1.745 459 324 169 999 826 281 696 186 924 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 1.745 459 324 169 999 826 281 696 186 924 2(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
1.745 459 324 169 999 826 281 696 186 924 2(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: 0.745 459 324 169 999 826 281 696 186 924 2.

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) 0.745 459 324 169 999 826 281 696 186 924 2 × 2 = 1 + 0.490 918 648 339 999 652 563 392 373 848 4;
  • 2) 0.490 918 648 339 999 652 563 392 373 848 4 × 2 = 0 + 0.981 837 296 679 999 305 126 784 747 696 8;
  • 3) 0.981 837 296 679 999 305 126 784 747 696 8 × 2 = 1 + 0.963 674 593 359 998 610 253 569 495 393 6;
  • 4) 0.963 674 593 359 998 610 253 569 495 393 6 × 2 = 1 + 0.927 349 186 719 997 220 507 138 990 787 2;
  • 5) 0.927 349 186 719 997 220 507 138 990 787 2 × 2 = 1 + 0.854 698 373 439 994 441 014 277 981 574 4;
  • 6) 0.854 698 373 439 994 441 014 277 981 574 4 × 2 = 1 + 0.709 396 746 879 988 882 028 555 963 148 8;
  • 7) 0.709 396 746 879 988 882 028 555 963 148 8 × 2 = 1 + 0.418 793 493 759 977 764 057 111 926 297 6;
  • 8) 0.418 793 493 759 977 764 057 111 926 297 6 × 2 = 0 + 0.837 586 987 519 955 528 114 223 852 595 2;
  • 9) 0.837 586 987 519 955 528 114 223 852 595 2 × 2 = 1 + 0.675 173 975 039 911 056 228 447 705 190 4;
  • 10) 0.675 173 975 039 911 056 228 447 705 190 4 × 2 = 1 + 0.350 347 950 079 822 112 456 895 410 380 8;
  • 11) 0.350 347 950 079 822 112 456 895 410 380 8 × 2 = 0 + 0.700 695 900 159 644 224 913 790 820 761 6;
  • 12) 0.700 695 900 159 644 224 913 790 820 761 6 × 2 = 1 + 0.401 391 800 319 288 449 827 581 641 523 2;
  • 13) 0.401 391 800 319 288 449 827 581 641 523 2 × 2 = 0 + 0.802 783 600 638 576 899 655 163 283 046 4;
  • 14) 0.802 783 600 638 576 899 655 163 283 046 4 × 2 = 1 + 0.605 567 201 277 153 799 310 326 566 092 8;
  • 15) 0.605 567 201 277 153 799 310 326 566 092 8 × 2 = 1 + 0.211 134 402 554 307 598 620 653 132 185 6;
  • 16) 0.211 134 402 554 307 598 620 653 132 185 6 × 2 = 0 + 0.422 268 805 108 615 197 241 306 264 371 2;
  • 17) 0.422 268 805 108 615 197 241 306 264 371 2 × 2 = 0 + 0.844 537 610 217 230 394 482 612 528 742 4;
  • 18) 0.844 537 610 217 230 394 482 612 528 742 4 × 2 = 1 + 0.689 075 220 434 460 788 965 225 057 484 8;
  • 19) 0.689 075 220 434 460 788 965 225 057 484 8 × 2 = 1 + 0.378 150 440 868 921 577 930 450 114 969 6;
  • 20) 0.378 150 440 868 921 577 930 450 114 969 6 × 2 = 0 + 0.756 300 881 737 843 155 860 900 229 939 2;
  • 21) 0.756 300 881 737 843 155 860 900 229 939 2 × 2 = 1 + 0.512 601 763 475 686 311 721 800 459 878 4;
  • 22) 0.512 601 763 475 686 311 721 800 459 878 4 × 2 = 1 + 0.025 203 526 951 372 623 443 600 919 756 8;
  • 23) 0.025 203 526 951 372 623 443 600 919 756 8 × 2 = 0 + 0.050 407 053 902 745 246 887 201 839 513 6;
  • 24) 0.050 407 053 902 745 246 887 201 839 513 6 × 2 = 0 + 0.100 814 107 805 490 493 774 403 679 027 2;
  • 25) 0.100 814 107 805 490 493 774 403 679 027 2 × 2 = 0 + 0.201 628 215 610 980 987 548 807 358 054 4;
  • 26) 0.201 628 215 610 980 987 548 807 358 054 4 × 2 = 0 + 0.403 256 431 221 961 975 097 614 716 108 8;
  • 27) 0.403 256 431 221 961 975 097 614 716 108 8 × 2 = 0 + 0.806 512 862 443 923 950 195 229 432 217 6;
