3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 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
3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 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: 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: 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 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) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 9 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 569 8;
  • 2) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 569 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 139 6;
  • 3) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 139 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 279 2;
  • 4) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 279 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 558 4;
  • 5) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 558 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 116 8;
  • 6) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 116 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 233 6;
  • 7) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 233 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 467 2;
  • 8) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 467 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 934 4;
  • 9) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 934 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 641 868 8;
  • 10) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 641 868 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 283 737 6;
  • 11) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 283 737 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 567 475 2;
  • 12) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 567 475 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 134 950 4;
  • 13) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 134 950 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 269 900 8;
  • 14) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 269 900 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 539 801 6;
  • 15) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 539 801 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 079 603 2;
  • 16) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 079 603 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 159 206 4;
  • 17) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 159 206 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 318 412 8;
  • 18) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 318 412 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 636 825 6;
  • 19) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 636 825 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 641 273 651 2;
  • 20) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 641 273 651 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 282 547 302 4;
  • 21) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 282 547 302 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 565 094 604 8;
  • 22) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 565 094 604 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 130 189 209 6;
  • 23) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 130 189 209 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 260 378 419 2;
  • 24) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 260 378 419 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 520 756 838 4;
  • 25) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 520 756 838 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 041 513 676 8;
  • 26) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 041 513 676 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 083 027 353 6;
  • 27) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 083 027 353 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 166 054 707 2;
  • 28) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 166 054 707 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 332 109 414 4;
  • 29) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 332 109 414 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 640 664 218 828 8;
  • 30) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 640 664 218 828 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 281 328 437 657 6;
  • 31) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 281 328 437 657 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 562 656 875 315 2;
  • 32) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 562 656 875 315 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 125 313 750 630 4;
  • 33) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 125 313 750 630 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 250 627 501 260 8;
  • 34) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 666 250 627 501 260 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 501 255 002 521 6;
  • 35) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 332 501 255 002 521 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 002 510 005 043 2;
  • 36) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 665 002 510 005 043 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 005 020 010 086 4;
  • 37) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 330 005 020 010 086 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 010 040 020 172 8;
  • 38) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 660 010 040 020 172 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 020 080 040 345 6;
  • 39) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 320 020 080 040 345 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 640 040 160 080 691 2;
  • 40) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 640 040 160 080 691 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 280 080 320 161 382 4;
  • 41) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 280 080 320 161 382 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 560 160 640 322 764 8;
  • 42) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 560 160 640 322 764 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 120 321 280 645 529 6;
  • 43) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 120 321 280 645 529 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 240 642 561 291 059 2;
  • 44) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 666 240 642 561 291 059 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 332 481 285 122 582 118 4;
  • 45) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 332 481 285 122 582 118 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 664 962 570 245 164 236 8;
  • 46) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 664 962 570 245 164 236 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 329 925 140 490 328 473 6;
  • 47) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 329 925 140 490 328 473 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 659 850 280 980 656 947 2;
  • 48) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 659 850 280 980 656 947 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 319 700 561 961 313 894 4;
  • 49) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 319 700 561 961 313 894 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 639 401 123 922 627 788 8;
  • 50) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 639 401 123 922 627 788 8 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 278 802 247 845 255 577 6;
  • 51) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 278 802 247 845 255 577 6 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 557 604 495 690 511 155 2;
  • 52) 0.666 666 666 666 666 666 666 666 666 666 666 666 666 557 604 495 690 511 155 2 × 2 = 1 + 0.333 333 333 333 333 333 333 333 333 333 333 333 333 115 208 991 381 022 310 4;
  • 53) 0.333 333 333 333 333 333 333 333 333 333 333 333 333 115 208 991 381 022 310 4 × 2 = 0 + 0.666 666 666 666 666 666 666 666 666 666 666 666 666 230 417 982 762 044 620 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:


0.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 9(10) =

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

5. Positive number before normalization:

3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 9(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:


3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 9(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 3.333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 333 284 9 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