1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 104 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 104 8(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
1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 104 8(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: 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 104 8.

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) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 104 8 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 209 6;
  • 2) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 209 6 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 419 2;
  • 3) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 419 2 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 838 4;
  • 4) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 838 4 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 676 8;
  • 5) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 676 8 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 353 6;
  • 6) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 353 6 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 707 2;
  • 7) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 707 2 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 414 4;
  • 8) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 414 4 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 442 828 8;
  • 9) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 442 828 8 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 885 657 6;
  • 10) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 885 657 6 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 771 315 2;
  • 11) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 771 315 2 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 542 630 4;
  • 12) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 542 630 4 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 085 260 8;
  • 13) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 085 260 8 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 170 521 6;
  • 14) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 170 521 6 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 341 043 2;
  • 15) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 341 043 2 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 682 086 4;
  • 16) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 682 086 4 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 364 172 8;
  • 17) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 364 172 8 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 554 728 345 6;
  • 18) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 554 728 345 6 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 109 456 691 2;
  • 19) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 109 456 691 2 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 218 913 382 4;
  • 20) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 218 913 382 4 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 437 826 764 8;
  • 21) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 437 826 764 8 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 875 653 529 6;
  • 22) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 875 653 529 6 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 751 307 059 2;
  • 23) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 751 307 059 2 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 502 614 118 4;
  • 24) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 502 614 118 4 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 005 228 236 8;
  • 25) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 005 228 236 8 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 010 456 473 6;
  • 26) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 010 456 473 6 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 020 912 947 2;
  • 27) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 020 912 947 2 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 041 825 894 4;
  • 28) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 041 825 894 4 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 776 083 651 788 8;
  • 29) 0.777 777 777 777 777 777 777 777 777 777 777 777 776 083 651 788 8 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 552 167 303 577 6;
  • 30) 0.555 555 555 555 555 555 555 555 555 555 555 555 552 167 303 577 6 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 104 334 607 155 2;
  • 31) 0.111 111 111 111 111 111 111 111 111 111 111 111 104 334 607 155 2 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 208 669 214 310 4;
  • 32) 0.222 222 222 222 222 222 222 222 222 222 222 222 208 669 214 310 4 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 417 338 428 620 8;
  • 33) 0.444 444 444 444 444 444 444 444 444 444 444 444 417 338 428 620 8 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 834 676 857 241 6;
  • 34) 0.888 888 888 888 888 888 888 888 888 888 888 888 834 676 857 241 6 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 669 353 714 483 2;
  • 35) 0.777 777 777 777 777 777 777 777 777 777 777 777 669 353 714 483 2 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 338 707 428 966 4;
  • 36) 0.555 555 555 555 555 555 555 555 555 555 555 555 338 707 428 966 4 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 110 677 414 857 932 8;
  • 37) 0.111 111 111 111 111 111 111 111 111 111 111 110 677 414 857 932 8 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 221 354 829 715 865 6;
  • 38) 0.222 222 222 222 222 222 222 222 222 222 222 221 354 829 715 865 6 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 442 709 659 431 731 2;
  • 39) 0.444 444 444 444 444 444 444 444 444 444 444 442 709 659 431 731 2 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 885 419 318 863 462 4;
  • 40) 0.888 888 888 888 888 888 888 888 888 888 888 885 419 318 863 462 4 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 770 838 637 726 924 8;
  • 41) 0.777 777 777 777 777 777 777 777 777 777 777 770 838 637 726 924 8 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 541 677 275 453 849 6;
  • 42) 0.555 555 555 555 555 555 555 555 555 555 555 541 677 275 453 849 6 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 083 354 550 907 699 2;
  • 43) 0.111 111 111 111 111 111 111 111 111 111 111 083 354 550 907 699 2 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 166 709 101 815 398 4;
  • 44) 0.222 222 222 222 222 222 222 222 222 222 222 166 709 101 815 398 4 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 333 418 203 630 796 8;
  • 45) 0.444 444 444 444 444 444 444 444 444 444 444 333 418 203 630 796 8 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 666 836 407 261 593 6;
  • 46) 0.888 888 888 888 888 888 888 888 888 888 888 666 836 407 261 593 6 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 333 672 814 523 187 2;
  • 47) 0.777 777 777 777 777 777 777 777 777 777 777 333 672 814 523 187 2 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 554 667 345 629 046 374 4;
  • 48) 0.555 555 555 555 555 555 555 555 555 555 554 667 345 629 046 374 4 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 109 334 691 258 092 748 8;
  • 49) 0.111 111 111 111 111 111 111 111 111 111 109 334 691 258 092 748 8 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 218 669 382 516 185 497 6;
  • 50) 0.222 222 222 222 222 222 222 222 222 222 218 669 382 516 185 497 6 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 437 338 765 032 370 995 2;
  • 51) 0.444 444 444 444 444 444 444 444 444 444 437 338 765 032 370 995 2 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 874 677 530 064 741 990 4;
  • 52) 0.888 888 888 888 888 888 888 888 888 888 874 677 530 064 741 990 4 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 749 355 060 129 483 980 8;
  • 53) 0.777 777 777 777 777 777 777 777 777 777 749 355 060 129 483 980 8 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 498 710 120 258 967 961 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:


0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 104 8(10) =

0.0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1(2)

5. Positive number before normalization:

1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 104 8(10) =

1.0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1(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:


1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 104 8(10) =

1.0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1(2) =

1.0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1(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):
1.0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1


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. 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1 =

0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001


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) =
0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001


Decimal number 1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 104 8 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001

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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