1.745 459 324 169 999 826 049 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 049(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
1.745 459 324 169 999 826 049(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: 0.745 459 324 169 999 826 049.

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) 0.745 459 324 169 999 826 049 × 2 = 1 + 0.490 918 648 339 999 652 098;
  • 2) 0.490 918 648 339 999 652 098 × 2 = 0 + 0.981 837 296 679 999 304 196;
  • 3) 0.981 837 296 679 999 304 196 × 2 = 1 + 0.963 674 593 359 998 608 392;
  • 4) 0.963 674 593 359 998 608 392 × 2 = 1 + 0.927 349 186 719 997 216 784;
  • 5) 0.927 349 186 719 997 216 784 × 2 = 1 + 0.854 698 373 439 994 433 568;
  • 6) 0.854 698 373 439 994 433 568 × 2 = 1 + 0.709 396 746 879 988 867 136;
  • 7) 0.709 396 746 879 988 867 136 × 2 = 1 + 0.418 793 493 759 977 734 272;
  • 8) 0.418 793 493 759 977 734 272 × 2 = 0 + 0.837 586 987 519 955 468 544;
  • 9) 0.837 586 987 519 955 468 544 × 2 = 1 + 0.675 173 975 039 910 937 088;
  • 10) 0.675 173 975 039 910 937 088 × 2 = 1 + 0.350 347 950 079 821 874 176;
  • 11) 0.350 347 950 079 821 874 176 × 2 = 0 + 0.700 695 900 159 643 748 352;
  • 12) 0.700 695 900 159 643 748 352 × 2 = 1 + 0.401 391 800 319 287 496 704;
  • 13) 0.401 391 800 319 287 496 704 × 2 = 0 + 0.802 783 600 638 574 993 408;
  • 14) 0.802 783 600 638 574 993 408 × 2 = 1 + 0.605 567 201 277 149 986 816;
  • 15) 0.605 567 201 277 149 986 816 × 2 = 1 + 0.211 134 402 554 299 973 632;
  • 16) 0.211 134 402 554 299 973 632 × 2 = 0 + 0.422 268 805 108 599 947 264;
  • 17) 0.422 268 805 108 599 947 264 × 2 = 0 + 0.844 537 610 217 199 894 528;
  • 18) 0.844 537 610 217 199 894 528 × 2 = 1 + 0.689 075 220 434 399 789 056;
  • 19) 0.689 075 220 434 399 789 056 × 2 = 1 + 0.378 150 440 868 799 578 112;
  • 20) 0.378 150 440 868 799 578 112 × 2 = 0 + 0.756 300 881 737 599 156 224;
  • 21) 0.756 300 881 737 599 156 224 × 2 = 1 + 0.512 601 763 475 198 312 448;
  • 22) 0.512 601 763 475 198 312 448 × 2 = 1 + 0.025 203 526 950 396 624 896;
  • 23) 0.025 203 526 950 396 624 896 × 2 = 0 + 0.050 407 053 900 793 249 792;
  • 24) 0.050 407 053 900 793 249 792 × 2 = 0 + 0.100 814 107 801 586 499 584;
  • 25) 0.100 814 107 801 586 499 584 × 2 = 0 + 0.201 628 215 603 172 999 168;
  • 26) 0.201 628 215 603 172 999 168 × 2 = 0 + 0.403 256 431 206 345 998 336;
  • 27) 0.403 256 431 206 345 998 336 × 2 = 0 + 0.806 512 862 412 691 996 672;
  • 28) 0.806 512 862 412 691 996 672 × 2 = 1 + 0.613 025 724 825 383 993 344;
  • 29) 0.613 025 724 825 383 993 344 × 2 = 1 + 0.226 051 449 650 767 986 688;
  • 30) 0.226 051 449 650 767 986 688 × 2 = 0 + 0.452 102 899 301 535 973 376;
  • 31) 0.452 102 899 301 535 973 376 × 2 = 0 + 0.904 205 798 603 071 946 752;
  • 32) 0.904 205 798 603 071 946 752 × 2 = 1 + 0.808 411 597 206 143 893 504;
  • 33) 0.808 411 597 206 143 893 504 × 2 = 1 + 0.616 823 194 412 287 787 008;
  • 34) 0.616 823 194 412 287 787 008 × 2 = 1 + 0.233 646 388 824 575 574 016;
  • 35) 0.233 646 388 824 575 574 016 × 2 = 0 + 0.467 292 777 649 151 148 032;
  • 36) 0.467 292 777 649 151 148 032 × 2 = 0 + 0.934 585 555 298 302 296 064;
  • 37) 0.934 585 555 298 302 296 064 × 2 = 1 + 0.869 171 110 596 604 592 128;
  • 38) 0.869 171 110 596 604 592 128 × 2 = 1 + 0.738 342 221 193 209 184 256;
  • 39) 0.738 342 221 193 209 184 256 × 2 = 1 + 0.476 684 442 386 418 368 512;
  • 40) 0.476 684 442 386 418 368 512 × 2 = 0 + 0.953 368 884 772 836 737 024;
  • 41) 0.953 368 884 772 836 737 024 × 2 = 1 + 0.906 737 769 545 673 474 048;
  • 42) 0.906 737 769 545 673 474 048 × 2 = 1 + 0.813 475 539 091 346 948 096;
  • 43) 0.813 475 539 091 346 948 096 × 2 = 1 + 0.626 951 078 182 693 896 192;
  • 44) 0.626 951 078 182 693 896 192 × 2 = 1 + 0.253 902 156 365 387 792 384;
  • 45) 0.253 902 156 365 387 792 384 × 2 = 0 + 0.507 804 312 730 775 584 768;
  • 46) 0.507 804 312 730 775 584 768 × 2 = 1 + 0.015 608 625 461 551 169 536;
  • 47) 0.015 608 625 461 551 169 536 × 2 = 0 + 0.031 217 250 923 102 339 072;
  • 48) 0.031 217 250 923 102 339 072 × 2 = 0 + 0.062 434 501 846 204 678 144;
  • 49) 0.062 434 501 846 204 678 144 × 2 = 0 + 0.124 869 003 692 409 356 288;
  • 50) 0.124 869 003 692 409 356 288 × 2 = 0 + 0.249 738 007 384 818 712 576;
  • 51) 0.249 738 007 384 818 712 576 × 2 = 0 + 0.499 476 014 769 637 425 152;
  • 52) 0.499 476 014 769 637 425 152 × 2 = 0 + 0.998 952 029 539 274 850 304;
  • 53) 0.998 952 029 539 274 850 304 × 2 = 1 + 0.997 904 059 078 549 700 608;

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:


0.745 459 324 169 999 826 049(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 049(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:


1.745 459 324 169 999 826 049(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 1.745 459 324 169 999 826 049 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