0.000 000 000 000 000 000 017 618 285 302 889 448 47 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 000 017 618 285 302 889 448 47(10) to 32 bit single precision IEEE 754 binary floating point representation standard (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 000 000 000 017 618 285 302 889 448 47(10) to 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

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

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.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 000 017 618 285 302 889 448 47.

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) 0.000 000 000 000 000 000 017 618 285 302 889 448 47 × 2 = 0 + 0.000 000 000 000 000 000 035 236 570 605 778 896 94;
  • 2) 0.000 000 000 000 000 000 035 236 570 605 778 896 94 × 2 = 0 + 0.000 000 000 000 000 000 070 473 141 211 557 793 88;
  • 3) 0.000 000 000 000 000 000 070 473 141 211 557 793 88 × 2 = 0 + 0.000 000 000 000 000 000 140 946 282 423 115 587 76;
  • 4) 0.000 000 000 000 000 000 140 946 282 423 115 587 76 × 2 = 0 + 0.000 000 000 000 000 000 281 892 564 846 231 175 52;
  • 5) 0.000 000 000 000 000 000 281 892 564 846 231 175 52 × 2 = 0 + 0.000 000 000 000 000 000 563 785 129 692 462 351 04;
  • 6) 0.000 000 000 000 000 000 563 785 129 692 462 351 04 × 2 = 0 + 0.000 000 000 000 000 001 127 570 259 384 924 702 08;
  • 7) 0.000 000 000 000 000 001 127 570 259 384 924 702 08 × 2 = 0 + 0.000 000 000 000 000 002 255 140 518 769 849 404 16;
  • 8) 0.000 000 000 000 000 002 255 140 518 769 849 404 16 × 2 = 0 + 0.000 000 000 000 000 004 510 281 037 539 698 808 32;
  • 9) 0.000 000 000 000 000 004 510 281 037 539 698 808 32 × 2 = 0 + 0.000 000 000 000 000 009 020 562 075 079 397 616 64;
  • 10) 0.000 000 000 000 000 009 020 562 075 079 397 616 64 × 2 = 0 + 0.000 000 000 000 000 018 041 124 150 158 795 233 28;
  • 11) 0.000 000 000 000 000 018 041 124 150 158 795 233 28 × 2 = 0 + 0.000 000 000 000 000 036 082 248 300 317 590 466 56;
  • 12) 0.000 000 000 000 000 036 082 248 300 317 590 466 56 × 2 = 0 + 0.000 000 000 000 000 072 164 496 600 635 180 933 12;
  • 13) 0.000 000 000 000 000 072 164 496 600 635 180 933 12 × 2 = 0 + 0.000 000 000 000 000 144 328 993 201 270 361 866 24;
  • 14) 0.000 000 000 000 000 144 328 993 201 270 361 866 24 × 2 = 0 + 0.000 000 000 000 000 288 657 986 402 540 723 732 48;
  • 15) 0.000 000 000 000 000 288 657 986 402 540 723 732 48 × 2 = 0 + 0.000 000 000 000 000 577 315 972 805 081 447 464 96;
  • 16) 0.000 000 000 000 000 577 315 972 805 081 447 464 96 × 2 = 0 + 0.000 000 000 000 001 154 631 945 610 162 894 929 92;
  • 17) 0.000 000 000 000 001 154 631 945 610 162 894 929 92 × 2 = 0 + 0.000 000 000 000 002 309 263 891 220 325 789 859 84;
  • 18) 0.000 000 000 000 002 309 263 891 220 325 789 859 84 × 2 = 0 + 0.000 000 000 000 004 618 527 782 440 651 579 719 68;
  • 19) 0.000 000 000 000 004 618 527 782 440 651 579 719 68 × 2 = 0 + 0.000 000 000 000 009 237 055 564 881 303 159 439 36;
