64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 10 653 532 162 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 10 653 532 162(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

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;
  • 10 653 532 162 ÷ 2 = 5 326 766 081 + 0;
  • 5 326 766 081 ÷ 2 = 2 663 383 040 + 1;
  • 2 663 383 040 ÷ 2 = 1 331 691 520 + 0;
  • 1 331 691 520 ÷ 2 = 665 845 760 + 0;
  • 665 845 760 ÷ 2 = 332 922 880 + 0;
  • 332 922 880 ÷ 2 = 166 461 440 + 0;
  • 166 461 440 ÷ 2 = 83 230 720 + 0;
  • 83 230 720 ÷ 2 = 41 615 360 + 0;
  • 41 615 360 ÷ 2 = 20 807 680 + 0;
  • 20 807 680 ÷ 2 = 10 403 840 + 0;
  • 10 403 840 ÷ 2 = 5 201 920 + 0;
  • 5 201 920 ÷ 2 = 2 600 960 + 0;
  • 2 600 960 ÷ 2 = 1 300 480 + 0;
  • 1 300 480 ÷ 2 = 650 240 + 0;
  • 650 240 ÷ 2 = 325 120 + 0;
  • 325 120 ÷ 2 = 162 560 + 0;
  • 162 560 ÷ 2 = 81 280 + 0;
  • 81 280 ÷ 2 = 40 640 + 0;
  • 40 640 ÷ 2 = 20 320 + 0;
  • 20 320 ÷ 2 = 10 160 + 0;
  • 10 160 ÷ 2 = 5 080 + 0;
  • 5 080 ÷ 2 = 2 540 + 0;
  • 2 540 ÷ 2 = 1 270 + 0;
  • 1 270 ÷ 2 = 635 + 0;
  • 635 ÷ 2 = 317 + 1;
  • 317 ÷ 2 = 158 + 1;
  • 158 ÷ 2 = 79 + 0;
  • 79 ÷ 2 = 39 + 1;
  • 39 ÷ 2 = 19 + 1;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number.

Take all the remainders starting from the bottom of the list constructed above.


10 653 532 162(10) =


10 0111 1011 0000 0000 0000 0000 0000 0010(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 33 positions to the left, so that only one non zero digit remains to the left of it:


10 653 532 162(10) =


10 0111 1011 0000 0000 0000 0000 0000 0010(2) =


10 0111 1011 0000 0000 0000 0000 0000 0010(2) × 20 =


1.0011 1101 1000 0000 0000 0000 0000 0001 0(2) × 233


4. 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): 33


Mantissa (not normalized):
1.0011 1101 1000 0000 0000 0000 0000 0001 0


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


33 + 2(11-1) - 1 =


(33 + 1 023)(10) =


1 056(10)


6. 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;
  • 1 056 ÷ 2 = 528 + 0;
  • 528 ÷ 2 = 264 + 0;
  • 264 ÷ 2 = 132 + 0;
  • 132 ÷ 2 = 66 + 0;
  • 66 ÷ 2 = 33 + 0;
  • 33 ÷ 2 = 16 + 1;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

7. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


1056(10) =


100 0010 0000(2)


8. 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 adding the necessary number of zeros to the right.


Mantissa (normalized) =


1. 0 0111 1011 0000 0000 0000 0000 0000 0010 000 0000 0000 0000 0000 =


0011 1101 1000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000


9. 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 0010 0000


Mantissa (52 bits) =
0011 1101 1000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000


The base ten decimal number 10 653 532 162 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 0010 0000 - 0011 1101 1000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000

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