-0.000 000 000 000 000 000 000 000 085 864 27 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 000 000 000 085 864 27(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
-0.000 000 000 000 000 000 000 000 085 864 27(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. Start with the positive version of the number:

|-0.000 000 000 000 000 000 000 000 085 864 27| = 0.000 000 000 000 000 000 000 000 085 864 27


2. 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;

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.

0(10) =

0(2)


4. Convert to binary (base 2) the fractional part: 0.000 000 000 000 000 000 000 000 085 864 27.

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) 0.000 000 000 000 000 000 000 000 085 864 27 × 2 = 0 + 0.000 000 000 000 000 000 000 000 171 728 54;
  • 2) 0.000 000 000 000 000 000 000 000 171 728 54 × 2 = 0 + 0.000 000 000 000 000 000 000 000 343 457 08;
  • 3) 0.000 000 000 000 000 000 000 000 343 457 08 × 2 = 0 + 0.000 000 000 000 000 000 000 000 686 914 16;
  • 4) 0.000 000 000 000 000 000 000 000 686 914 16 × 2 = 0 + 0.000 000 000 000 000 000 000 001 373 828 32;
  • 5) 0.000 000 000 000 000 000 000 001 373 828 32 × 2 = 0 + 0.000 000 000 000 000 000 000 002 747 656 64;
  • 6) 0.000 000 000 000 000 000 000 002 747 656 64 × 2 = 0 + 0.000 000 000 000 000 000 000 005 495 313 28;
  • 7) 0.000 000 000 000 000 000 000 005 495 313 28 × 2 = 0 + 0.000 000 000 000 000 000 000 010 990 626 56;
  • 8) 0.000 000 000 000 000 000 000 010 990 626 56 × 2 = 0 + 0.000 000 000 000 000 000 000 021 981 253 12;
  • 9) 0.000 000 000 000 000 000 000 021 981 253 12 × 2 = 0 + 0.000 000 000 000 000 000 000 043 962 506 24;
  • 10) 0.000 000 000 000 000 000 000 043 962 506 24 × 2 = 0 + 0.000 000 000 000 000 000 000 087 925 012 48;
  • 11) 0.000 000 000 000 000 000 000 087 925 012 48 × 2 = 0 + 0.000 000 000 000 000 000 000 175 850 024 96;
  • 12) 0.000 000 000 000 000 000 000 175 850 024 96 × 2 = 0 + 0.000 000 000 000 000 000 000 351 700 049 92;
  • 13) 0.000 000 000 000 000 000 000 351 700 049 92 × 2 = 0 + 0.000 000 000 000 000 000 000 703 400 099 84;
  • 14) 0.000 000 000 000 000 000 000 703 400 099 84 × 2 = 0 + 0.000 000 000 000 000 000 001 406 800 199 68;
  • 15) 0.000 000 000 000 000 000 001 406 800 199 68 × 2 = 0 + 0.000 000 000 000 000 000 002 813 600 399 36;
  • 16) 0.000 000 000 000 000 000 002 813 600 399 36 × 2 = 0 + 0.000 000 000 000 000 000 005 627 200 798 72;
  • 17) 0.000 000 000 000 000 000 005 627 200 798 72 × 2 = 0 + 0.000 000 000 000 000 000 011 254 401 597 44;
  • 18) 0.000 000 000 000 000 000 011 254 401 597 44 × 2 = 0 + 0.000 000 000 000 000 000 022 508 803 194 88;
  • 19) 0.000 000 000 000 000 000 022 508 803 194 88 × 2 = 0 + 0.000 000 000 000 000 000 045 017 606 389 76;
  • 20) 0.000 000 000 000 000 000 045 017 606 389 76 × 2 = 0 + 0.000 000 000 000 000 000 090 035 212 779 52;
  • 21) 0.000 000 000 000 000 000 090 035 212 779 52 × 2 = 0 + 0.000 000 000 000 000 000 180 070 425 559 04;
  • 22) 0.000 000 000 000 000 000 180 070 425 559 04 × 2 = 0 + 0.000 000 000 000 000 000 360 140 851 118 08;
  • 23) 0.000 000 000 000 000 000 360 140 851 118 08 × 2 = 0 + 0.000 000 000 000 000 000 720 281 702 236 16;
  • 24) 0.000 000 000 000 000 000 720 281 702 236 16 × 2 = 0 + 0.000 000 000 000 000 001 440 563 404 472 32;
  • 25) 0.000 000 000 000 000 001 440 563 404 472 32 × 2 = 0 + 0.000 000 000 000 000 002 881 126 808 944 64;