  • 20) 0.000 000 000 000 009 237 055 564 881 303 159 439 36 × 2 = 0 + 0.000 000 000 000 018 474 111 129 762 606 318 878 72;
  • 21) 0.000 000 000 000 018 474 111 129 762 606 318 878 72 × 2 = 0 + 0.000 000 000 000 036 948 222 259 525 212 637 757 44;
  • 22) 0.000 000 000 000 036 948 222 259 525 212 637 757 44 × 2 = 0 + 0.000 000 000 000 073 896 444 519 050 425 275 514 88;
  • 23) 0.000 000 000 000 073 896 444 519 050 425 275 514 88 × 2 = 0 + 0.000 000 000 000 147 792 889 038 100 850 551 029 76;
  • 24) 0.000 000 000 000 147 792 889 038 100 850 551 029 76 × 2 = 0 + 0.000 000 000 000 295 585 778 076 201 701 102 059 52;
  • 25) 0.000 000 000 000 295 585 778 076 201 701 102 059 52 × 2 = 0 + 0.000 000 000 000 591 171 556 152 403 402 204 119 04;
  • 26) 0.000 000 000 000 591 171 556 152 403 402 204 119 04 × 2 = 0 + 0.000 000 000 001 182 343 112 304 806 804 408 238 08;
  • 27) 0.000 000 000 001 182 343 112 304 806 804 408 238 08 × 2 = 0 + 0.000 000 000 002 364 686 224 609 613 608 816 476 16;
  • 28) 0.000 000 000 002 364 686 224 609 613 608 816 476 16 × 2 = 0 + 0.000 000 000 004 729 372 449 219 227 217 632 952 32;
  • 29) 0.000 000 000 004 729 372 449 219 227 217 632 952 32 × 2 = 0 + 0.000 000 000 009 458 744 898 438 454 435 265 904 64;
  • 30) 0.000 000 000 009 458 744 898 438 454 435 265 904 64 × 2 = 0 + 0.000 000 000 018 917 489 796 876 908 870 531 809 28;
  • 31) 0.000 000 000 018 917 489 796 876 908 870 531 809 28 × 2 = 0 + 0.000 000 000 037 834 979 593 753 817 741 063 618 56;
  • 32) 0.000 000 000 037 834 979 593 753 817 741 063 618 56 × 2 = 0 + 0.000 000 000 075 669 959 187 507 635 482 127 237 12;
  • 33) 0.000 000 000 075 669 959 187 507 635 482 127 237 12 × 2 = 0 + 0.000 000 000 151 339 918 375 015 270 964 254 474 24;
  • 34) 0.000 000 000 151 339 918 375 015 270 964 254 474 24 × 2 = 0 + 0.000 000 000 302 679 836 750 030 541 928 508 948 48;
  • 35) 0.000 000 000 302 679 836 750 030 541 928 508 948 48 × 2 = 0 + 0.000 000 000 605 359 673 500 061 083 857 017 896 96;
  • 36) 0.000 000 000 605 359 673 500 061 083 857 017 896 96 × 2 = 0 + 0.000 000 001 210 719 347 000 122 167 714 035 793 92;
  • 37) 0.000 000 001 210 719 347 000 122 167 714 035 793 92 × 2 = 0 + 0.000 000 002 421 438 694 000 244 335 428 071 587 84;
  • 38) 0.000 000 002 421 438 694 000 244 335 428 071 587 84 × 2 = 0 + 0.000 000 004 842 877 388 000 488 670 856 143 175 68;
  • 39) 0.000 000 004 842 877 388 000 488 670 856 143 175 68 × 2 = 0 + 0.000 000 009 685 754 776 000 977 341 712 286 351 36;
  • 40) 0.000 000 009 685 754 776 000 977 341 712 286 351 36 × 2 = 0 + 0.000 000 019 371 509 552 001 954 683 424 572 702 72;
  • 41) 0.000 000 019 371 509 552 001 954 683 424 572 702 72 × 2 = 0 + 0.000 000 038 743 019 104 003 909 366 849 145 405 44;
  • 42) 0.000 000 038 743 019 104 003 909 366 849 145 405 44 × 2 = 0 + 0.000 000 077 486 038 208 007 818 733 698 290 810 88;
