1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 103 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 103 9(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
1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 103 9(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: 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 103 9.

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) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 103 9 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 207 8;
  • 2) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 207 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 415 6;
  • 3) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 415 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 831 2;
  • 4) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 831 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 662 4;
  • 5) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 662 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 324 8;
  • 6) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 324 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 649 6;
  • 7) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 649 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 299 2;
  • 8) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 299 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 442 598 4;
  • 9) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 442 598 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 885 196 8;
  • 10) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 885 196 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 770 393 6;
  • 11) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 770 393 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 540 787 2;
  • 12) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 555 540 787 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 081 574 4;
  • 13) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 081 574 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 163 148 8;
  • 14) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 222 163 148 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 326 297 6;
  • 15) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 444 326 297 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 652 595 2;
  • 16) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 888 652 595 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 305 190 4;
  • 17) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 777 305 190 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 554 610 380 8;
  • 18) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 554 610 380 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 111 109 220 761 6;
  • 19) 0.111 111 111 111 111 111 111 111 111 111 111 111 111 109 220 761 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 222 218 441 523 2;
  • 20) 0.222 222 222 222 222 222 222 222 222 222 222 222 222 218 441 523 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 444 436 883 046 4;
  • 21) 0.444 444 444 444 444 444 444 444 444 444 444 444 444 436 883 046 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 888 873 766 092 8;
  • 22) 0.888 888 888 888 888 888 888 888 888 888 888 888 888 873 766 092 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 777 747 532 185 6;
  • 23) 0.777 777 777 777 777 777 777 777 777 777 777 777 777 747 532 185 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 555 495 064 371 2;
  • 24) 0.555 555 555 555 555 555 555 555 555 555 555 555 555 495 064 371 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 110 990 128 742 4;
  • 25) 0.111 111 111 111 111 111 111 111 111 111 111 111 110 990 128 742 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 221 980 257 484 8;
  • 26) 0.222 222 222 222 222 222 222 222 222 222 222 222 221 980 257 484 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 443 960 514 969 6;
  • 27) 0.444 444 444 444 444 444 444 444 444 444 444 444 443 960 514 969 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 887 921 029 939 2;
  • 28) 0.888 888 888 888 888 888 888 888 888 888 888 888 887 921 029 939 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 775 842 059 878 4;
  • 29) 0.777 777 777 777 777 777 777 777 777 777 777 777 775 842 059 878 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 551 684 119 756 8;
  • 30) 0.555 555 555 555 555 555 555 555 555 555 555 555 551 684 119 756 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 111 103 368 239 513 6;
  • 31) 0.111 111 111 111 111 111 111 111 111 111 111 111 103 368 239 513 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 222 206 736 479 027 2;
  • 32) 0.222 222 222 222 222 222 222 222 222 222 222 222 206 736 479 027 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 444 413 472 958 054 4;
  • 33) 0.444 444 444 444 444 444 444 444 444 444 444 444 413 472 958 054 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 888 826 945 916 108 8;
  • 34) 0.888 888 888 888 888 888 888 888 888 888 888 888 826 945 916 108 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 777 653 891 832 217 6;
  • 35) 0.777 777 777 777 777 777 777 777 777 777 777 777 653 891 832 217 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 555 307 783 664 435 2;
  • 36) 0.555 555 555 555 555 555 555 555 555 555 555 555 307 783 664 435 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 110 615 567 328 870 4;
  • 37) 0.111 111 111 111 111 111 111 111 111 111 111 110 615 567 328 870 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 221 231 134 657 740 8;
  • 38) 0.222 222 222 222 222 222 222 222 222 222 222 221 231 134 657 740 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 442 462 269 315 481 6;
  • 39) 0.444 444 444 444 444 444 444 444 444 444 444 442 462 269 315 481 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 884 924 538 630 963 2;
  • 40) 0.888 888 888 888 888 888 888 888 888 888 888 884 924 538 630 963 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 769 849 077 261 926 4;
  • 41) 0.777 777 777 777 777 777 777 777 777 777 777 769 849 077 261 926 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 555 539 698 154 523 852 8;
  • 42) 0.555 555 555 555 555 555 555 555 555 555 555 539 698 154 523 852 8 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 111 079 396 309 047 705 6;
  • 43) 0.111 111 111 111 111 111 111 111 111 111 111 079 396 309 047 705 6 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 222 158 792 618 095 411 2;
  • 44) 0.222 222 222 222 222 222 222 222 222 222 222 158 792 618 095 411 2 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 444 317 585 236 190 822 4;
  • 45) 0.444 444 444 444 444 444 444 444 444 444 444 317 585 236 190 822 4 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 888 635 170 472 381 644 8;
  • 46) 0.888 888 888 888 888 888 888 888 888 888 888 635 170 472 381 644 8 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 777 270 340 944 763 289 6;
  • 47) 0.777 777 777 777 777 777 777 777 777 777 777 270 340 944 763 289 6 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 554 540 681 889 526 579 2;
  • 48) 0.555 555 555 555 555 555 555 555 555 555 554 540 681 889 526 579 2 × 2 = 1 + 0.111 111 111 111 111 111 111 111 111 111 109 081 363 779 053 158 4;
  • 49) 0.111 111 111 111 111 111 111 111 111 111 109 081 363 779 053 158 4 × 2 = 0 + 0.222 222 222 222 222 222 222 222 222 222 218 162 727 558 106 316 8;
  • 50) 0.222 222 222 222 222 222 222 222 222 222 218 162 727 558 106 316 8 × 2 = 0 + 0.444 444 444 444 444 444 444 444 444 444 436 325 455 116 212 633 6;
  • 51) 0.444 444 444 444 444 444 444 444 444 444 436 325 455 116 212 633 6 × 2 = 0 + 0.888 888 888 888 888 888 888 888 888 888 872 650 910 232 425 267 2;
  • 52) 0.888 888 888 888 888 888 888 888 888 888 872 650 910 232 425 267 2 × 2 = 1 + 0.777 777 777 777 777 777 777 777 777 777 745 301 820 464 850 534 4;
  • 53) 0.777 777 777 777 777 777 777 777 777 777 745 301 820 464 850 534 4 × 2 = 1 + 0.555 555 555 555 555 555 555 555 555 555 490 603 640 929 701 068 8;

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:


0.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 103 9(10) =

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

5. Positive number before normalization:

1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 103 9(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:


1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 103 9(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 1.111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 103 9 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