3.141 592 653 589 793 238 462 196 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.141 592 653 589 793 238 462 196(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
3.141 592 653 589 793 238 462 196(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: 0.141 592 653 589 793 238 462 196.

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) 0.141 592 653 589 793 238 462 196 × 2 = 0 + 0.283 185 307 179 586 476 924 392;
  • 2) 0.283 185 307 179 586 476 924 392 × 2 = 0 + 0.566 370 614 359 172 953 848 784;
  • 3) 0.566 370 614 359 172 953 848 784 × 2 = 1 + 0.132 741 228 718 345 907 697 568;
  • 4) 0.132 741 228 718 345 907 697 568 × 2 = 0 + 0.265 482 457 436 691 815 395 136;
  • 5) 0.265 482 457 436 691 815 395 136 × 2 = 0 + 0.530 964 914 873 383 630 790 272;
  • 6) 0.530 964 914 873 383 630 790 272 × 2 = 1 + 0.061 929 829 746 767 261 580 544;
  • 7) 0.061 929 829 746 767 261 580 544 × 2 = 0 + 0.123 859 659 493 534 523 161 088;
  • 8) 0.123 859 659 493 534 523 161 088 × 2 = 0 + 0.247 719 318 987 069 046 322 176;
  • 9) 0.247 719 318 987 069 046 322 176 × 2 = 0 + 0.495 438 637 974 138 092 644 352;
  • 10) 0.495 438 637 974 138 092 644 352 × 2 = 0 + 0.990 877 275 948 276 185 288 704;
  • 11) 0.990 877 275 948 276 185 288 704 × 2 = 1 + 0.981 754 551 896 552 370 577 408;
  • 12) 0.981 754 551 896 552 370 577 408 × 2 = 1 + 0.963 509 103 793 104 741 154 816;
  • 13) 0.963 509 103 793 104 741 154 816 × 2 = 1 + 0.927 018 207 586 209 482 309 632;
  • 14) 0.927 018 207 586 209 482 309 632 × 2 = 1 + 0.854 036 415 172 418 964 619 264;
  • 15) 0.854 036 415 172 418 964 619 264 × 2 = 1 + 0.708 072 830 344 837 929 238 528;
  • 16) 0.708 072 830 344 837 929 238 528 × 2 = 1 + 0.416 145 660 689 675 858 477 056;
  • 17) 0.416 145 660 689 675 858 477 056 × 2 = 0 + 0.832 291 321 379 351 716 954 112;
  • 18) 0.832 291 321 379 351 716 954 112 × 2 = 1 + 0.664 582 642 758 703 433 908 224;
  • 19) 0.664 582 642 758 703 433 908 224 × 2 = 1 + 0.329 165 285 517 406 867 816 448;
  • 20) 0.329 165 285 517 406 867 816 448 × 2 = 0 + 0.658 330 571 034 813 735 632 896;
  • 21) 0.658 330 571 034 813 735 632 896 × 2 = 1 + 0.316 661 142 069 627 471 265 792;
  • 22) 0.316 661 142 069 627 471 265 792 × 2 = 0 + 0.633 322 284 139 254 942 531 584;
  • 23) 0.633 322 284 139 254 942 531 584 × 2 = 1 + 0.266 644 568 278 509 885 063 168;
  • 24) 0.266 644 568 278 509 885 063 168 × 2 = 0 + 0.533 289 136 557 019 770 126 336;
  • 25) 0.533 289 136 557 019 770 126 336 × 2 = 1 + 0.066 578 273 114 039 540 252 672;
  • 26) 0.066 578 273 114 039 540 252 672 × 2 = 0 + 0.133 156 546 228 079 080 505 344;
  • 27) 0.133 156 546 228 079 080 505 344 × 2 = 0 + 0.266 313 092 456 158 161 010 688;
  • 28) 0.266 313 092 456 158 161 010 688 × 2 = 0 + 0.532 626 184 912 316 322 021 376;
  • 29) 0.532 626 184 912 316 322 021 376 × 2 = 1 + 0.065 252 369 824 632 644 042 752;
  • 30) 0.065 252 369 824 632 644 042 752 × 2 = 0 + 0.130 504 739 649 265 288 085 504;
  • 31) 0.130 504 739 649 265 288 085 504 × 2 = 0 + 0.261 009 479 298 530 576 171 008;