  • 26) 0.000 000 000 000 000 002 881 126 808 944 64 × 2 = 0 + 0.000 000 000 000 000 005 762 253 617 889 28;
  • 27) 0.000 000 000 000 000 005 762 253 617 889 28 × 2 = 0 + 0.000 000 000 000 000 011 524 507 235 778 56;
  • 28) 0.000 000 000 000 000 011 524 507 235 778 56 × 2 = 0 + 0.000 000 000 000 000 023 049 014 471 557 12;
  • 29) 0.000 000 000 000 000 023 049 014 471 557 12 × 2 = 0 + 0.000 000 000 000 000 046 098 028 943 114 24;
  • 30) 0.000 000 000 000 000 046 098 028 943 114 24 × 2 = 0 + 0.000 000 000 000 000 092 196 057 886 228 48;
  • 31) 0.000 000 000 000 000 092 196 057 886 228 48 × 2 = 0 + 0.000 000 000 000 000 184 392 115 772 456 96;
  • 32) 0.000 000 000 000 000 184 392 115 772 456 96 × 2 = 0 + 0.000 000 000 000 000 368 784 231 544 913 92;
  • 33) 0.000 000 000 000 000 368 784 231 544 913 92 × 2 = 0 + 0.000 000 000 000 000 737 568 463 089 827 84;
  • 34) 0.000 000 000 000 000 737 568 463 089 827 84 × 2 = 0 + 0.000 000 000 000 001 475 136 926 179 655 68;
  • 35) 0.000 000 000 000 001 475 136 926 179 655 68 × 2 = 0 + 0.000 000 000 000 002 950 273 852 359 311 36;
  • 36) 0.000 000 000 000 002 950 273 852 359 311 36 × 2 = 0 + 0.000 000 000 000 005 900 547 704 718 622 72;
  • 37) 0.000 000 000 000 005 900 547 704 718 622 72 × 2 = 0 + 0.000 000 000 000 011 801 095 409 437 245 44;
  • 38) 0.000 000 000 000 011 801 095 409 437 245 44 × 2 = 0 + 0.000 000 000 000 023 602 190 818 874 490 88;
  • 39) 0.000 000 000 000 023 602 190 818 874 490 88 × 2 = 0 + 0.000 000 000 000 047 204 381 637 748 981 76;
  • 40) 0.000 000 000 000 047 204 381 637 748 981 76 × 2 = 0 + 0.000 000 000 000 094 408 763 275 497 963 52;
  • 41) 0.000 000 000 000 094 408 763 275 497 963 52 × 2 = 0 + 0.000 000 000 000 188 817 526 550 995 927 04;
  • 42) 0.000 000 000 000 188 817 526 550 995 927 04 × 2 = 0 + 0.000 000 000 000 377 635 053 101 991 854 08;
  • 43) 0.000 000 000 000 377 635 053 101 991 854 08 × 2 = 0 + 0.000 000 000 000 755 270 106 203 983 708 16;
  • 44) 0.000 000 000 000 755 270 106 203 983 708 16 × 2 = 0 + 0.000 000 000 001 510 540 212 407 967 416 32;
  • 45) 0.000 000 000 001 510 540 212 407 967 416 32 × 2 = 0 + 0.000 000 000 003 021 080 424 815 934 832 64;
  • 46) 0.000 000 000 003 021 080 424 815 934 832 64 × 2 = 0 + 0.000 000 000 006 042 160 849 631 869 665 28;
  • 47) 0.000 000 000 006 042 160 849 631 869 665 28 × 2 = 0 + 0.000 000 000 012 084 321 699 263 739 330 56;
  • 48) 0.000 000 000 012 084 321 699 263 739 330 56 × 2 = 0 + 0.000 000 000 024 168 643 398 527 478 661 12;
  • 49) 0.000 000 000 024 168 643 398 527 478 661 12 × 2 = 0 + 0.000 000 000 048 337 286 797 054 957 322 24;
  • 50) 0.000 000 000 048 337 286 797 054 957 322 24 × 2 = 0 + 0.000 000 000 096 674 573 594 109 914 644 48;
  • 51) 0.000 000 000 096 674 573 594 109 914 644 48 × 2 = 0 + 0.000 000 000 193 349 147 188 219 829 288 96;
  • 52) 0.000 000 000 193 349 147 188 219 829 288 96 × 2 = 0 + 0.000 000 000 386 698 294 376 439 658 577 92;
  • 53) 0.000 000 000 386 698 294 376 439 658 577 92 × 2 = 0 + 0.000 000 000 773 396 588 752 879 317 155 84;
  • 54) 0.000 000 000 773 396 588 752 879 317 155 84 × 2 = 0 + 0.000 000 001 546 793 177 505 758 634 311 68;
  • 55) 0.000 000 001 546 793 177 505 758 634 311 68 × 2 = 0 + 0.000 000 003 093 586 355 011 517 268 623 36;
  • 56) 0.000 000 003 093 586 355 011 517 268 623 36 × 2 = 0 + 0.000 000 006 187 172 710 023 034 537 246 72;
  • 57) 0.000 000 006 187 172 710 023 034 537 246 72 × 2 = 0 + 0.000 000 012 374 345 420 046 069 074 493 44;
