1.745 459 324 169 999 826 281 697 16 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 697 16(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
1.745 459 324 169 999 826 281 697 16(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: 0.745 459 324 169 999 826 281 697 16.

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) 0.745 459 324 169 999 826 281 697 16 × 2 = 1 + 0.490 918 648 339 999 652 563 394 32;
  • 2) 0.490 918 648 339 999 652 563 394 32 × 2 = 0 + 0.981 837 296 679 999 305 126 788 64;
  • 3) 0.981 837 296 679 999 305 126 788 64 × 2 = 1 + 0.963 674 593 359 998 610 253 577 28;
  • 4) 0.963 674 593 359 998 610 253 577 28 × 2 = 1 + 0.927 349 186 719 997 220 507 154 56;
  • 5) 0.927 349 186 719 997 220 507 154 56 × 2 = 1 + 0.854 698 373 439 994 441 014 309 12;
  • 6) 0.854 698 373 439 994 441 014 309 12 × 2 = 1 + 0.709 396 746 879 988 882 028 618 24;
  • 7) 0.709 396 746 879 988 882 028 618 24 × 2 = 1 + 0.418 793 493 759 977 764 057 236 48;
  • 8) 0.418 793 493 759 977 764 057 236 48 × 2 = 0 + 0.837 586 987 519 955 528 114 472 96;
  • 9) 0.837 586 987 519 955 528 114 472 96 × 2 = 1 + 0.675 173 975 039 911 056 228 945 92;
  • 10) 0.675 173 975 039 911 056 228 945 92 × 2 = 1 + 0.350 347 950 079 822 112 457 891 84;
  • 11) 0.350 347 950 079 822 112 457 891 84 × 2 = 0 + 0.700 695 900 159 644 224 915 783 68;
  • 12) 0.700 695 900 159 644 224 915 783 68 × 2 = 1 + 0.401 391 800 319 288 449 831 567 36;
  • 13) 0.401 391 800 319 288 449 831 567 36 × 2 = 0 + 0.802 783 600 638 576 899 663 134 72;
  • 14) 0.802 783 600 638 576 899 663 134 72 × 2 = 1 + 0.605 567 201 277 153 799 326 269 44;
  • 15) 0.605 567 201 277 153 799 326 269 44 × 2 = 1 + 0.211 134 402 554 307 598 652 538 88;
  • 16) 0.211 134 402 554 307 598 652 538 88 × 2 = 0 + 0.422 268 805 108 615 197 305 077 76;
  • 17) 0.422 268 805 108 615 197 305 077 76 × 2 = 0 + 0.844 537 610 217 230 394 610 155 52;
  • 18) 0.844 537 610 217 230 394 610 155 52 × 2 = 1 + 0.689 075 220 434 460 789 220 311 04;
  • 19) 0.689 075 220 434 460 789 220 311 04 × 2 = 1 + 0.378 150 440 868 921 578 440 622 08;
  • 20) 0.378 150 440 868 921 578 440 622 08 × 2 = 0 + 0.756 300 881 737 843 156 881 244 16;
  • 21) 0.756 300 881 737 843 156 881 244 16 × 2 = 1 + 0.512 601 763 475 686 313 762 488 32;
  • 22) 0.512 601 763 475 686 313 762 488 32 × 2 = 1 + 0.025 203 526 951 372 627 524 976 64;
  • 23) 0.025 203 526 951 372 627 524 976 64 × 2 = 0 + 0.050 407 053 902 745 255 049 953 28;
  • 24) 0.050 407 053 902 745 255 049 953 28 × 2 = 0 + 0.100 814 107 805 490 510 099 906 56;
  • 25) 0.100 814 107 805 490 510 099 906 56 × 2 = 0 + 0.201 628 215 610 981 020 199 813 12;
  • 26) 0.201 628 215 610 981 020 199 813 12 × 2 = 0 + 0.403 256 431 221 962 040 399 626 24;
  • 27) 0.403 256 431 221 962 040 399 626 24 × 2 = 0 + 0.806 512 862 443 924 080 799 252 48;
  • 28) 0.806 512 862 443 924 080 799 252 48 × 2 = 1 + 0.613 025 724 887 848 161 598 504 96;
  • 29) 0.613 025 724 887 848 161 598 504 96 × 2 = 1 + 0.226 051 449 775 696 323 197 009 92;
  • 30) 0.226 051 449 775 696 323 197 009 92 × 2 = 0 + 0.452 102 899 551 392 646 394 019 84;