  • 28) 0.806 512 862 443 923 950 195 229 432 217 6 × 2 = 1 + 0.613 025 724 887 847 900 390 458 864 435 2;
  • 29) 0.613 025 724 887 847 900 390 458 864 435 2 × 2 = 1 + 0.226 051 449 775 695 800 780 917 728 870 4;
  • 30) 0.226 051 449 775 695 800 780 917 728 870 4 × 2 = 0 + 0.452 102 899 551 391 601 561 835 457 740 8;
  • 31) 0.452 102 899 551 391 601 561 835 457 740 8 × 2 = 0 + 0.904 205 799 102 783 203 123 670 915 481 6;
  • 32) 0.904 205 799 102 783 203 123 670 915 481 6 × 2 = 1 + 0.808 411 598 205 566 406 247 341 830 963 2;
  • 33) 0.808 411 598 205 566 406 247 341 830 963 2 × 2 = 1 + 0.616 823 196 411 132 812 494 683 661 926 4;
  • 34) 0.616 823 196 411 132 812 494 683 661 926 4 × 2 = 1 + 0.233 646 392 822 265 624 989 367 323 852 8;
  • 35) 0.233 646 392 822 265 624 989 367 323 852 8 × 2 = 0 + 0.467 292 785 644 531 249 978 734 647 705 6;
  • 36) 0.467 292 785 644 531 249 978 734 647 705 6 × 2 = 0 + 0.934 585 571 289 062 499 957 469 295 411 2;
  • 37) 0.934 585 571 289 062 499 957 469 295 411 2 × 2 = 1 + 0.869 171 142 578 124 999 914 938 590 822 4;
  • 38) 0.869 171 142 578 124 999 914 938 590 822 4 × 2 = 1 + 0.738 342 285 156 249 999 829 877 181 644 8;
  • 39) 0.738 342 285 156 249 999 829 877 181 644 8 × 2 = 1 + 0.476 684 570 312 499 999 659 754 363 289 6;
  • 40) 0.476 684 570 312 499 999 659 754 363 289 6 × 2 = 0 + 0.953 369 140 624 999 999 319 508 726 579 2;
  • 41) 0.953 369 140 624 999 999 319 508 726 579 2 × 2 = 1 + 0.906 738 281 249 999 998 639 017 453 158 4;
  • 42) 0.906 738 281 249 999 998 639 017 453 158 4 × 2 = 1 + 0.813 476 562 499 999 997 278 034 906 316 8;
  • 43) 0.813 476 562 499 999 997 278 034 906 316 8 × 2 = 1 + 0.626 953 124 999 999 994 556 069 812 633 6;
  • 44) 0.626 953 124 999 999 994 556 069 812 633 6 × 2 = 1 + 0.253 906 249 999 999 989 112 139 625 267 2;
  • 45) 0.253 906 249 999 999 989 112 139 625 267 2 × 2 = 0 + 0.507 812 499 999 999 978 224 279 250 534 4;
  • 46) 0.507 812 499 999 999 978 224 279 250 534 4 × 2 = 1 + 0.015 624 999 999 999 956 448 558 501 068 8;
  • 47) 0.015 624 999 999 999 956 448 558 501 068 8 × 2 = 0 + 0.031 249 999 999 999 912 897 117 002 137 6;
  • 48) 0.031 249 999 999 999 912 897 117 002 137 6 × 2 = 0 + 0.062 499 999 999 999 825 794 234 004 275 2;
  • 49) 0.062 499 999 999 999 825 794 234 004 275 2 × 2 = 0 + 0.124 999 999 999 999 651 588 468 008 550 4;
  • 50) 0.124 999 999 999 999 651 588 468 008 550 4 × 2 = 0 + 0.249 999 999 999 999 303 176 936 017 100 8;
  • 51) 0.249 999 999 999 999 303 176 936 017 100 8 × 2 = 0 + 0.499 999 999 999 998 606 353 872 034 201 6;
  • 52) 0.499 999 999 999 998 606 353 872 034 201 6 × 2 = 0 + 0.999 999 999 999 997 212 707 744 068 403 2;
  • 53) 0.999 999 999 999 997 212 707 744 068 403 2 × 2 = 1 + 0.999 999 999 999 994 425 415 488 136 806 4;

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:


0.745 459 324 169 999 826 281 696 186 924 2(10) =

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

5. Positive number before normalization:

1.745 459 324 169 999 826 281 696 186 924 2(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:


1.745 459 324 169 999 826 281 696 186 924 2(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 1.745 459 324 169 999 826 281 696 186 924 2 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