7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 741 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 741(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
7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 741(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: 7.
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;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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.

7(10) =

111(2)


3. Convert to binary (base 2) the fractional part: 0.200 000 000 000 000 177 635 683 940 025 046 467 781 066 741.

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) 0.200 000 000 000 000 177 635 683 940 025 046 467 781 066 741 × 2 = 0 + 0.400 000 000 000 000 355 271 367 880 050 092 935 562 133 482;
  • 2) 0.400 000 000 000 000 355 271 367 880 050 092 935 562 133 482 × 2 = 0 + 0.800 000 000 000 000 710 542 735 760 100 185 871 124 266 964;
  • 3) 0.800 000 000 000 000 710 542 735 760 100 185 871 124 266 964 × 2 = 1 + 0.600 000 000 000 001 421 085 471 520 200 371 742 248 533 928;
  • 4) 0.600 000 000 000 001 421 085 471 520 200 371 742 248 533 928 × 2 = 1 + 0.200 000 000 000 002 842 170 943 040 400 743 484 497 067 856;
  • 5) 0.200 000 000 000 002 842 170 943 040 400 743 484 497 067 856 × 2 = 0 + 0.400 000 000 000 005 684 341 886 080 801 486 968 994 135 712;
  • 6) 0.400 000 000 000 005 684 341 886 080 801 486 968 994 135 712 × 2 = 0 + 0.800 000 000 000 011 368 683 772 161 602 973 937 988 271 424;
  • 7) 0.800 000 000 000 011 368 683 772 161 602 973 937 988 271 424 × 2 = 1 + 0.600 000 000 000 022 737 367 544 323 205 947 875 976 542 848;
  • 8) 0.600 000 000 000 022 737 367 544 323 205 947 875 976 542 848 × 2 = 1 + 0.200 000 000 000 045 474 735 088 646 411 895 751 953 085 696;
  • 9) 0.200 000 000 000 045 474 735 088 646 411 895 751 953 085 696 × 2 = 0 + 0.400 000 000 000 090 949 470 177 292 823 791 503 906 171 392;
  • 10) 0.400 000 000 000 090 949 470 177 292 823 791 503 906 171 392 × 2 = 0 + 0.800 000 000 000 181 898 940 354 585 647 583 007 812 342 784;
  • 11) 0.800 000 000 000 181 898 940 354 585 647 583 007 812 342 784 × 2 = 1 + 0.600 000 000 000 363 797 880 709 171 295 166 015 624 685 568;
  • 12) 0.600 000 000 000 363 797 880 709 171 295 166 015 624 685 568 × 2 = 1 + 0.200 000 000 000 727 595 761 418 342 590 332 031 249 371 136;
  • 13) 0.200 000 000 000 727 595 761 418 342 590 332 031 249 371 136 × 2 = 0 + 0.400 000 000 001 455 191 522 836 685 180 664 062 498 742 272;
  • 14) 0.400 000 000 001 455 191 522 836 685 180 664 062 498 742 272 × 2 = 0 + 0.800 000 000 002 910 383 045 673 370 361 328 124 997 484 544;
  • 15) 0.800 000 000 002 910 383 045 673 370 361 328 124 997 484 544 × 2 = 1 + 0.600 000 000 005 820 766 091 346 740 722 656 249 994 969 088;
  • 16) 0.600 000 000 005 820 766 091 346 740 722 656 249 994 969 088 × 2 = 1 + 0.200 000 000 011 641 532 182 693 481 445 312 499 989 938 176;
  • 17) 0.200 000 000 011 641 532 182 693 481 445 312 499 989 938 176 × 2 = 0 + 0.400 000 000 023 283 064 365 386 962 890 624 999 979 876 352;
  • 18) 0.400 000 000 023 283 064 365 386 962 890 624 999 979 876 352 × 2 = 0 + 0.800 000 000 046 566 128 730 773 925 781 249 999 959 752 704;
  • 19) 0.800 000 000 046 566 128 730 773 925 781 249 999 959 752 704 × 2 = 1 + 0.600 000 000 093 132 257 461 547 851 562 499 999 919 505 408;
