1.745 459 324 169 999 826 212 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 212(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
1.745 459 324 169 999 826 212(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: 0.745 459 324 169 999 826 212.

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) 0.745 459 324 169 999 826 212 × 2 = 1 + 0.490 918 648 339 999 652 424;
  • 2) 0.490 918 648 339 999 652 424 × 2 = 0 + 0.981 837 296 679 999 304 848;
  • 3) 0.981 837 296 679 999 304 848 × 2 = 1 + 0.963 674 593 359 998 609 696;
  • 4) 0.963 674 593 359 998 609 696 × 2 = 1 + 0.927 349 186 719 997 219 392;
  • 5) 0.927 349 186 719 997 219 392 × 2 = 1 + 0.854 698 373 439 994 438 784;
  • 6) 0.854 698 373 439 994 438 784 × 2 = 1 + 0.709 396 746 879 988 877 568;
  • 7) 0.709 396 746 879 988 877 568 × 2 = 1 + 0.418 793 493 759 977 755 136;
  • 8) 0.418 793 493 759 977 755 136 × 2 = 0 + 0.837 586 987 519 955 510 272;
  • 9) 0.837 586 987 519 955 510 272 × 2 = 1 + 0.675 173 975 039 911 020 544;
  • 10) 0.675 173 975 039 911 020 544 × 2 = 1 + 0.350 347 950 079 822 041 088;
  • 11) 0.350 347 950 079 822 041 088 × 2 = 0 + 0.700 695 900 159 644 082 176;
  • 12) 0.700 695 900 159 644 082 176 × 2 = 1 + 0.401 391 800 319 288 164 352;
  • 13) 0.401 391 800 319 288 164 352 × 2 = 0 + 0.802 783 600 638 576 328 704;
  • 14) 0.802 783 600 638 576 328 704 × 2 = 1 + 0.605 567 201 277 152 657 408;
  • 15) 0.605 567 201 277 152 657 408 × 2 = 1 + 0.211 134 402 554 305 314 816;
  • 16) 0.211 134 402 554 305 314 816 × 2 = 0 + 0.422 268 805 108 610 629 632;
  • 17) 0.422 268 805 108 610 629 632 × 2 = 0 + 0.844 537 610 217 221 259 264;
  • 18) 0.844 537 610 217 221 259 264 × 2 = 1 + 0.689 075 220 434 442 518 528;
  • 19) 0.689 075 220 434 442 518 528 × 2 = 1 + 0.378 150 440 868 885 037 056;
  • 20) 0.378 150 440 868 885 037 056 × 2 = 0 + 0.756 300 881 737 770 074 112;
  • 21) 0.756 300 881 737 770 074 112 × 2 = 1 + 0.512 601 763 475 540 148 224;
  • 22) 0.512 601 763 475 540 148 224 × 2 = 1 + 0.025 203 526 951 080 296 448;
  • 23) 0.025 203 526 951 080 296 448 × 2 = 0 + 0.050 407 053 902 160 592 896;
  • 24) 0.050 407 053 902 160 592 896 × 2 = 0 + 0.100 814 107 804 321 185 792;
  • 25) 0.100 814 107 804 321 185 792 × 2 = 0 + 0.201 628 215 608 642 371 584;
  • 26) 0.201 628 215 608 642 371 584 × 2 = 0 + 0.403 256 431 217 284 743 168;
  • 27) 0.403 256 431 217 284 743 168 × 2 = 0 + 0.806 512 862 434 569 486 336;
  • 28) 0.806 512 862 434 569 486 336 × 2 = 1 + 0.613 025 724 869 138 972 672;
  • 29) 0.613 025 724 869 138 972 672 × 2 = 1 + 0.226 051 449 738 277 945 344;
  • 30) 0.226 051 449 738 277 945 344 × 2 = 0 + 0.452 102 899 476 555 890 688;
  • 31) 0.452 102 899 476 555 890 688 × 2 = 0 + 0.904 205 798 953 111 781 376;
  • 32) 0.904 205 798 953 111 781 376 × 2 = 1 + 0.808 411 597 906 223 562 752;
  • 33) 0.808 411 597 906 223 562 752 × 2 = 1 + 0.616 823 195 812 447 125 504;
  • 34) 0.616 823 195 812 447 125 504 × 2 = 1 + 0.233 646 391 624 894 251 008;
  • 35) 0.233 646 391 624 894 251 008 × 2 = 0 + 0.467 292 783 249 788 502 016;
  • 36) 0.467 292 783 249 788 502 016 × 2 = 0 + 0.934 585 566 499 577 004 032;
  • 37) 0.934 585 566 499 577 004 032 × 2 = 1 + 0.869 171 132 999 154 008 064;
  • 38) 0.869 171 132 999 154 008 064 × 2 = 1 + 0.738 342 265 998 308 016 128;
  • 39) 0.738 342 265 998 308 016 128 × 2 = 1 + 0.476 684 531 996 616 032 256;
  • 40) 0.476 684 531 996 616 032 256 × 2 = 0 + 0.953 369 063 993 232 064 512;
  • 41) 0.953 369 063 993 232 064 512 × 2 = 1 + 0.906 738 127 986 464 129 024;
  • 42) 0.906 738 127 986 464 129 024 × 2 = 1 + 0.813 476 255 972 928 258 048;
  • 43) 0.813 476 255 972 928 258 048 × 2 = 1 + 0.626 952 511 945 856 516 096;
  • 44) 0.626 952 511 945 856 516 096 × 2 = 1 + 0.253 905 023 891 713 032 192;
  • 45) 0.253 905 023 891 713 032 192 × 2 = 0 + 0.507 810 047 783 426 064 384;
  • 46) 0.507 810 047 783 426 064 384 × 2 = 1 + 0.015 620 095 566 852 128 768;
  • 47) 0.015 620 095 566 852 128 768 × 2 = 0 + 0.031 240 191 133 704 257 536;
  • 48) 0.031 240 191 133 704 257 536 × 2 = 0 + 0.062 480 382 267 408 515 072;
  • 49) 0.062 480 382 267 408 515 072 × 2 = 0 + 0.124 960 764 534 817 030 144;
  • 50) 0.124 960 764 534 817 030 144 × 2 = 0 + 0.249 921 529 069 634 060 288;
  • 51) 0.249 921 529 069 634 060 288 × 2 = 0 + 0.499 843 058 139 268 120 576;
  • 52) 0.499 843 058 139 268 120 576 × 2 = 0 + 0.999 686 116 278 536 241 152;
  • 53) 0.999 686 116 278 536 241 152 × 2 = 1 + 0.999 372 232 557 072 482 304;

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:


0.745 459 324 169 999 826 212(10) =

0.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1(2)

5. Positive number before normalization:

1.745 459 324 169 999 826 212(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 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:


1.745 459 324 169 999 826 212(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1(2) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 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):
1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 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. 1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000 1 =

1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000


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) =
1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000


Decimal number 1.745 459 324 169 999 826 212 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1111 - 1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0000

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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