.894 084 352;
  • 11) 0.894 084 352 × 2 = 1 + 0.788 168 704;
  • 12) 0.788 168 704 × 2 = 1 + 0.576 337 408;
  • 13) 0.576 337 408 × 2 = 1 + 0.152 674 816;
  • 14) 0.152 674 816 × 2 = 0 + 0.305 349 632;
  • 15) 0.305 349 632 × 2 = 0 + 0.610 699 264;
  • 16) 0.610 699 264 × 2 = 1 + 0.221 398 528;
  • 17) 0.221 398 528 × 2 = 0 + 0.442 797 056;
  • 18) 0.442 797 056 × 2 = 0 + 0.885 594 112;
  • 19) 0.885 594 112 × 2 = 1 + 0.771 188 224;
  • 20) 0.771 188 224 × 2 = 1 + 0.542 376 448;
  • 21) 0.542 376 448 × 2 = 1 + 0.084 752 896;
  • 22) 0.084 752 896 × 2 = 0 + 0.169 505 792;
  • 23) 0.169 505 792 × 2 = 0 + 0.339 011 584;
  • 24) 0.339 011 584 × 2 = 0 + 0.678 023 168;
  • 25) 0.678 023 168 × 2 = 1 + 0.356 046 336;
  • 26) 0.356 046 336 × 2 = 0 + 0.712 092 672;
  • 27) 0.712 092 672 × 2 = 1 + 0.424 185 344;
  • 28) 0.424 185 344 × 2 = 0 + 0.848 370 688;
  • 29) 0.848 370 688 × 2 = 1 + 0.696 741 376;
  • 30) 0.696 741 376 × 2 = 1 + 0.393 482 752;
  • 31) 0.393 482 752 × 2 = 0 + 0.786 965 504;
  • 32) 0.786 965 504 × 2 = 1 + 0.573 931 008;
  • 33) 0.573 931 008 × 2 = 1 + 0.147 862 016;
  • 34) 0.147 862 016 × 2 = 0 + 0.295 724 032;
  • 35) 0.295 724 032 × 2 = 0 + 0.591 448 064;
  • 36) 0.591 448 064 × 2 = 1 + 0.182 896 128;
  • 37) 0.182 896 128 × 2 = 0 + 0.365 792 256;
  • 38) 0.365 792 256 × 2 = 0 + 0.731 584 512;
  • 39) 0.731 584 512 × 2 = 1 + 0.463 169 024;
  • 40) 0.463 169 024 × 2 = 0 + 0.926 338 048;
  • 41) 0.926 338 048 × 2 = 1 + 0.852 676 096;
  • 42) 0.852 676 096 × 2 = 1 + 0.705 352 192;
  • 43) 0.705 352 192 × 2 = 1 + 0.410 704 384;
  • 44) 0.410 704 384 × 2 = 0 + 0.821 408 768;
  • 45) 0.821 408 768 × 2 = 1 + 0.642 817 536;
  • 46) 0.642 817 536 × 2 = 1 + 0.285 635 072;
  • 47) 0.285 635 072 × 2 = 0 + 0.571 270 144;
  • 48) 0.571 270 144 × 2 = 1 + 0.142 540 288;
  • 49) 0.142 540 288 × 2 = 0 + 0.285 080 576;
  • 50) 0.285 080 576 × 2 = 0 + 0.570 161 152;
  • 51) 0.570 161 152 × 2 = 1 + 0.140 322 304;
  • 52) 0.140 322 304 × 2 = 0 + 0.280 644 608;
  • 53) 0.280 644 608 × 2 = 0 + 0.561 289 216;

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:


0.833 880 941 75(10) =

0.1101 0101 0111 1001 0011 1000 1010 1101 1001 0010 1110 1101 0010 0(2)

6. Positive number before normalization:

1.833 880 941 75(10) =

1.1101 0101 0111 1001 0011 1000 1010 1101 1001 0010 1110 1101 0010 0(2)

7. 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:


1.833 880 941 75(10) =

1.1101 0101 0111 1001 0011 1000 1010 1101 1001 0010 1110 1101 0010 0(2) =

1.1101 0101 0111 1001 0011 1000 1010 1101 1001 0010 1110 1101 0010 0(2) × 20


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): 0

Mantissa (not normalized):
1.1101 0101 0111 1001 0011 1000 1010 1101 1001 0010 1110 1101 0010 0


9. 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)


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;
  • 1 023 ÷ 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;

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) =

1023(10) =

011 1111 1111(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, 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. 1101 0101 0111 1001 0011 1000 1010 1101 1001 0010 1110 1101 0010 0 =

1101 0101 0111 1001 0011 1000 1010 1101 1001 0010 1110 1101 0010


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 1111

Mantissa (52 bits) =
1101 0101 0111 1001 0011 1000 1010 1101 1001 0010 1110 1101 0010


Decimal number -1.833 880 941 75 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 011 1111 1111 - 1101 0101 0111 1001 0011 1000 1010 1101 1001 0010 1110 1101 0010

» Convert decimal -1.833 880 941 87 to 64 bit double precision IEEE 754 binary floating point representation standard

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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