  • 20) 0.600 000 000 093 132 257 461 547 851 562 499 999 919 505 408 × 2 = 1 + 0.200 000 000 186 264 514 923 095 703 124 999 999 839 010 816;
  • 21) 0.200 000 000 186 264 514 923 095 703 124 999 999 839 010 816 × 2 = 0 + 0.400 000 000 372 529 029 846 191 406 249 999 999 678 021 632;
  • 22) 0.400 000 000 372 529 029 846 191 406 249 999 999 678 021 632 × 2 = 0 + 0.800 000 000 745 058 059 692 382 812 499 999 999 356 043 264;
  • 23) 0.800 000 000 745 058 059 692 382 812 499 999 999 356 043 264 × 2 = 1 + 0.600 000 001 490 116 119 384 765 624 999 999 998 712 086 528;
  • 24) 0.600 000 001 490 116 119 384 765 624 999 999 998 712 086 528 × 2 = 1 + 0.200 000 002 980 232 238 769 531 249 999 999 997 424 173 056;
  • 25) 0.200 000 002 980 232 238 769 531 249 999 999 997 424 173 056 × 2 = 0 + 0.400 000 005 960 464 477 539 062 499 999 999 994 848 346 112;
  • 26) 0.400 000 005 960 464 477 539 062 499 999 999 994 848 346 112 × 2 = 0 + 0.800 000 011 920 928 955 078 124 999 999 999 989 696 692 224;
  • 27) 0.800 000 011 920 928 955 078 124 999 999 999 989 696 692 224 × 2 = 1 + 0.600 000 023 841 857 910 156 249 999 999 999 979 393 384 448;
  • 28) 0.600 000 023 841 857 910 156 249 999 999 999 979 393 384 448 × 2 = 1 + 0.200 000 047 683 715 820 312 499 999 999 999 958 786 768 896;
  • 29) 0.200 000 047 683 715 820 312 499 999 999 999 958 786 768 896 × 2 = 0 + 0.400 000 095 367 431 640 624 999 999 999 999 917 573 537 792;
  • 30) 0.400 000 095 367 431 640 624 999 999 999 999 917 573 537 792 × 2 = 0 + 0.800 000 190 734 863 281 249 999 999 999 999 835 147 075 584;
  • 31) 0.800 000 190 734 863 281 249 999 999 999 999 835 147 075 584 × 2 = 1 + 0.600 000 381 469 726 562 499 999 999 999 999 670 294 151 168;
  • 32) 0.600 000 381 469 726 562 499 999 999 999 999 670 294 151 168 × 2 = 1 + 0.200 000 762 939 453 124 999 999 999 999 999 340 588 302 336;
  • 33) 0.200 000 762 939 453 124 999 999 999 999 999 340 588 302 336 × 2 = 0 + 0.400 001 525 878 906 249 999 999 999 999 998 681 176 604 672;
  • 34) 0.400 001 525 878 906 249 999 999 999 999 998 681 176 604 672 × 2 = 0 + 0.800 003 051 757 812 499 999 999 999 999 997 362 353 209 344;
  • 35) 0.800 003 051 757 812 499 999 999 999 999 997 362 353 209 344 × 2 = 1 + 0.600 006 103 515 624 999 999 999 999 999 994 724 706 418 688;
  • 36) 0.600 006 103 515 624 999 999 999 999 999 994 724 706 418 688 × 2 = 1 + 0.200 012 207 031 249 999 999 999 999 999 989 449 412 837 376;
  • 37) 0.200 012 207 031 249 999 999 999 999 999 989 449 412 837 376 × 2 = 0 + 0.400 024 414 062 499 999 999 999 999 999 978 898 825 674 752;
  • 38) 0.400 024 414 062 499 999 999 999 999 999 978 898 825 674 752 × 2 = 0 + 0.800 048 828 124 999 999 999 999 999 999 957 797 651 349 504;
  • 39) 0.800 048 828 124 999 999 999 999 999 999 957 797 651 349 504 × 2 = 1 + 0.600 097 656 249 999 999 999 999 999 999 915 595 302 699 008;
  • 40) 0.600 097 656 249 999 999 999 999 999 999 915 595 302 699 008 × 2 = 1 + 0.200 195 312 499 999 999 999 999 999 999 831 190 605 398 016;
  • 41) 0.200 195 312 499 999 999 999 999 999 999 831 190 605 398 016 × 2 = 0 + 0.400 390 624 999 999 999 999 999 999 999 662 381 210 796 032;
  • 42) 0.400 390 624 999 999 999 999 999 999 999 662 381 210 796 032 × 2 = 0 + 0.800 781 249 999 999 999 999 999 999 999 324 762 421 592 064;
  • 43) 0.800 781 249 999 999 999 999 999 999 999 324 762 421 592 064 × 2 = 1 + 0.601 562 499 999 999 999 999 999 999 998 649 524 843 184 128;