  • 32) 0.261 009 479 298 530 576 171 008 × 2 = 0 + 0.522 018 958 597 061 152 342 016;
  • 33) 0.522 018 958 597 061 152 342 016 × 2 = 1 + 0.044 037 917 194 122 304 684 032;
  • 34) 0.044 037 917 194 122 304 684 032 × 2 = 0 + 0.088 075 834 388 244 609 368 064;
  • 35) 0.088 075 834 388 244 609 368 064 × 2 = 0 + 0.176 151 668 776 489 218 736 128;
  • 36) 0.176 151 668 776 489 218 736 128 × 2 = 0 + 0.352 303 337 552 978 437 472 256;
  • 37) 0.352 303 337 552 978 437 472 256 × 2 = 0 + 0.704 606 675 105 956 874 944 512;
  • 38) 0.704 606 675 105 956 874 944 512 × 2 = 1 + 0.409 213 350 211 913 749 889 024;
  • 39) 0.409 213 350 211 913 749 889 024 × 2 = 0 + 0.818 426 700 423 827 499 778 048;
  • 40) 0.818 426 700 423 827 499 778 048 × 2 = 1 + 0.636 853 400 847 654 999 556 096;
  • 41) 0.636 853 400 847 654 999 556 096 × 2 = 1 + 0.273 706 801 695 309 999 112 192;
  • 42) 0.273 706 801 695 309 999 112 192 × 2 = 0 + 0.547 413 603 390 619 998 224 384;
  • 43) 0.547 413 603 390 619 998 224 384 × 2 = 1 + 0.094 827 206 781 239 996 448 768;
  • 44) 0.094 827 206 781 239 996 448 768 × 2 = 0 + 0.189 654 413 562 479 992 897 536;
  • 45) 0.189 654 413 562 479 992 897 536 × 2 = 0 + 0.379 308 827 124 959 985 795 072;
  • 46) 0.379 308 827 124 959 985 795 072 × 2 = 0 + 0.758 617 654 249 919 971 590 144;
  • 47) 0.758 617 654 249 919 971 590 144 × 2 = 1 + 0.517 235 308 499 839 943 180 288;
  • 48) 0.517 235 308 499 839 943 180 288 × 2 = 1 + 0.034 470 616 999 679 886 360 576;
  • 49) 0.034 470 616 999 679 886 360 576 × 2 = 0 + 0.068 941 233 999 359 772 721 152;
  • 50) 0.068 941 233 999 359 772 721 152 × 2 = 0 + 0.137 882 467 998 719 545 442 304;
  • 51) 0.137 882 467 998 719 545 442 304 × 2 = 0 + 0.275 764 935 997 439 090 884 608;
  • 52) 0.275 764 935 997 439 090 884 608 × 2 = 0 + 0.551 529 871 994 878 181 769 216;
  • 53) 0.551 529 871 994 878 181 769 216 × 2 = 1 + 0.103 059 743 989 756 363 538 432;

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:


0.141 592 653 589 793 238 462 196(10) =

0.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2)

5. Positive number before normalization:

3.141 592 653 589 793 238 462 196(10) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 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:


3.141 592 653 589 793 238 462 196(10) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2) =

11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011 0000 1(2) × 20 =

1.1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 01(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 0100 0100 0100 0010 1101 0001 1000 01


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 0100 0100 0100 0010 1101 0001 1000 01 =

1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000


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 0100 0100 0100 0010 1101 0001 1000


Decimal number 3.141 592 653 589 793 238 462 196 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0000 - 1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000

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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