  • 31) 0.452 102 899 551 392 646 394 019 84 × 2 = 0 + 0.904 205 799 102 785 292 788 039 68;
  • 32) 0.904 205 799 102 785 292 788 039 68 × 2 = 1 + 0.808 411 598 205 570 585 576 079 36;
  • 33) 0.808 411 598 205 570 585 576 079 36 × 2 = 1 + 0.616 823 196 411 141 171 152 158 72;
  • 34) 0.616 823 196 411 141 171 152 158 72 × 2 = 1 + 0.233 646 392 822 282 342 304 317 44;
  • 35) 0.233 646 392 822 282 342 304 317 44 × 2 = 0 + 0.467 292 785 644 564 684 608 634 88;
  • 36) 0.467 292 785 644 564 684 608 634 88 × 2 = 0 + 0.934 585 571 289 129 369 217 269 76;
  • 37) 0.934 585 571 289 129 369 217 269 76 × 2 = 1 + 0.869 171 142 578 258 738 434 539 52;
  • 38) 0.869 171 142 578 258 738 434 539 52 × 2 = 1 + 0.738 342 285 156 517 476 869 079 04;
  • 39) 0.738 342 285 156 517 476 869 079 04 × 2 = 1 + 0.476 684 570 313 034 953 738 158 08;
  • 40) 0.476 684 570 313 034 953 738 158 08 × 2 = 0 + 0.953 369 140 626 069 907 476 316 16;
  • 41) 0.953 369 140 626 069 907 476 316 16 × 2 = 1 + 0.906 738 281 252 139 814 952 632 32;
  • 42) 0.906 738 281 252 139 814 952 632 32 × 2 = 1 + 0.813 476 562 504 279 629 905 264 64;
  • 43) 0.813 476 562 504 279 629 905 264 64 × 2 = 1 + 0.626 953 125 008 559 259 810 529 28;
  • 44) 0.626 953 125 008 559 259 810 529 28 × 2 = 1 + 0.253 906 250 017 118 519 621 058 56;
  • 45) 0.253 906 250 017 118 519 621 058 56 × 2 = 0 + 0.507 812 500 034 237 039 242 117 12;
  • 46) 0.507 812 500 034 237 039 242 117 12 × 2 = 1 + 0.015 625 000 068 474 078 484 234 24;
  • 47) 0.015 625 000 068 474 078 484 234 24 × 2 = 0 + 0.031 250 000 136 948 156 968 468 48;
  • 48) 0.031 250 000 136 948 156 968 468 48 × 2 = 0 + 0.062 500 000 273 896 313 936 936 96;
  • 49) 0.062 500 000 273 896 313 936 936 96 × 2 = 0 + 0.125 000 000 547 792 627 873 873 92;
  • 50) 0.125 000 000 547 792 627 873 873 92 × 2 = 0 + 0.250 000 001 095 585 255 747 747 84;
  • 51) 0.250 000 001 095 585 255 747 747 84 × 2 = 0 + 0.500 000 002 191 170 511 495 495 68;
  • 52) 0.500 000 002 191 170 511 495 495 68 × 2 = 1 + 0.000 000 004 382 341 022 990 991 36;
  • 53) 0.000 000 004 382 341 022 990 991 36 × 2 = 0 + 0.000 000 008 764 682 045 981 982 72;

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:


0.745 459 324 169 999 826 281 697 16(10) =

0.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(2)

5. Positive number before normalization:

1.745 459 324 169 999 826 281 697 16(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(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:


1.745 459 324 169 999 826 281 697 16(10) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(2) =

1.1011 1110 1101 0110 0110 1100 0001 1001 1100 1110 1111 0100 0001 0(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 0001 0


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 0001 0 =

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


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 0001


Decimal number 1.745 459 324 169 999 826 281 697 16 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 0001

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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