1.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 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
1.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 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: 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: 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 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) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 3 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 988 6;
  • 2) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 988 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 977 2;
  • 3) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 977 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 954 4;
  • 4) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 954 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 908 8;
  • 5) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 908 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 817 6;
  • 6) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 817 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 635 2;
  • 7) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 635 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 270 4;
  • 8) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 270 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 540 8;
  • 9) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 540 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 081 6;
  • 10) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 081 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 163 2;
  • 11) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 163 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 988 326 4;
  • 12) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 988 326 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 976 652 8;
  • 13) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 976 652 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 953 305 6;
  • 14) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 953 305 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 906 611 2;
  • 15) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 906 611 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 813 222 4;
  • 16) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 813 222 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 626 444 8;
  • 17) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 626 444 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 252 889 6;
  • 18) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 252 889 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 505 779 2;
  • 19) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 505 779 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 011 558 4;
  • 20) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 011 558 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 023 116 8;
  • 21) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 023 116 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 988 046 233 6;
  • 22) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 988 046 233 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 976 092 467 2;
  • 23) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 976 092 467 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 952 184 934 4;
  • 24) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 952 184 934 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 904 369 868 8;
  • 25) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 904 369 868 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 808 739 737 6;
  • 26) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 808 739 737 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 617 479 475 2;
  • 27) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 617 479 475 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 234 958 950 4;
  • 28) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 234 958 950 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 469 917 900 8;
  • 29) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 469 917 900 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 996 939 835 801 6;
  • 30) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 996 939 835 801 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 993 879 671 603 2;
  • 31) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 993 879 671 603 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 759 343 206 4;
  • 32) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 759 343 206 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 975 518 686 412 8;
  • 33) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 975 518 686 412 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 951 037 372 825 6;
  • 34) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 951 037 372 825 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 902 074 745 651 2;
  • 35) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 902 074 745 651 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 804 149 491 302 4;
  • 36) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 804 149 491 302 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 608 298 982 604 8;
  • 37) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 608 298 982 604 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 216 597 965 209 6;
  • 38) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 216 597 965 209 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 433 195 930 419 2;
  • 39) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 433 195 930 419 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 996 866 391 860 838 4;
  • 40) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 996 866 391 860 838 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 993 732 783 721 676 8;
  • 41) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 993 732 783 721 676 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 465 567 443 353 6;
  • 42) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 465 567 443 353 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 974 931 134 886 707 2;
  • 43) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 974 931 134 886 707 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 949 862 269 773 414 4;
  • 44) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 949 862 269 773 414 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 899 724 539 546 828 8;
  • 45) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 899 724 539 546 828 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 799 449 079 093 657 6;
  • 46) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 799 449 079 093 657 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 598 898 158 187 315 2;
  • 47) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 598 898 158 187 315 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 197 796 316 374 630 4;
  • 48) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 197 796 316 374 630 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 395 592 632 749 260 8;
  • 49) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 395 592 632 749 260 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 996 791 185 265 498 521 6;
  • 50) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 996 791 185 265 498 521 6 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 993 582 370 530 997 043 2;
  • 51) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 993 582 370 530 997 043 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 164 741 061 994 086 4;
  • 52) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 164 741 061 994 086 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 974 329 482 123 988 172 8;
  • 53) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 974 329 482 123 988 172 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 948 658 964 247 976 345 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:


0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 3(10) =

0.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1(2)

5. Positive number before normalization:

1.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 3(10) =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 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:


1.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 3(10) =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1(2) =

1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 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):
1.1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 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. 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1 =

1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


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) =
1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111


Decimal number 1.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 994 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111

Share

» Convert decimal 1.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 986 4 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 08, 2026 [August]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: August - by our visitors

» Improve your social skills: Are You Using Communication by Design, or by Default? Gemini Notebook YouTube Video of 6.33 minutes

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