3.141 592 741 012 573 242 071 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.141 592 741 012 573 242 071(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
3.141 592 741 012 573 242 071(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: 0.141 592 741 012 573 242 071.

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) 0.141 592 741 012 573 242 071 × 2 = 0 + 0.283 185 482 025 146 484 142;
  • 2) 0.283 185 482 025 146 484 142 × 2 = 0 + 0.566 370 964 050 292 968 284;
  • 3) 0.566 370 964 050 292 968 284 × 2 = 1 + 0.132 741 928 100 585 936 568;
  • 4) 0.132 741 928 100 585 936 568 × 2 = 0 + 0.265 483 856 201 171 873 136;
  • 5) 0.265 483 856 201 171 873 136 × 2 = 0 + 0.530 967 712 402 343 746 272;
  • 6) 0.530 967 712 402 343 746 272 × 2 = 1 + 0.061 935 424 804 687 492 544;
  • 7) 0.061 935 424 804 687 492 544 × 2 = 0 + 0.123 870 849 609 374 985 088;
  • 8) 0.123 870 849 609 374 985 088 × 2 = 0 + 0.247 741 699 218 749 970 176;
  • 9) 0.247 741 699 218 749 970 176 × 2 = 0 + 0.495 483 398 437 499 940 352;
  • 10) 0.495 483 398 437 499 940 352 × 2 = 0 + 0.990 966 796 874 999 880 704;
  • 11) 0.990 966 796 874 999 880 704 × 2 = 1 + 0.981 933 593 749 999 761 408;
  • 12) 0.981 933 593 749 999 761 408 × 2 = 1 + 0.963 867 187 499 999 522 816;
  • 13) 0.963 867 187 499 999 522 816 × 2 = 1 + 0.927 734 374 999 999 045 632;
  • 14) 0.927 734 374 999 999 045 632 × 2 = 1 + 0.855 468 749 999 998 091 264;
  • 15) 0.855 468 749 999 998 091 264 × 2 = 1 + 0.710 937 499 999 996 182 528;
  • 16) 0.710 937 499 999 996 182 528 × 2 = 1 + 0.421 874 999 999 992 365 056;
  • 17) 0.421 874 999 999 992 365 056 × 2 = 0 + 0.843 749 999 999 984 730 112;
  • 18) 0.843 749 999 999 984 730 112 × 2 = 1 + 0.687 499 999 999 969 460 224;
  • 19) 0.687 499 999 999 969 460 224 × 2 = 1 + 0.374 999 999 999 938 920 448;
  • 20) 0.374 999 999 999 938 920 448 × 2 = 0 + 0.749 999 999 999 877 840 896;
  • 21) 0.749 999 999 999 877 840 896 × 2 = 1 + 0.499 999 999 999 755 681 792;
  • 22) 0.499 999 999 999 755 681 792 × 2 = 0 + 0.999 999 999 999 511 363 584;
  • 23) 0.999 999 999 999 511 363 584 × 2 = 1 + 0.999 999 999 999 022 727 168;
  • 24) 0.999 999 999 999 022 727 168 × 2 = 1 + 0.999 999 999 998 045 454 336;
  • 25) 0.999 999 999 998 045 454 336 × 2 = 1 + 0.999 999 999 996 090 908 672;
  • 26) 0.999 999 999 996 090 908 672 × 2 = 1 + 0.999 999 999 992 181 817 344;
  • 27) 0.999 999 999 992 181 817 344 × 2 = 1 + 0.999 999 999 984 363 634 688;
  • 28) 0.999 999 999 984 363 634 688 × 2 = 1 + 0.999 999 999 968 727 269 376;
  • 29) 0.999 999 999 968 727 269 376 × 2 = 1 + 0.999 999 999 937 454 538 752;
  • 30) 0.999 999 999 937 454 538 752 × 2 = 1 + 0.999 999 999 874 909 077 504;
  • 31) 0.999 999 999 874 909 077 504 × 2 = 1 + 0.999 999 999 749 818 155 008;
  • 32) 0.999 999 999 749 818 155 008 × 2 = 1 + 0.999 999 999 499 636 310 016;
  • 33) 0.999 999 999 499 636 310 016 × 2 = 1 + 0.999 999 998 999 272 620 032;
  • 34) 0.999 999 998 999 272 620 032 × 2 = 1 + 0.999 999 997 998 545 240 064;
  • 35) 0.999 999 997 998 545 240 064 × 2 = 1 + 0.999 999 995 997 090 480 128;
  • 36) 0.999 999 995 997 090 480 128 × 2 = 1 + 0.999 999 991 994 180 960 256;
  • 37) 0.999 999 991 994 180 960 256 × 2 = 1 + 0.999 999 983 988 361 920 512;
  • 38) 0.999 999 983 988 361 920 512 × 2 = 1 + 0.999 999 967 976 723 841 024;
  • 39) 0.999 999 967 976 723 841 024 × 2 = 1 + 0.999 999 935 953 447 682 048;
  • 40) 0.999 999 935 953 447 682 048 × 2 = 1 + 0.999 999 871 906 895 364 096;
  • 41) 0.999 999 871 906 895 364 096 × 2 = 1 + 0.999 999 743 813 790 728 192;
  • 42) 0.999 999 743 813 790 728 192 × 2 = 1 + 0.999 999 487 627 581 456 384;
  • 43) 0.999 999 487 627 581 456 384 × 2 = 1 + 0.999 998 975 255 162 912 768;
  • 44) 0.999 998 975 255 162 912 768 × 2 = 1 + 0.999 997 950 510 325 825 536;
  • 45) 0.999 997 950 510 325 825 536 × 2 = 1 + 0.999 995 901 020 651 651 072;
  • 46) 0.999 995 901 020 651 651 072 × 2 = 1 + 0.999 991 802 041 303 302 144;
  • 47) 0.999 991 802 041 303 302 144 × 2 = 1 + 0.999 983 604 082 606 604 288;
  • 48) 0.999 983 604 082 606 604 288 × 2 = 1 + 0.999 967 208 165 213 208 576;
  • 49) 0.999 967 208 165 213 208 576 × 2 = 1 + 0.999 934 416 330 426 417 152;
  • 50) 0.999 934 416 330 426 417 152 × 2 = 1 + 0.999 868 832 660 852 834 304;
  • 51) 0.999 868 832 660 852 834 304 × 2 = 1 + 0.999 737 665 321 705 668 608;
  • 52) 0.999 737 665 321 705 668 608 × 2 = 1 + 0.999 475 330 643 411 337 216;
  • 53) 0.999 475 330 643 411 337 216 × 2 = 1 + 0.998 950 661 286 822 674 432;