  • 43) 0.000 000 077 486 038 208 007 818 733 698 290 810 88 × 2 = 0 + 0.000 000 154 972 076 416 015 637 467 396 581 621 76;
  • 44) 0.000 000 154 972 076 416 015 637 467 396 581 621 76 × 2 = 0 + 0.000 000 309 944 152 832 031 274 934 793 163 243 52;
  • 45) 0.000 000 309 944 152 832 031 274 934 793 163 243 52 × 2 = 0 + 0.000 000 619 888 305 664 062 549 869 586 326 487 04;
  • 46) 0.000 000 619 888 305 664 062 549 869 586 326 487 04 × 2 = 0 + 0.000 001 239 776 611 328 125 099 739 172 652 974 08;
  • 47) 0.000 001 239 776 611 328 125 099 739 172 652 974 08 × 2 = 0 + 0.000 002 479 553 222 656 250 199 478 345 305 948 16;
  • 48) 0.000 002 479 553 222 656 250 199 478 345 305 948 16 × 2 = 0 + 0.000 004 959 106 445 312 500 398 956 690 611 896 32;
  • 49) 0.000 004 959 106 445 312 500 398 956 690 611 896 32 × 2 = 0 + 0.000 009 918 212 890 625 000 797 913 381 223 792 64;
  • 50) 0.000 009 918 212 890 625 000 797 913 381 223 792 64 × 2 = 0 + 0.000 019 836 425 781 250 001 595 826 762 447 585 28;
  • 51) 0.000 019 836 425 781 250 001 595 826 762 447 585 28 × 2 = 0 + 0.000 039 672 851 562 500 003 191 653 524 895 170 56;
  • 52) 0.000 039 672 851 562 500 003 191 653 524 895 170 56 × 2 = 0 + 0.000 079 345 703 125 000 006 383 307 049 790 341 12;
  • 53) 0.000 079 345 703 125 000 006 383 307 049 790 341 12 × 2 = 0 + 0.000 158 691 406 250 000 012 766 614 099 580 682 24;
  • 54) 0.000 158 691 406 250 000 012 766 614 099 580 682 24 × 2 = 0 + 0.000 317 382 812 500 000 025 533 228 199 161 364 48;
  • 55) 0.000 317 382 812 500 000 025 533 228 199 161 364 48 × 2 = 0 + 0.000 634 765 625 000 000 051 066 456 398 322 728 96;
  • 56) 0.000 634 765 625 000 000 051 066 456 398 322 728 96 × 2 = 0 + 0.001 269 531 250 000 000 102 132 912 796 645 457 92;
  • 57) 0.001 269 531 250 000 000 102 132 912 796 645 457 92 × 2 = 0 + 0.002 539 062 500 000 000 204 265 825 593 290 915 84;
  • 58) 0.002 539 062 500 000 000 204 265 825 593 290 915 84 × 2 = 0 + 0.005 078 125 000 000 000 408 531 651 186 581 831 68;
  • 59) 0.005 078 125 000 000 000 408 531 651 186 581 831 68 × 2 = 0 + 0.010 156 250 000 000 000 817 063 302 373 163 663 36;
  • 60) 0.010 156 250 000 000 000 817 063 302 373 163 663 36 × 2 = 0 + 0.020 312 500 000 000 001 634 126 604 746 327 326 72;
  • 61) 0.020 312 500 000 000 001 634 126 604 746 327 326 72 × 2 = 0 + 0.040 625 000 000 000 003 268 253 209 492 654 653 44;
  • 62) 0.040 625 000 000 000 003 268 253 209 492 654 653 44 × 2 = 0 + 0.081 250 000 000 000 006 536 506 418 985 309 306 88;
  • 63) 0.081 250 000 000 000 006 536 506 418 985 309 306 88 × 2 = 0 + 0.162 500 000 000 000 013 073 012 837 970 618 613 76;
  • 64) 0.162 500 000 000 000 013 073 012 837 970 618 613 76 × 2 = 0 + 0.325 000 000 000 000 026 146 025 675 941 237 227 52;
  • 65) 0.325 000 000 000 000 026 146 025 675 941 237 227 52 × 2 = 0 + 0.650 000 000 000 000 052 292 051 351 882 474 455 04;
  • 66) 0.650 000 000 000 000 052 292 051 351 882 474 455 04 × 2 = 1 + 0.300 000 000 000 000 104 584 102 703 764 948 910 08;
