1.745 459 324 169 999 825 926 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 825 926(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
1.745 459 324 169 999 825 926(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: 0.745 459 324 169 999 825 926.

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) 0.745 459 324 169 999 825 926 × 2 = 1 + 0.490 918 648 339 999 651 852;
  • 2) 0.490 918 648 339 999 651 852 × 2 = 0 + 0.981 837 296 679 999 303 704;
  • 3) 0.981 837 296 679 999 303 704 × 2 = 1 + 0.963 674 593 359 998 607 408;
  • 4) 0.963 674 593 359 998 607 408 × 2 = 1 + 0.927 349 186 719 997 214 816;
  • 5) 0.927 349 186 719 997 214 816 × 2 = 1 + 0.854 698 373 439 994 429 632;
  • 6) 0.854 698 373 439 994 429 632 × 2 = 1 + 0.709 396 746 879 988 859 264;
  • 7) 0.709 396 746 879 988 859 264 × 2 = 1 + 0.418 793 493 759 977 718 528;
  • 8) 0.418 793 493 759 977 718 528 × 2 = 0 + 0.837 586 987 519 955 437 056;
  • 9) 0.837 586 987 519 955 437 056 × 2 = 1 + 0.675 173 975 039 910 874 112;
  • 10) 0.675 173 975 039 910 874 112 × 2 = 1 + 0.350 347 950 079 821 748 224;
  • 11) 0.350 347 950 079 821 748 224 × 2 = 0 + 0.700 695 900 159 643 496 448;
  • 12) 0.700 695 900 159 643 496 448 × 2 = 1 + 0.401 391 800 319 286 992 896;
  • 13) 0.401 391 800 319 286 992 896 × 2 = 0 + 0.802 783 600 638 573 985 792;
  • 14) 0.802 783 600 638 573 985 792 × 2 = 1 + 0.605 567 201 277 147 971 584;
  • 15) 0.605 567 201 277 147 971 584 × 2 = 1 + 0.211 134 402 554 295 943 168;
  • 16) 0.211 134 402 554 295 943 168 × 2 = 0 + 0.422 268 805 108 591 886 336;
  • 17) 0.422 268 805 108 591 886 336 × 2 = 0 + 0.844 537 610 217 183 772 672;
  • 18) 0.844 537 610 217 183 772 672 × 2 = 1 + 0.689 075 220 434 367 545 344;
  • 19) 0.689 075 220 434 367 545 344 × 2 = 1 + 0.378 150 440 868 735 090 688;
  • 20) 0.378 150 440 868 735 090 688 × 2 = 0 + 0.756 300 881 737 470 181 376;
  • 21) 0.756 300 881 737 470 181 376 × 2 = 1 + 0.512 601 763 474 940 362 752;
  • 22) 0.512 601 763 474 940 362 752 × 2 = 1 + 0.025 203 526 949 880 725 504;
  • 23) 0.025 203 526 949 880 725 504 × 2 = 0 + 0.050 407 053 899 761 451 008;
  • 24) 0.050 407 053 899 761 451 008 × 2 = 0 + 0.100 814 107 799 522 902 016;
  • 25) 0.100 814 107 799 522 902 016 × 2 = 0 + 0.201 628 215 599 045 804 032;
  • 26) 0.201 628 215 599 045 804 032 × 2 = 0 + 0.403 256 431 198 091 608 064;
  • 27) 0.403 256 431 198 091 608 064 × 2 = 0 + 0.806 512 862 396 183 216 128;
  • 28) 0.806 512 862 396 183 216 128 × 2 = 1 + 0.613 025 724 792 366 432 256;
  • 29) 0.613 025 724 792 366 432 256 × 2 = 1 + 0.226 051 449 584 732 864 512;
  • 30) 0.226 051 449 584 732 864 512 × 2 = 0 + 0.452 102 899 169 465 729 024;
  • 31) 0.452 102 899 169 465 729 024 × 2 = 0 + 0.904 205 798 338 931 458 048;
  • 32) 0.904 205 798 338 931 458 048 × 2 = 1 + 0.808 411 596 677 862 916 096;
  • 33) 0.808 411 596 677 862 916 096 × 2 = 1 + 0.616 823 193 355 725 832 192;
  • 34) 0.616 823 193 355 725 832 192 × 2 = 1 + 0.233 646 386 711 451 664 384;
  • 35) 0.233 646 386 711 451 664 384 × 2 = 0 + 0.467 292 773 422 903 328 768;
  • 36) 0.467 292 773 422 903 328 768 × 2 = 0 + 0.934 585 546 845 806 657 536;
  • 37) 0.934 585 546 845 806 657 536 × 2 = 1 + 0.869 171 093 691 613 315 072;
  • 38) 0.869 171 093 691 613 315 072 × 2 = 1 + 0.738 342 187 383 226 630 144;
  • 39) 0.738 342 187 383 226 630 144 × 2 = 1 + 0.476 684 374 766 453 260 288;
  • 40) 0.476 684 374 766 453 260 288 × 2 = 0 + 0.953 368 749 532 906 520 576;
  • 41) 0.953 368 749 532 906 520 576 × 2 = 1 + 0.906 737 499 065 813 041 152;
  • 42) 0.906 737 499 065 813 041 152 × 2 = 1 + 0.813 474 998 131 626 082 304;
  • 43) 0.813 474 998 131 626 082 304 × 2 = 1 + 0.626 949 996 263 252 164 608;
  • 44) 0.626 949 996 263 252 164 608 × 2 = 1 + 0.253 899 992 526 504 329 216;
  • 45) 0.253 899 992 526 504 329 216 × 2 = 0 + 0.507 799 985 053 008 658 432;
  • 46) 0.507 799 985 053 008 658 432 × 2 = 1 + 0.015 599 970 106 017 316 864;
  • 47) 0.015 599 970 106 017 316 864 × 2 = 0 + 0.031 199 940 212 034 633 728;
  • 48) 0.031 199 940 212 034 633 728 × 2 = 0 + 0.062 399 880 424 069 267 456;
  • 49) 0.062 399 880 424 069 267 456 × 2 = 0 + 0.124 799 760 848 138 534 912;
  • 50) 0.124 799 760 848 138 534 912 × 2 = 0 + 0.249 599 521 696 277 069 824;
  • 51) 0.249 599 521 696 277 069 824 × 2 = 0 + 0.499 199 043 392 554 139 648;
  • 52) 0.499 199 043 392 554 139 648 × 2 = 0 + 0.998 398 086 785 108 279 296;
  • 53) 0.998 398 086 785 108 279 296 × 2 = 1 + 0.996 796 173 570 216 558 592;

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:


0.745 459 324 169 999 825 926(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 825 926(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:


1.745 459 324 169 999 825 926(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 1.745 459 324 169 999 825 926 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