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:


0.141 592 741 012 573 242 071(10) =

0.0010 0100 0011 1111 0110 1011 1111 1111 1111 1111 1111 1111 1111 1(2)

5. Positive number before normalization:

3.141 592 741 012 573 242 071(10) =

11.0010 0100 0011 1111 0110 1011 1111 1111 1111 1111 1111 1111 1111 1(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:


3.141 592 741 012 573 242 071(10) =

11.0010 0100 0011 1111 0110 1011 1111 1111 1111 1111 1111 1111 1111 1(2) =

11.0010 0100 0011 1111 0110 1011 1111 1111 1111 1111 1111 1111 1111 1(2) × 20 =

1.1001 0010 0001 1111 1011 0101 1111 1111 1111 1111 1111 1111 1111 11(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):
1.1001 0010 0001 1111 1011 0101 1111 1111 1111 1111 1111 1111 1111 11


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. 1001 0010 0001 1111 1011 0101 1111 1111 1111 1111 1111 1111 1111 11 =

1001 0010 0001 1111 1011 0101 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) =
100 0000 0000

Mantissa (52 bits) =
1001 0010 0001 1111 1011 0101 1111 1111 1111 1111 1111 1111 1111


Decimal number 3.141 592 741 012 573 242 071 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 1001 0010 0001 1111 1011 0101 1111 1111 1111 1111 1111 1111 1111

Share

» Convert decimal 3.141 592 741 012 573 241 976 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 07, 2026 [July]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: July - by our visitors

» Improve your social skills: Your Greatest Competitor, Your Most Powerful Ally. NotebookLM YouTube Video of 7.54 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