-19.981 999 999 999 999 317 878 973 670 303 821 563 716 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -19.981 999 999 999 999 317 878 973 670 303 821 563 716 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
-19.981 999 999 999 999 317 878 973 670 303 821 563 716 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. Start with the positive version of the number:

|-19.981 999 999 999 999 317 878 973 670 303 821 563 716 2| = 19.981 999 999 999 999 317 878 973 670 303 821 563 716 2


2. First, convert to binary (in base 2) the integer part: 19.
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;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

3. 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.

19(10) =

1 0011(2)


4. Convert to binary (base 2) the fractional part: 0.981 999 999 999 999 317 878 973 670 303 821 563 716 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) 0.981 999 999 999 999 317 878 973 670 303 821 563 716 2 × 2 = 1 + 0.963 999 999 999 998 635 757 947 340 607 643 127 432 4;
  • 2) 0.963 999 999 999 998 635 757 947 340 607 643 127 432 4 × 2 = 1 + 0.927 999 999 999 997 271 515 894 681 215 286 254 864 8;
  • 3) 0.927 999 999 999 997 271 515 894 681 215 286 254 864 8 × 2 = 1 + 0.855 999 999 999 994 543 031 789 362 430 572 509 729 6;
  • 4) 0.855 999 999 999 994 543 031 789 362 430 572 509 729 6 × 2 = 1 + 0.711 999 999 999 989 086 063 578 724 861 145 019 459 2;
  • 5) 0.711 999 999 999 989 086 063 578 724 861 145 019 459 2 × 2 = 1 + 0.423 999 999 999 978 172 127 157 449 722 290 038 918 4;
  • 6) 0.423 999 999 999 978 172 127 157 449 722 290 038 918 4 × 2 = 0 + 0.847 999 999 999 956 344 254 314 899 444 580 077 836 8;
  • 7) 0.847 999 999 999 956 344 254 314 899 444 580 077 836 8 × 2 = 1 + 0.695 999 999 999 912 688 508 629 798 889 160 155 673 6;
  • 8) 0.695 999 999 999 912 688 508 629 798 889 160 155 673 6 × 2 = 1 + 0.391 999 999 999 825 377 017 259 597 778 320 311 347 2;
  • 9) 0.391 999 999 999 825 377 017 259 597 778 320 311 347 2 × 2 = 0 + 0.783 999 999 999 650 754 034 519 195 556 640 622 694 4;
  • 10) 0.783 999 999 999 650 754 034 519 195 556 640 622 694 4 × 2 = 1 + 0.567 999 999 999 301 508 069 038 391 113 281 245 388 8;
  • 11) 0.567 999 999 999 301 508 069 038 391 113 281 245 388 8 × 2 = 1 + 0.135 999 999 998 603 016 138 076 782 226 562 490 777 6;
  • 12) 0.135 999 999 998 603 016 138 076 782 226 562 490 777 6 × 2 = 0 + 0.271 999 999 997 206 032 276 153 564 453 124 981 555 2;
  • 13) 0.271 999 999 997 206 032 276 153 564 453 124 981 555 2 × 2 = 0 + 0.543 999 999 994 412 064 552 307 128 906 249 963 110 4;
  • 14) 0.543 999 999 994 412 064 552 307 128 906 249 963 110 4 × 2 = 1 + 0.087 999 999 988 824 129 104 614 257 812 499 926 220 8;
  • 15) 0.087 999 999 988 824 129 104 614 257 812 499 926 220 8 × 2 = 0 + 0.175 999 999 977 648 258 209 228 515 624 999 852 441 6;
  • 16) 0.175 999 999 977 648 258 209 228 515 624 999 852 441 6 × 2 = 0 + 0.351 999 999 955 296 516 418 457 031 249 999 704 883 2;
  • 17) 0.351 999 999 955 296 516 418 457 031 249 999 704 883 2 × 2 = 0 + 0.703 999 999 910 593 032 836 914 062 499 999 409 766 4;
  • 18) 0.703 999 999 910 593 032 836 914 062 499 999 409 766 4 × 2 = 1 + 0.407 999 999 821 186 065 673 828 124 999 998 819 532 8;
  • 19) 0.407 999 999 821 186 065 673 828 124 999 998 819 532 8 × 2 = 0 + 0.815 999 999 642 372 131 347 656 249 999 997 639 065 6;
  • 20) 0.815 999 999 642 372 131 347 656 249 999 997 639 065 6 × 2 = 1 + 0.631 999 999 284 744 262 695 312 499 999 995 278 131 2;
  • 21) 0.631 999 999 284 744 262 695 312 499 999 995 278 131 2 × 2 = 1 + 0.263 999 998 569 488 525 390 624 999 999 990 556 262 4;
  • 22) 0.263 999 998 569 488 525 390 624 999 999 990 556 262 4 × 2 = 0 + 0.527 999 997 138 977 050 781 249 999 999 981 112 524 8;
  • 23) 0.527 999 997 138 977 050 781 249 999 999 981 112 524 8 × 2 = 1 + 0.055 999 994 277 954 101 562 499 999 999 962 225 049 6;
