64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 1 157 890 338 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 1 157 890 338(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;
  • 1 157 890 338 ÷ 2 = 578 945 169 + 0;
  • 578 945 169 ÷ 2 = 289 472 584 + 1;
  • 289 472 584 ÷ 2 = 144 736 292 + 0;
  • 144 736 292 ÷ 2 = 72 368 146 + 0;
  • 72 368 146 ÷ 2 = 36 184 073 + 0;
  • 36 184 073 ÷ 2 = 18 092 036 + 1;
  • 18 092 036 ÷ 2 = 9 046 018 + 0;
  • 9 046 018 ÷ 2 = 4 523 009 + 0;
  • 4 523 009 ÷ 2 = 2 261 504 + 1;
  • 2 261 504 ÷ 2 = 1 130 752 + 0;
  • 1 130 752 ÷ 2 = 565 376 + 0;
  • 565 376 ÷ 2 = 282 688 + 0;
  • 282 688 ÷ 2 = 141 344 + 0;
  • 141 344 ÷ 2 = 70 672 + 0;
  • 70 672 ÷ 2 = 35 336 + 0;
  • 35 336 ÷ 2 = 17 668 + 0;
  • 17 668 ÷ 2 = 8 834 + 0;
  • 8 834 ÷ 2 = 4 417 + 0;
  • 4 417 ÷ 2 = 2 208 + 1;
  • 2 208 ÷ 2 = 1 104 + 0;
  • 1 104 ÷ 2 = 552 + 0;
  • 552 ÷ 2 = 276 + 0;
  • 276 ÷ 2 = 138 + 0;
  • 138 ÷ 2 = 69 + 0;
  • 69 ÷ 2 = 34 + 1;
  • 34 ÷ 2 = 17 + 0;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 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.


1 157 890 338(10) =


100 0101 0000 0100 0000 0001 0010 0010(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 30 positions to the left, so that only one non zero digit remains to the left of it:


1 157 890 338(10) =


100 0101 0000 0100 0000 0001 0010 0010(2) =


100 0101 0000 0100 0000 0001 0010 0010(2) × 20 =


1.0001 0100 0001 0000 0000 0100 1000 10(2) × 230


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): 30


Mantissa (not normalized):
1.0001 0100 0001 0000 0000 0100 1000 10


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


30 + 2(11-1) - 1 =


(30 + 1 023)(10) =


1 053(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;
  • 1 053 ÷ 2 = 526 + 1;
  • 526 ÷ 2 = 263 + 0;
  • 263 ÷ 2 = 131 + 1;
  • 131 ÷ 2 = 65 + 1;
  • 65 ÷ 2 = 32 + 1;
  • 32 ÷ 2 = 16 + 0;
  • 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) =


1053(10) =


100 0001 1101(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. 00 0101 0000 0100 0000 0001 0010 0010 00 0000 0000 0000 0000 0000 =


0001 0100 0001 0000 0000 0100 1000 1000 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 0001 1101


Mantissa (52 bits) =
0001 0100 0001 0000 0000 0100 1000 1000 0000 0000 0000 0000 0000


The base ten decimal number 1 157 890 338 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 0001 1101 - 0001 0100 0001 0000 0000 0100 1000 1000 0000 0000 0000 0000 0000

The latest decimal numbers converted from base ten to 64 bit double precision IEEE 754 floating point binary standard representation