3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 3(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
3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 3(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: 3.
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;
  • 3 ÷ 2 = 1 + 1;
  • 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.

3(10) =

11(2)


3. Convert to binary (base 2) the fractional part: 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 3.

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) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 3 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 622 6;
  • 2) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 622 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 245 2;
  • 3) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 245 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 490 4;
  • 4) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 490 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 980 8;
  • 5) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 980 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 961 6;
  • 6) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 961 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 923 2;
  • 7) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 923 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 846 4;
  • 8) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 846 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 692 8;
  • 9) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 692 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 655 385 6;
  • 10) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 655 385 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 310 771 2;
  • 11) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 310 771 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 621 542 4;
  • 12) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 621 542 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 243 084 8;
  • 13) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 243 084 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 486 169 6;
  • 14) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 486 169 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 972 339 2;
  • 15) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 972 339 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 944 678 4;
  • 16) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 944 678 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 889 356 8;
  • 17) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 889 356 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 778 713 6;
  • 18) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 778 713 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 557 427 2;
  • 19) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 557 427 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 655 114 854 4;
  • 20) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 655 114 854 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 310 229 708 8;
  • 21) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 310 229 708 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 620 459 417 6;
  • 22) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 620 459 417 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 240 918 835 2;
  • 23) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 240 918 835 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 481 837 670 4;
  • 24) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 481 837 670 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 963 675 340 8;
  • 25) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 963 675 340 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 927 350 681 6;
  • 26) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 927 350 681 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 854 701 363 2;
  • 27) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 854 701 363 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 709 402 726 4;
  • 28) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 709 402 726 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 418 805 452 8;
  • 29) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 418 805 452 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 654 837 610 905 6;
  • 30) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 654 837 610 905 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 309 675 221 811 2;
  • 31) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 309 675 221 811 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 619 350 443 622 4;
  • 32) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 619 350 443 622 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 238 700 887 244 8;
  • 33) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 238 700 887 244 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 477 401 774 489 6;
  • 34) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 477 401 774 489 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 954 803 548 979 2;
  • 35) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 954 803 548 979 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 909 607 097 958 4;
  • 36) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 909 607 097 958 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 819 214 195 916 8;
  • 37) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 331 819 214 195 916 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 638 428 391 833 6;
  • 38) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 663 638 428 391 833 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 276 856 783 667 2;
  • 39) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 327 276 856 783 667 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 654 553 713 567 334 4;
  • 40) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 654 553 713 567 334 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 309 107 427 134 668 8;
  • 41) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 309 107 427 134 668 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 618 214 854 269 337 6;
  • 42) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 618 214 854 269 337 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 236 429 708 538 675 2;
  • 43) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 236 429 708 538 675 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 472 859 417 077 350 4;
  • 44) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 472 859 417 077 350 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 332 945 718 834 154 700 8;
  • 45) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 332 945 718 834 154 700 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 665 891 437 668 309 401 6;
  • 46) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 665 891 437 668 309 401 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 331 782 875 336 618 803 2;
  • 47) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 331 782 875 336 618 803 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 663 565 750 673 237 606 4;
  • 48) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 663 565 750 673 237 606 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 327 131 501 346 475 212 8;
  • 49) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 327 131 501 346 475 212 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 654 263 002 692 950 425 6;
  • 50) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 654 263 002 692 950 425 6 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 308 526 005 385 900 851 2;
  • 51) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 308 526 005 385 900 851 2 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 617 052 010 771 801 702 4;
  • 52) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 617 052 010 771 801 702 4 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 234 104 021 543 603 404 8;
  • 53) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 234 104 021 543 603 404 8 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 468 208 043 087 206 809 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:


0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 3(10) =

0.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0(2)

5. Positive number before normalization:

3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 3(10) =

11.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 1 positions to the left, so that only one non zero digit remains to the left of it:


3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 3(10) =

11.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0(2) =

11.0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0(2) × 20 =

1.1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 10(2) × 21


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): 1

Mantissa (not normalized):
1.1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 10


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

1 + 2(11-1) - 1 =

(1 + 1 023)(10) =

1 024(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;
  • 1 024 ÷ 2 = 512 + 0;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

1024(10) =

100 0000 0000(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. 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 10 =

1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010


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) =
100 0000 0000

Mantissa (52 bits) =
1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010


Decimal number 3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 311 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010

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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