3.141 592 653 589 793 237 721 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.141 592 653 589 793 237 721(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
3.141 592 653 589 793 237 721(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: 0.141 592 653 589 793 237 721.

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) 0.141 592 653 589 793 237 721 × 2 = 0 + 0.283 185 307 179 586 475 442;
  • 2) 0.283 185 307 179 586 475 442 × 2 = 0 + 0.566 370 614 359 172 950 884;
  • 3) 0.566 370 614 359 172 950 884 × 2 = 1 + 0.132 741 228 718 345 901 768;
  • 4) 0.132 741 228 718 345 901 768 × 2 = 0 + 0.265 482 457 436 691 803 536;
  • 5) 0.265 482 457 436 691 803 536 × 2 = 0 + 0.530 964 914 873 383 607 072;
  • 6) 0.530 964 914 873 383 607 072 × 2 = 1 + 0.061 929 829 746 767 214 144;
  • 7) 0.061 929 829 746 767 214 144 × 2 = 0 + 0.123 859 659 493 534 428 288;
  • 8) 0.123 859 659 493 534 428 288 × 2 = 0 + 0.247 719 318 987 068 856 576;
  • 9) 0.247 719 318 987 068 856 576 × 2 = 0 + 0.495 438 637 974 137 713 152;
  • 10) 0.495 438 637 974 137 713 152 × 2 = 0 + 0.990 877 275 948 275 426 304;
  • 11) 0.990 877 275 948 275 426 304 × 2 = 1 + 0.981 754 551 896 550 852 608;
  • 12) 0.981 754 551 896 550 852 608 × 2 = 1 + 0.963 509 103 793 101 705 216;
  • 13) 0.963 509 103 793 101 705 216 × 2 = 1 + 0.927 018 207 586 203 410 432;
  • 14) 0.927 018 207 586 203 410 432 × 2 = 1 + 0.854 036 415 172 406 820 864;
  • 15) 0.854 036 415 172 406 820 864 × 2 = 1 + 0.708 072 830 344 813 641 728;
  • 16) 0.708 072 830 344 813 641 728 × 2 = 1 + 0.416 145 660 689 627 283 456;
  • 17) 0.416 145 660 689 627 283 456 × 2 = 0 + 0.832 291 321 379 254 566 912;
  • 18) 0.832 291 321 379 254 566 912 × 2 = 1 + 0.664 582 642 758 509 133 824;
  • 19) 0.664 582 642 758 509 133 824 × 2 = 1 + 0.329 165 285 517 018 267 648;
  • 20) 0.329 165 285 517 018 267 648 × 2 = 0 + 0.658 330 571 034 036 535 296;
  • 21) 0.658 330 571 034 036 535 296 × 2 = 1 + 0.316 661 142 068 073 070 592;
  • 22) 0.316 661 142 068 073 070 592 × 2 = 0 + 0.633 322 284 136 146 141 184;
  • 23) 0.633 322 284 136 146 141 184 × 2 = 1 + 0.266 644 568 272 292 282 368;
  • 24) 0.266 644 568 272 292 282 368 × 2 = 0 + 0.533 289 136 544 584 564 736;
  • 25) 0.533 289 136 544 584 564 736 × 2 = 1 + 0.066 578 273 089 169 129 472;
  • 26) 0.066 578 273 089 169 129 472 × 2 = 0 + 0.133 156 546 178 338 258 944;
  • 27) 0.133 156 546 178 338 258 944 × 2 = 0 + 0.266 313 092 356 676 517 888;
  • 28) 0.266 313 092 356 676 517 888 × 2 = 0 + 0.532 626 184 713 353 035 776;
  • 29) 0.532 626 184 713 353 035 776 × 2 = 1 + 0.065 252 369 426 706 071 552;
  • 30) 0.065 252 369 426 706 071 552 × 2 = 0 + 0.130 504 738 853 412 143 104;
  • 31) 0.130 504 738 853 412 143 104 × 2 = 0 + 0.261 009 477 706 824 286 208;
  • 32) 0.261 009 477 706 824 286 208 × 2 = 0 + 0.522 018 955 413 648 572 416;
  • 33) 0.522 018 955 413 648 572 416 × 2 = 1 + 0.044 037 910 827 297 144 832;
  • 34) 0.044 037 910 827 297 144 832 × 2 = 0 + 0.088 075 821 654 594 289 664;
  • 35) 0.088 075 821 654 594 289 664 × 2 = 0 + 0.176 151 643 309 188 579 328;
  • 36) 0.176 151 643 309 188 579 328 × 2 = 0 + 0.352 303 286 618 377 158 656;
  • 37) 0.352 303 286 618 377 158 656 × 2 = 0 + 0.704 606 573 236 754 317 312;
  • 38) 0.704 606 573 236 754 317 312 × 2 = 1 + 0.409 213 146 473 508 634 624;
  • 39) 0.409 213 146 473 508 634 624 × 2 = 0 + 0.818 426 292 947 017 269 248;
  • 40) 0.818 426 292 947 017 269 248 × 2 = 1 + 0.636 852 585 894 034 538 496;
  • 41) 0.636 852 585 894 034 538 496 × 2 = 1 + 0.273 705 171 788 069 076 992;
  • 42) 0.273 705 171 788 069 076 992 × 2 = 0 + 0.547 410 343 576 138 153 984;
  • 43) 0.547 410 343 576 138 153 984 × 2 = 1 + 0.094 820 687 152 276 307 968;
  • 44) 0.094 820 687 152 276 307 968 × 2 = 0 + 0.189 641 374 304 552 615 936;
  • 45) 0.189 641 374 304 552 615 936 × 2 = 0 + 0.379 282 748 609 105 231 872;
  • 46) 0.379 282 748 609 105 231 872 × 2 = 0 + 0.758 565 497 218 210 463 744;
  • 47) 0.758 565 497 218 210 463 744 × 2 = 1 + 0.517 130 994 436 420 927 488;
  • 48) 0.517 130 994 436 420 927 488 × 2 = 1 + 0.034 261 988 872 841 854 976;
  • 49) 0.034 261 988 872 841 854 976 × 2 = 0 + 0.068 523 977 745 683 709 952;
  • 50) 0.068 523 977 745 683 709 952 × 2 = 0 + 0.137 047 955 491 367 419 904;
  • 51) 0.137 047 955 491 367 419 904 × 2 = 0 + 0.274 095 910 982 734 839 808;
  • 52) 0.274 095 910 982 734 839 808 × 2 = 0 + 0.548 191 821 965 469 679 616;
  • 53) 0.548 191 821 965 469 679 616 × 2 = 1 + 0.096 383 643 930 939 359 232;

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:


0.141 592 653 589 793 237 721(10) =

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

5. Positive number before normalization:

3.141 592 653 589 793 237 721(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:


3.141 592 653 589 793 237 721(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 3.141 592 653 589 793 237 721 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