  • 24) 0.055 999 994 277 954 101 562 499 999 999 962 225 049 6 × 2 = 0 + 0.111 999 988 555 908 203 124 999 999 999 924 450 099 2;
  • 25) 0.111 999 988 555 908 203 124 999 999 999 924 450 099 2 × 2 = 0 + 0.223 999 977 111 816 406 249 999 999 999 848 900 198 4;
  • 26) 0.223 999 977 111 816 406 249 999 999 999 848 900 198 4 × 2 = 0 + 0.447 999 954 223 632 812 499 999 999 999 697 800 396 8;
  • 27) 0.447 999 954 223 632 812 499 999 999 999 697 800 396 8 × 2 = 0 + 0.895 999 908 447 265 624 999 999 999 999 395 600 793 6;
  • 28) 0.895 999 908 447 265 624 999 999 999 999 395 600 793 6 × 2 = 1 + 0.791 999 816 894 531 249 999 999 999 998 791 201 587 2;
  • 29) 0.791 999 816 894 531 249 999 999 999 998 791 201 587 2 × 2 = 1 + 0.583 999 633 789 062 499 999 999 999 997 582 403 174 4;
  • 30) 0.583 999 633 789 062 499 999 999 999 997 582 403 174 4 × 2 = 1 + 0.167 999 267 578 124 999 999 999 999 995 164 806 348 8;
  • 31) 0.167 999 267 578 124 999 999 999 999 995 164 806 348 8 × 2 = 0 + 0.335 998 535 156 249 999 999 999 999 990 329 612 697 6;
  • 32) 0.335 998 535 156 249 999 999 999 999 990 329 612 697 6 × 2 = 0 + 0.671 997 070 312 499 999 999 999 999 980 659 225 395 2;
  • 33) 0.671 997 070 312 499 999 999 999 999 980 659 225 395 2 × 2 = 1 + 0.343 994 140 624 999 999 999 999 999 961 318 450 790 4;
  • 34) 0.343 994 140 624 999 999 999 999 999 961 318 450 790 4 × 2 = 0 + 0.687 988 281 249 999 999 999 999 999 922 636 901 580 8;
  • 35) 0.687 988 281 249 999 999 999 999 999 922 636 901 580 8 × 2 = 1 + 0.375 976 562 499 999 999 999 999 999 845 273 803 161 6;
  • 36) 0.375 976 562 499 999 999 999 999 999 845 273 803 161 6 × 2 = 0 + 0.751 953 124 999 999 999 999 999 999 690 547 606 323 2;
  • 37) 0.751 953 124 999 999 999 999 999 999 690 547 606 323 2 × 2 = 1 + 0.503 906 249 999 999 999 999 999 999 381 095 212 646 4;
  • 38) 0.503 906 249 999 999 999 999 999 999 381 095 212 646 4 × 2 = 1 + 0.007 812 499 999 999 999 999 999 998 762 190 425 292 8;
  • 39) 0.007 812 499 999 999 999 999 999 998 762 190 425 292 8 × 2 = 0 + 0.015 624 999 999 999 999 999 999 997 524 380 850 585 6;
  • 40) 0.015 624 999 999 999 999 999 999 997 524 380 850 585 6 × 2 = 0 + 0.031 249 999 999 999 999 999 999 995 048 761 701 171 2;
  • 41) 0.031 249 999 999 999 999 999 999 995 048 761 701 171 2 × 2 = 0 + 0.062 499 999 999 999 999 999 999 990 097 523 402 342 4;
  • 42) 0.062 499 999 999 999 999 999 999 990 097 523 402 342 4 × 2 = 0 + 0.124 999 999 999 999 999 999 999 980 195 046 804 684 8;
  • 43) 0.124 999 999 999 999 999 999 999 980 195 046 804 684 8 × 2 = 0 + 0.249 999 999 999 999 999 999 999 960 390 093 609 369 6;
  • 44) 0.249 999 999 999 999 999 999 999 960 390 093 609 369 6 × 2 = 0 + 0.499 999 999 999 999 999 999 999 920 780 187 218 739 2;
  • 45) 0.499 999 999 999 999 999 999 999 920 780 187 218 739 2 × 2 = 0 + 0.999 999 999 999 999 999 999 999 841 560 374 437 478 4;
  • 46) 0.999 999 999 999 999 999 999 999 841 560 374 437 478 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 683 120 748 874 956 8;
  • 47) 0.999 999 999 999 999 999 999 999 683 120 748 874 956 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 366 241 497 749 913 6;
  • 48) 0.999 999 999 999 999 999 999 999 366 241 497 749 913 6 × 2 = 1 + 0.999 999 999 999 999 999 999 998 732 482 995 499 827 2;
  • 49) 0.999 999 999 999 999 999 999 998 732 482 995 499 827 2 × 2 = 1 + 0.999 999 999 999 999 999 999 997 464 965 990 999 654 4;
  • 50) 0.999 999 999 999 999 999 999 997 464 965 990 999 654 4 × 2 = 1 + 0.999 999 999 999 999 999 999 994 929 931 981 999 308 8;
  • 51) 0.999 999 999 999 999 999 999 994 929 931 981 999 308 8 × 2 = 1 + 0.999 999 999 999 999 999 999 989 859 863 963 998 617 6;
  • 52) 0.999 999 999 999 999 999 999 989 859 863 963 998 617 6 × 2 = 1 + 0.999 999 999 999 999 999 999 979 719 727 927 997 235 2;
  • 53) 0.999 999 999 999 999 999 999 979 719 727 927 997 235 2 × 2 = 1 + 0.999 999 999 999 999 999 999 959 439 455 855 994 470 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).