  • 44) 0.601 562 499 999 999 999 999 999 999 998 649 524 843 184 128 × 2 = 1 + 0.203 124 999 999 999 999 999 999 999 997 299 049 686 368 256;
  • 45) 0.203 124 999 999 999 999 999 999 999 997 299 049 686 368 256 × 2 = 0 + 0.406 249 999 999 999 999 999 999 999 994 598 099 372 736 512;
  • 46) 0.406 249 999 999 999 999 999 999 999 994 598 099 372 736 512 × 2 = 0 + 0.812 499 999 999 999 999 999 999 999 989 196 198 745 473 024;
  • 47) 0.812 499 999 999 999 999 999 999 999 989 196 198 745 473 024 × 2 = 1 + 0.624 999 999 999 999 999 999 999 999 978 392 397 490 946 048;
  • 48) 0.624 999 999 999 999 999 999 999 999 978 392 397 490 946 048 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 956 784 794 981 892 096;
  • 49) 0.249 999 999 999 999 999 999 999 999 956 784 794 981 892 096 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 913 569 589 963 784 192;
  • 50) 0.499 999 999 999 999 999 999 999 999 913 569 589 963 784 192 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 827 139 179 927 568 384;
  • 51) 0.999 999 999 999 999 999 999 999 999 827 139 179 927 568 384 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 654 278 359 855 136 768;
  • 52) 0.999 999 999 999 999 999 999 999 999 654 278 359 855 136 768 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 308 556 719 710 273 536;
  • 53) 0.999 999 999 999 999 999 999 999 999 308 556 719 710 273 536 × 2 = 1 + 0.999 999 999 999 999 999 999 999 998 617 113 439 420 547 072;

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:


0.200 000 000 000 000 177 635 683 940 025 046 467 781 066 741(10) =

0.0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 1(2)

5. Positive number before normalization:

7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 741(10) =

111.0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 2 positions to the left, so that only one non zero digit remains to the left of it:


7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 741(10) =

111.0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 1(2) =

111.0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 1(2) × 20 =

1.1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 111(2) × 22


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): 2

Mantissa (not normalized):
1.1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

2 + 2(11-1) - 1 =

(2 + 1 023)(10) =

1 025(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;
  • 1 025 ÷ 2 = 512 + 1;
  • 512 ÷ 2 = 256 + 0;
  • 256 ÷ 2 = 128 + 0;
  • 128 ÷ 2 = 64 + 0;
  • 64 ÷ 2 = 32 + 0;
  • 32 ÷ 2 = 16 + 0;
  • 16 ÷ 2 = 8 + 0;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 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) =

1025(10) =

100 0000 0001(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. 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 111 =

1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100


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) =
100 0000 0001

Mantissa (52 bits) =
1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100


Decimal number 7.200 000 000 000 000 177 635 683 940 025 046 467 781 066 741 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 0000 0001 - 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100

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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