  • 67) 0.300 000 000 000 000 104 584 102 703 764 948 910 08 × 2 = 0 + 0.600 000 000 000 000 209 168 205 407 529 897 820 16;
  • 68) 0.600 000 000 000 000 209 168 205 407 529 897 820 16 × 2 = 1 + 0.200 000 000 000 000 418 336 410 815 059 795 640 32;
  • 69) 0.200 000 000 000 000 418 336 410 815 059 795 640 32 × 2 = 0 + 0.400 000 000 000 000 836 672 821 630 119 591 280 64;
  • 70) 0.400 000 000 000 000 836 672 821 630 119 591 280 64 × 2 = 0 + 0.800 000 000 000 001 673 345 643 260 239 182 561 28;
  • 71) 0.800 000 000 000 001 673 345 643 260 239 182 561 28 × 2 = 1 + 0.600 000 000 000 003 346 691 286 520 478 365 122 56;
  • 72) 0.600 000 000 000 003 346 691 286 520 478 365 122 56 × 2 = 1 + 0.200 000 000 000 006 693 382 573 040 956 730 245 12;
  • 73) 0.200 000 000 000 006 693 382 573 040 956 730 245 12 × 2 = 0 + 0.400 000 000 000 013 386 765 146 081 913 460 490 24;
  • 74) 0.400 000 000 000 013 386 765 146 081 913 460 490 24 × 2 = 0 + 0.800 000 000 000 026 773 530 292 163 826 920 980 48;
  • 75) 0.800 000 000 000 026 773 530 292 163 826 920 980 48 × 2 = 1 + 0.600 000 000 000 053 547 060 584 327 653 841 960 96;
  • 76) 0.600 000 000 000 053 547 060 584 327 653 841 960 96 × 2 = 1 + 0.200 000 000 000 107 094 121 168 655 307 683 921 92;
  • 77) 0.200 000 000 000 107 094 121 168 655 307 683 921 92 × 2 = 0 + 0.400 000 000 000 214 188 242 337 310 615 367 843 84;
  • 78) 0.400 000 000 000 214 188 242 337 310 615 367 843 84 × 2 = 0 + 0.800 000 000 000 428 376 484 674 621 230 735 687 68;
  • 79) 0.800 000 000 000 428 376 484 674 621 230 735 687 68 × 2 = 1 + 0.600 000 000 000 856 752 969 349 242 461 471 375 36;
  • 80) 0.600 000 000 000 856 752 969 349 242 461 471 375 36 × 2 = 1 + 0.200 000 000 001 713 505 938 698 484 922 942 750 72;
  • 81) 0.200 000 000 001 713 505 938 698 484 922 942 750 72 × 2 = 0 + 0.400 000 000 003 427 011 877 396 969 845 885 501 44;
  • 82) 0.400 000 000 003 427 011 877 396 969 845 885 501 44 × 2 = 0 + 0.800 000 000 006 854 023 754 793 939 691 771 002 88;
  • 83) 0.800 000 000 006 854 023 754 793 939 691 771 002 88 × 2 = 1 + 0.600 000 000 013 708 047 509 587 879 383 542 005 76;
  • 84) 0.600 000 000 013 708 047 509 587 879 383 542 005 76 × 2 = 1 + 0.200 000 000 027 416 095 019 175 758 767 084 011 52;
  • 85) 0.200 000 000 027 416 095 019 175 758 767 084 011 52 × 2 = 0 + 0.400 000 000 054 832 190 038 351 517 534 168 023 04;
  • 86) 0.400 000 000 054 832 190 038 351 517 534 168 023 04 × 2 = 0 + 0.800 000 000 109 664 380 076 703 035 068 336 046 08;
  • 87) 0.800 000 000 109 664 380 076 703 035 068 336 046 08 × 2 = 1 + 0.600 000 000 219 328 760 153 406 070 136 672 092 16;
  • 88) 0.600 000 000 219 328 760 153 406 070 136 672 092 16 × 2 = 1 + 0.200 000 000 438 657 520 306 812 140 273 344 184 32;
  • 89) 0.200 000 000 438 657 520 306 812 140 273 344 184 32 × 2 = 0 + 0.400 000 000 877 315 040 613 624 280 546 688 368 64;

