3.141 592 653 589 793 238 462 042 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 3.141 592 653 589 793 238 462 042(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
3.141 592 653 589 793 238 462 042(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: 0.141 592 653 589 793 238 462 042.

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) 0.141 592 653 589 793 238 462 042 × 2 = 0 + 0.283 185 307 179 586 476 924 084;
  • 2) 0.283 185 307 179 586 476 924 084 × 2 = 0 + 0.566 370 614 359 172 953 848 168;
  • 3) 0.566 370 614 359 172 953 848 168 × 2 = 1 + 0.132 741 228 718 345 907 696 336;
  • 4) 0.132 741 228 718 345 907 696 336 × 2 = 0 + 0.265 482 457 436 691 815 392 672;
  • 5) 0.265 482 457 436 691 815 392 672 × 2 = 0 + 0.530 964 914 873 383 630 785 344;
  • 6) 0.530 964 914 873 383 630 785 344 × 2 = 1 + 0.061 929 829 746 767 261 570 688;
  • 7) 0.061 929 829 746 767 261 570 688 × 2 = 0 + 0.123 859 659 493 534 523 141 376;
  • 8) 0.123 859 659 493 534 523 141 376 × 2 = 0 + 0.247 719 318 987 069 046 282 752;
  • 9) 0.247 719 318 987 069 046 282 752 × 2 = 0 + 0.495 438 637 974 138 092 565 504;
  • 10) 0.495 438 637 974 138 092 565 504 × 2 = 0 + 0.990 877 275 948 276 185 131 008;
  • 11) 0.990 877 275 948 276 185 131 008 × 2 = 1 + 0.981 754 551 896 552 370 262 016;
  • 12) 0.981 754 551 896 552 370 262 016 × 2 = 1 + 0.963 509 103 793 104 740 524 032;
  • 13) 0.963 509 103 793 104 740 524 032 × 2 = 1 + 0.927 018 207 586 209 481 048 064;
  • 14) 0.927 018 207 586 209 481 048 064 × 2 = 1 + 0.854 036 415 172 418 962 096 128;
  • 15) 0.854 036 415 172 418 962 096 128 × 2 = 1 + 0.708 072 830 344 837 924 192 256;
  • 16) 0.708 072 830 344 837 924 192 256 × 2 = 1 + 0.416 145 660 689 675 848 384 512;
  • 17) 0.416 145 660 689 675 848 384 512 × 2 = 0 + 0.832 291 321 379 351 696 769 024;
  • 18) 0.832 291 321 379 351 696 769 024 × 2 = 1 + 0.664 582 642 758 703 393 538 048;
  • 19) 0.664 582 642 758 703 393 538 048 × 2 = 1 + 0.329 165 285 517 406 787 076 096;
  • 20) 0.329 165 285 517 406 787 076 096 × 2 = 0 + 0.658 330 571 034 813 574 152 192;
  • 21) 0.658 330 571 034 813 574 152 192 × 2 = 1 + 0.316 661 142 069 627 148 304 384;
  • 22) 0.316 661 142 069 627 148 304 384 × 2 = 0 + 0.633 322 284 139 254 296 608 768;
  • 23) 0.633 322 284 139 254 296 608 768 × 2 = 1 + 0.266 644 568 278 508 593 217 536;
  • 24) 0.266 644 568 278 508 593 217 536 × 2 = 0 + 0.533 289 136 557 017 186 435 072;
  • 25) 0.533 289 136 557 017 186 435 072 × 2 = 1 + 0.066 578 273 114 034 372 870 144;
  • 26) 0.066 578 273 114 034 372 870 144 × 2 = 0 + 0.133 156 546 228 068 745 740 288;
  • 27) 0.133 156 546 228 068 745 740 288 × 2 = 0 + 0.266 313 092 456 137 491 480 576;
  • 28) 0.266 313 092 456 137 491 480 576 × 2 = 0 + 0.532 626 184 912 274 982 961 152;
  • 29) 0.532 626 184 912 274 982 961 152 × 2 = 1 + 0.065 252 369 824 549 965 922 304;
  • 30) 0.065 252 369 824 549 965 922 304 × 2 = 0 + 0.130 504 739 649 099 931 844 608;
  • 31) 0.130 504 739 649 099 931 844 608 × 2 = 0 + 0.261 009 479 298 199 863 689 216;
  • 32) 0.261 009 479 298 199 863 689 216 × 2 = 0 + 0.522 018 958 596 399 727 378 432;
  • 33) 0.522 018 958 596 399 727 378 432 × 2 = 1 + 0.044 037 917 192 799 454 756 864;
  • 34) 0.044 037 917 192 799 454 756 864 × 2 = 0 + 0.088 075 834 385 598 909 513 728;
  • 35) 0.088 075 834 385 598 909 513 728 × 2 = 0 + 0.176 151 668 771 197 819 027 456;
  • 36) 0.176 151 668 771 197 819 027 456 × 2 = 0 + 0.352 303 337 542 395 638 054 912;
  • 37) 0.352 303 337 542 395 638 054 912 × 2 = 0 + 0.704 606 675 084 791 276 109 824;
  • 38) 0.704 606 675 084 791 276 109 824 × 2 = 1 + 0.409 213 350 169 582 552 219 648;
  • 39) 0.409 213 350 169 582 552 219 648 × 2 = 0 + 0.818 426 700 339 165 104 439 296;
  • 40) 0.818 426 700 339 165 104 439 296 × 2 = 1 + 0.636 853 400 678 330 208 878 592;
  • 41) 0.636 853 400 678 330 208 878 592 × 2 = 1 + 0.273 706 801 356 660 417 757 184;
  • 42) 0.273 706 801 356 660 417 757 184 × 2 = 0 + 0.547 413 602 713 320 835 514 368;
  • 43) 0.547 413 602 713 320 835 514 368 × 2 = 1 + 0.094 827 205 426 641 671 028 736;
  • 44) 0.094 827 205 426 641 671 028 736 × 2 = 0 + 0.189 654 410 853 283 342 057 472;
  • 45) 0.189 654 410 853 283 342 057 472 × 2 = 0 + 0.379 308 821 706 566 684 114 944;
  • 46) 0.379 308 821 706 566 684 114 944 × 2 = 0 + 0.758 617 643 413 133 368 229 888;
  • 47) 0.758 617 643 413 133 368 229 888 × 2 = 1 + 0.517 235 286 826 266 736 459 776;
  • 48) 0.517 235 286 826 266 736 459 776 × 2 = 1 + 0.034 470 573 652 533 472 919 552;
  • 49) 0.034 470 573 652 533 472 919 552 × 2 = 0 + 0.068 941 147 305 066 945 839 104;
  • 50) 0.068 941 147 305 066 945 839 104 × 2 = 0 + 0.137 882 294 610 133 891 678 208;
  • 51) 0.137 882 294 610 133 891 678 208 × 2 = 0 + 0.275 764 589 220 267 783 356 416;
  • 52) 0.275 764 589 220 267 783 356 416 × 2 = 0 + 0.551 529 178 440 535 566 712 832;
  • 53) 0.551 529 178 440 535 566 712 832 × 2 = 1 + 0.103 058 356 881 071 133 425 664;

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:


0.141 592 653 589 793 238 462 042(10) =

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

5. Positive number before normalization:

3.141 592 653 589 793 238 462 042(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:


3.141 592 653 589 793 238 462 042(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 3.141 592 653 589 793 238 462 042 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