1.745 459 324 169 999 826 167 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 167(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
1.745 459 324 169 999 826 167(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: 0.745 459 324 169 999 826 167.

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) 0.745 459 324 169 999 826 167 × 2 = 1 + 0.490 918 648 339 999 652 334;
  • 2) 0.490 918 648 339 999 652 334 × 2 = 0 + 0.981 837 296 679 999 304 668;
  • 3) 0.981 837 296 679 999 304 668 × 2 = 1 + 0.963 674 593 359 998 609 336;
  • 4) 0.963 674 593 359 998 609 336 × 2 = 1 + 0.927 349 186 719 997 218 672;
  • 5) 0.927 349 186 719 997 218 672 × 2 = 1 + 0.854 698 373 439 994 437 344;
  • 6) 0.854 698 373 439 994 437 344 × 2 = 1 + 0.709 396 746 879 988 874 688;
  • 7) 0.709 396 746 879 988 874 688 × 2 = 1 + 0.418 793 493 759 977 749 376;
  • 8) 0.418 793 493 759 977 749 376 × 2 = 0 + 0.837 586 987 519 955 498 752;
  • 9) 0.837 586 987 519 955 498 752 × 2 = 1 + 0.675 173 975 039 910 997 504;
  • 10) 0.675 173 975 039 910 997 504 × 2 = 1 + 0.350 347 950 079 821 995 008;
  • 11) 0.350 347 950 079 821 995 008 × 2 = 0 + 0.700 695 900 159 643 990 016;
  • 12) 0.700 695 900 159 643 990 016 × 2 = 1 + 0.401 391 800 319 287 980 032;
  • 13) 0.401 391 800 319 287 980 032 × 2 = 0 + 0.802 783 600 638 575 960 064;
  • 14) 0.802 783 600 638 575 960 064 × 2 = 1 + 0.605 567 201 277 151 920 128;
  • 15) 0.605 567 201 277 151 920 128 × 2 = 1 + 0.211 134 402 554 303 840 256;
  • 16) 0.211 134 402 554 303 840 256 × 2 = 0 + 0.422 268 805 108 607 680 512;
  • 17) 0.422 268 805 108 607 680 512 × 2 = 0 + 0.844 537 610 217 215 361 024;
  • 18) 0.844 537 610 217 215 361 024 × 2 = 1 + 0.689 075 220 434 430 722 048;
  • 19) 0.689 075 220 434 430 722 048 × 2 = 1 + 0.378 150 440 868 861 444 096;
  • 20) 0.378 150 440 868 861 444 096 × 2 = 0 + 0.756 300 881 737 722 888 192;
  • 21) 0.756 300 881 737 722 888 192 × 2 = 1 + 0.512 601 763 475 445 776 384;
  • 22) 0.512 601 763 475 445 776 384 × 2 = 1 + 0.025 203 526 950 891 552 768;
  • 23) 0.025 203 526 950 891 552 768 × 2 = 0 + 0.050 407 053 901 783 105 536;
  • 24) 0.050 407 053 901 783 105 536 × 2 = 0 + 0.100 814 107 803 566 211 072;
  • 25) 0.100 814 107 803 566 211 072 × 2 = 0 + 0.201 628 215 607 132 422 144;
  • 26) 0.201 628 215 607 132 422 144 × 2 = 0 + 0.403 256 431 214 264 844 288;
  • 27) 0.403 256 431 214 264 844 288 × 2 = 0 + 0.806 512 862 428 529 688 576;
  • 28) 0.806 512 862 428 529 688 576 × 2 = 1 + 0.613 025 724 857 059 377 152;
  • 29) 0.613 025 724 857 059 377 152 × 2 = 1 + 0.226 051 449 714 118 754 304;
  • 30) 0.226 051 449 714 118 754 304 × 2 = 0 + 0.452 102 899 428 237 508 608;
  • 31) 0.452 102 899 428 237 508 608 × 2 = 0 + 0.904 205 798 856 475 017 216;
  • 32) 0.904 205 798 856 475 017 216 × 2 = 1 + 0.808 411 597 712 950 034 432;
  • 33) 0.808 411 597 712 950 034 432 × 2 = 1 + 0.616 823 195 425 900 068 864;
  • 34) 0.616 823 195 425 900 068 864 × 2 = 1 + 0.233 646 390 851 800 137 728;
  • 35) 0.233 646 390 851 800 137 728 × 2 = 0 + 0.467 292 781 703 600 275 456;
  • 36) 0.467 292 781 703 600 275 456 × 2 = 0 + 0.934 585 563 407 200 550 912;
  • 37) 0.934 585 563 407 200 550 912 × 2 = 1 + 0.869 171 126 814 401 101 824;
  • 38) 0.869 171 126 814 401 101 824 × 2 = 1 + 0.738 342 253 628 802 203 648;
  • 39) 0.738 342 253 628 802 203 648 × 2 = 1 + 0.476 684 507 257 604 407 296;
  • 40) 0.476 684 507 257 604 407 296 × 2 = 0 + 0.953 369 014 515 208 814 592;
  • 41) 0.953 369 014 515 208 814 592 × 2 = 1 + 0.906 738 029 030 417 629 184;
  • 42) 0.906 738 029 030 417 629 184 × 2 = 1 + 0.813 476 058 060 835 258 368;
  • 43) 0.813 476 058 060 835 258 368 × 2 = 1 + 0.626 952 116 121 670 516 736;
  • 44) 0.626 952 116 121 670 516 736 × 2 = 1 + 0.253 904 232 243 341 033 472;
  • 45) 0.253 904 232 243 341 033 472 × 2 = 0 + 0.507 808 464 486 682 066 944;
  • 46) 0.507 808 464 486 682 066 944 × 2 = 1 + 0.015 616 928 973 364 133 888;
  • 47) 0.015 616 928 973 364 133 888 × 2 = 0 + 0.031 233 857 946 728 267 776;
  • 48) 0.031 233 857 946 728 267 776 × 2 = 0 + 0.062 467 715 893 456 535 552;
  • 49) 0.062 467 715 893 456 535 552 × 2 = 0 + 0.124 935 431 786 913 071 104;
  • 50) 0.124 935 431 786 913 071 104 × 2 = 0 + 0.249 870 863 573 826 142 208;
  • 51) 0.249 870 863 573 826 142 208 × 2 = 0 + 0.499 741 727 147 652 284 416;
  • 52) 0.499 741 727 147 652 284 416 × 2 = 0 + 0.999 483 454 295 304 568 832;
  • 53) 0.999 483 454 295 304 568 832 × 2 = 1 + 0.998 966 908 590 609 137 664;

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:


0.745 459 324 169 999 826 167(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 167(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:


1.745 459 324 169 999 826 167(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 1.745 459 324 169 999 826 167 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