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:


0.000 000 000 000 000 000 017 618 285 302 889 448 47(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0011 0011 0011 0011 0011 0(2)

5. Positive number before normalization:

0.000 000 000 000 000 000 017 618 285 302 889 448 47(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0011 0011 0011 0011 0011 0(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 66 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 017 618 285 302 889 448 47(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0011 0011 0011 0011 0011 0(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0011 0011 0011 0011 0011 0(2) × 20 =

1.0100 1100 1100 1100 1100 110(2) × 2-66


7. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -66

Mantissa (not normalized):
1.0100 1100 1100 1100 1100 110


8. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(8-1) - 1 =

-66 + 2(8-1) - 1 =

(-66 + 127)(10) =

61(10)


9. Convert the adjusted exponent from the decimal (base 10) to 8 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

61(10) =

0011 1101(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 23 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 010 0110 0110 0110 0110 0110 =

010 0110 0110 0110 0110 0110


12. The three elements that make up the number's 32 bit single precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (8 bits) =
0011 1101

Mantissa (23 bits) =
010 0110 0110 0110 0110 0110


Decimal number 0.000 000 000 000 000 000 017 618 285 302 889 448 47 converted to 32 bit single precision IEEE 754 binary floating point representation:

0 - 0011 1101 - 010 0110 0110 0110 0110 0110

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How to convert decimal numbers from base ten to 32 bit single precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 32 bit single 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 base ten 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 of the previous dividing 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 previous multiplying operations, starting from the top of the constructed list 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, by shifting the decimal point (or if you prefer, the decimal mark) "n" positions either to the left or to the right, so that only one non zero digit remains to the left of the decimal point.
  • 7. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign if the case) and adjust its length to 23 bits, either by removing the excess bits from the right (losing precision...) or by adding extra '0' bits 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 -25.347 from decimal system (base ten) to 32 bit single precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-25.347| = 25.347

  • 2. First convert the integer part, 25. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 25 ÷ 2 = 12 + 1;
    • 12 ÷ 2 = 6 + 0;
    • 6 ÷ 2 = 3 + 0;
    • 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:

    25(10) = 1 1001(2)

  • 4. Then convert the fractional part, 0.347. 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) 0.347 × 2 = 0 + 0.694;
    • 2) 0.694 × 2 = 1 + 0.388;
    • 3) 0.388 × 2 = 0 + 0.776;
    • 4) 0.776 × 2 = 1 + 0.552;
    • 5) 0.552 × 2 = 1 + 0.104;
    • 6) 0.104 × 2 = 0 + 0.208;
    • 7) 0.208 × 2 = 0 + 0.416;
    • 8) 0.416 × 2 = 0 + 0.832;
    • 9) 0.832 × 2 = 1 + 0.664;
    • 10) 0.664 × 2 = 1 + 0.328;
    • 11) 0.328 × 2 = 0 + 0.656;
    • 12) 0.656 × 2 = 1 + 0.312;
    • 13) 0.312 × 2 = 0 + 0.624;
    • 14) 0.624 × 2 = 1 + 0.248;
    • 15) 0.248 × 2 = 0 + 0.496;
    • 16) 0.496 × 2 = 0 + 0.992;
    • 17) 0.992 × 2 = 1 + 0.984;
    • 18) 0.984 × 2 = 1 + 0.968;
    • 19) 0.968 × 2 = 1 + 0.936;
    • 20) 0.936 × 2 = 1 + 0.872;
    • 21) 0.872 × 2 = 1 + 0.744;
    • 22) 0.744 × 2 = 1 + 0.488;
    • 23) 0.488 × 2 = 0 + 0.976;
    • 24) 0.976 × 2 = 1 + 0.952;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 23) 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:

    0.347(10) = 0.0101 1000 1101 0100 1111 1101(2)

  • 6. Summarizing - the positive number before normalization:

    25.347(10) = 1 1001.0101 1000 1101 0100 1111 1101(2)

  • 7. Normalize the binary representation of the number, shifting the decimal point 4 positions to the left so that only one non-zero digit stays to the left of the decimal point:

    25.347(10) =
    1 1001.0101 1000 1101 0100 1111 1101(2) =
    1 1001.0101 1000 1101 0100 1111 1101(2) × 20 =
    1.1001 0101 1000 1101 0100 1111 1101(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 32 bit single precision IEEE 754 binary floating point:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

  • 9. Adjust the exponent in 8 bit excess/bias notation and then convert it from decimal (base 10) to 8 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as already demonstrated above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(8-1) - 1 = (4 + 127)(10) = 131(10) =
    1000 0011(2)

  • 10. Normalize the mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal point) and adjust its length to 23 bits, by removing the excess bits from the right (losing precision...):

    Mantissa (not-normalized): 1.1001 0101 1000 1101 0100 1111 1101

    Mantissa (normalized): 100 1010 1100 0110 1010 0111

  • Conclusion:

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

    Exponent (8 bits) = 1000 0011

    Mantissa (23 bits) = 100 1010 1100 0110 1010 0111

  • Number -25.347, converted from the decimal system (base 10) to 32 bit single precision IEEE 754 binary floating point =
    1 - 1000 0011 - 100 1010 1100 0110 1010 0111