5. 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:


0.981 999 999 999 999 317 878 973 670 303 821 563 716 2(10) =

0.1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1(2)

6. Positive number before normalization:

19.981 999 999 999 999 317 878 973 670 303 821 563 716 2(10) =

1 0011.1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 4 positions to the left, so that only one non zero digit remains to the left of it:


19.981 999 999 999 999 317 878 973 670 303 821 563 716 2(10) =

1 0011.1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1(2) =

1 0011.1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1(2) × 20 =

1.0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1(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):
1.0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1111 1


9. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

4 + 2(11-1) - 1 =

(4 + 1 023)(10) =

1 027(10)


10. 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;
  • 1 027 ÷ 2 = 513 + 1;
  • 513 ÷ 2 = 256 + 1;
  • 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;

11. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1027(10) =

100 0000 0011(2)


12. 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. 0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111 1 1111 =

0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111


13. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
1 (a negative number)

Exponent (11 bits) =
100 0000 0011

Mantissa (52 bits) =
0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111


Decimal number -19.981 999 999 999 999 317 878 973 670 303 821 563 716 2 converted to 64 bit double precision IEEE 754 binary floating point representation:

1 - 100 0000 0011 - 0011 1111 1011 0110 0100 0101 1010 0001 1100 1010 1100 0000 0111

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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