1.745 459 324 169 999 826 214 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 214(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
1.745 459 324 169 999 826 214(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: 0.745 459 324 169 999 826 214.

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) 0.745 459 324 169 999 826 214 × 2 = 1 + 0.490 918 648 339 999 652 428;
  • 2) 0.490 918 648 339 999 652 428 × 2 = 0 + 0.981 837 296 679 999 304 856;
  • 3) 0.981 837 296 679 999 304 856 × 2 = 1 + 0.963 674 593 359 998 609 712;
  • 4) 0.963 674 593 359 998 609 712 × 2 = 1 + 0.927 349 186 719 997 219 424;
  • 5) 0.927 349 186 719 997 219 424 × 2 = 1 + 0.854 698 373 439 994 438 848;
  • 6) 0.854 698 373 439 994 438 848 × 2 = 1 + 0.709 396 746 879 988 877 696;
  • 7) 0.709 396 746 879 988 877 696 × 2 = 1 + 0.418 793 493 759 977 755 392;
  • 8) 0.418 793 493 759 977 755 392 × 2 = 0 + 0.837 586 987 519 955 510 784;
  • 9) 0.837 586 987 519 955 510 784 × 2 = 1 + 0.675 173 975 039 911 021 568;
  • 10) 0.675 173 975 039 911 021 568 × 2 = 1 + 0.350 347 950 079 822 043 136;
  • 11) 0.350 347 950 079 822 043 136 × 2 = 0 + 0.700 695 900 159 644 086 272;
  • 12) 0.700 695 900 159 644 086 272 × 2 = 1 + 0.401 391 800 319 288 172 544;
  • 13) 0.401 391 800 319 288 172 544 × 2 = 0 + 0.802 783 600 638 576 345 088;
  • 14) 0.802 783 600 638 576 345 088 × 2 = 1 + 0.605 567 201 277 152 690 176;
  • 15) 0.605 567 201 277 152 690 176 × 2 = 1 + 0.211 134 402 554 305 380 352;
  • 16) 0.211 134 402 554 305 380 352 × 2 = 0 + 0.422 268 805 108 610 760 704;
  • 17) 0.422 268 805 108 610 760 704 × 2 = 0 + 0.844 537 610 217 221 521 408;
  • 18) 0.844 537 610 217 221 521 408 × 2 = 1 + 0.689 075 220 434 443 042 816;
  • 19) 0.689 075 220 434 443 042 816 × 2 = 1 + 0.378 150 440 868 886 085 632;
  • 20) 0.378 150 440 868 886 085 632 × 2 = 0 + 0.756 300 881 737 772 171 264;
  • 21) 0.756 300 881 737 772 171 264 × 2 = 1 + 0.512 601 763 475 544 342 528;
  • 22) 0.512 601 763 475 544 342 528 × 2 = 1 + 0.025 203 526 951 088 685 056;
  • 23) 0.025 203 526 951 088 685 056 × 2 = 0 + 0.050 407 053 902 177 370 112;
  • 24) 0.050 407 053 902 177 370 112 × 2 = 0 + 0.100 814 107 804 354 740 224;
  • 25) 0.100 814 107 804 354 740 224 × 2 = 0 + 0.201 628 215 608 709 480 448;
  • 26) 0.201 628 215 608 709 480 448 × 2 = 0 + 0.403 256 431 217 418 960 896;
  • 27) 0.403 256 431 217 418 960 896 × 2 = 0 + 0.806 512 862 434 837 921 792;
  • 28) 0.806 512 862 434 837 921 792 × 2 = 1 + 0.613 025 724 869 675 843 584;
  • 29) 0.613 025 724 869 675 843 584 × 2 = 1 + 0.226 051 449 739 351 687 168;
  • 30) 0.226 051 449 739 351 687 168 × 2 = 0 + 0.452 102 899 478 703 374 336;
  • 31) 0.452 102 899 478 703 374 336 × 2 = 0 + 0.904 205 798 957 406 748 672;
  • 32) 0.904 205 798 957 406 748 672 × 2 = 1 + 0.808 411 597 914 813 497 344;
  • 33) 0.808 411 597 914 813 497 344 × 2 = 1 + 0.616 823 195 829 626 994 688;
  • 34) 0.616 823 195 829 626 994 688 × 2 = 1 + 0.233 646 391 659 253 989 376;
  • 35) 0.233 646 391 659 253 989 376 × 2 = 0 + 0.467 292 783 318 507 978 752;
  • 36) 0.467 292 783 318 507 978 752 × 2 = 0 + 0.934 585 566 637 015 957 504;
  • 37) 0.934 585 566 637 015 957 504 × 2 = 1 + 0.869 171 133 274 031 915 008;
  • 38) 0.869 171 133 274 031 915 008 × 2 = 1 + 0.738 342 266 548 063 830 016;
  • 39) 0.738 342 266 548 063 830 016 × 2 = 1 + 0.476 684 533 096 127 660 032;
  • 40) 0.476 684 533 096 127 660 032 × 2 = 0 + 0.953 369 066 192 255 320 064;
  • 41) 0.953 369 066 192 255 320 064 × 2 = 1 + 0.906 738 132 384 510 640 128;
  • 42) 0.906 738 132 384 510 640 128 × 2 = 1 + 0.813 476 264 769 021 280 256;
  • 43) 0.813 476 264 769 021 280 256 × 2 = 1 + 0.626 952 529 538 042 560 512;
  • 44) 0.626 952 529 538 042 560 512 × 2 = 1 + 0.253 905 059 076 085 121 024;
  • 45) 0.253 905 059 076 085 121 024 × 2 = 0 + 0.507 810 118 152 170 242 048;
  • 46) 0.507 810 118 152 170 242 048 × 2 = 1 + 0.015 620 236 304 340 484 096;
  • 47) 0.015 620 236 304 340 484 096 × 2 = 0 + 0.031 240 472 608 680 968 192;
  • 48) 0.031 240 472 608 680 968 192 × 2 = 0 + 0.062 480 945 217 361 936 384;
  • 49) 0.062 480 945 217 361 936 384 × 2 = 0 + 0.124 961 890 434 723 872 768;
  • 50) 0.124 961 890 434 723 872 768 × 2 = 0 + 0.249 923 780 869 447 745 536;
  • 51) 0.249 923 780 869 447 745 536 × 2 = 0 + 0.499 847 561 738 895 491 072;
  • 52) 0.499 847 561 738 895 491 072 × 2 = 0 + 0.999 695 123 477 790 982 144;
  • 53) 0.999 695 123 477 790 982 144 × 2 = 1 + 0.999 390 246 955 581 964 288;

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:


0.745 459 324 169 999 826 214(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 214(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:


1.745 459 324 169 999 826 214(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 1.745 459 324 169 999 826 214 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