  • 58) 0.000 000 012 374 345 420 046 069 074 493 44 × 2 = 0 + 0.000 000 024 748 690 840 092 138 148 986 88;
  • 59) 0.000 000 024 748 690 840 092 138 148 986 88 × 2 = 0 + 0.000 000 049 497 381 680 184 276 297 973 76;
  • 60) 0.000 000 049 497 381 680 184 276 297 973 76 × 2 = 0 + 0.000 000 098 994 763 360 368 552 595 947 52;
  • 61) 0.000 000 098 994 763 360 368 552 595 947 52 × 2 = 0 + 0.000 000 197 989 526 720 737 105 191 895 04;
  • 62) 0.000 000 197 989 526 720 737 105 191 895 04 × 2 = 0 + 0.000 000 395 979 053 441 474 210 383 790 08;
  • 63) 0.000 000 395 979 053 441 474 210 383 790 08 × 2 = 0 + 0.000 000 791 958 106 882 948 420 767 580 16;
  • 64) 0.000 000 791 958 106 882 948 420 767 580 16 × 2 = 0 + 0.000 001 583 916 213 765 896 841 535 160 32;
  • 65) 0.000 001 583 916 213 765 896 841 535 160 32 × 2 = 0 + 0.000 003 167 832 427 531 793 683 070 320 64;
  • 66) 0.000 003 167 832 427 531 793 683 070 320 64 × 2 = 0 + 0.000 006 335 664 855 063 587 366 140 641 28;
  • 67) 0.000 006 335 664 855 063 587 366 140 641 28 × 2 = 0 + 0.000 012 671 329 710 127 174 732 281 282 56;
  • 68) 0.000 012 671 329 710 127 174 732 281 282 56 × 2 = 0 + 0.000 025 342 659 420 254 349 464 562 565 12;
  • 69) 0.000 025 342 659 420 254 349 464 562 565 12 × 2 = 0 + 0.000 050 685 318 840 508 698 929 125 130 24;
  • 70) 0.000 050 685 318 840 508 698 929 125 130 24 × 2 = 0 + 0.000 101 370 637 681 017 397 858 250 260 48;
  • 71) 0.000 101 370 637 681 017 397 858 250 260 48 × 2 = 0 + 0.000 202 741 275 362 034 795 716 500 520 96;
  • 72) 0.000 202 741 275 362 034 795 716 500 520 96 × 2 = 0 + 0.000 405 482 550 724 069 591 433 001 041 92;
  • 73) 0.000 405 482 550 724 069 591 433 001 041 92 × 2 = 0 + 0.000 810 965 101 448 139 182 866 002 083 84;
  • 74) 0.000 810 965 101 448 139 182 866 002 083 84 × 2 = 0 + 0.001 621 930 202 896 278 365 732 004 167 68;
  • 75) 0.001 621 930 202 896 278 365 732 004 167 68 × 2 = 0 + 0.003 243 860 405 792 556 731 464 008 335 36;
  • 76) 0.003 243 860 405 792 556 731 464 008 335 36 × 2 = 0 + 0.006 487 720 811 585 113 462 928 016 670 72;
  • 77) 0.006 487 720 811 585 113 462 928 016 670 72 × 2 = 0 + 0.012 975 441 623 170 226 925 856 033 341 44;
  • 78) 0.012 975 441 623 170 226 925 856 033 341 44 × 2 = 0 + 0.025 950 883 246 340 453 851 712 066 682 88;
  • 79) 0.025 950 883 246 340 453 851 712 066 682 88 × 2 = 0 + 0.051 901 766 492 680 907 703 424 133 365 76;
  • 80) 0.051 901 766 492 680 907 703 424 133 365 76 × 2 = 0 + 0.103 803 532 985 361 815 406 848 266 731 52;
  • 81) 0.103 803 532 985 361 815 406 848 266 731 52 × 2 = 0 + 0.207 607 065 970 723 630 813 696 533 463 04;
  • 82) 0.207 607 065 970 723 630 813 696 533 463 04 × 2 = 0 + 0.415 214 131 941 447 261 627 393 066 926 08;
  • 83) 0.415 214 131 941 447 261 627 393 066 926 08 × 2 = 0 + 0.830 428 263 882 894 523 254 786 133 852 16;
  • 84) 0.830 428 263 882 894 523 254 786 133 852 16 × 2 = 1 + 0.660 856 527 765 789 046 509 572 267 704 32;
  • 85) 0.660 856 527 765 789 046 509 572 267 704 32 × 2 = 1 + 0.321 713 055 531 578 093 019 144 535 408 64;
  • 86) 0.321 713 055 531 578 093 019 144 535 408 64 × 2 = 0 + 0.643 426 111 063 156 186 038 289 070 817 28;
  • 87) 0.643 426 111 063 156 186 038 289 070 817 28 × 2 = 1 + 0.286 852 222 126 312 372 076 578 141 634 56;
  • 88) 0.286 852 222 126 312 372 076 578 141 634 56 × 2 = 0 + 0.573 704 444 252 624 744 153 156 283 269 12;
  • 89) 0.573 704 444 252 624 744 153 156 283 269 12 × 2 = 1 + 0.147 408 888 505 249 488 306 312 566 538 24;
  • 90) 0.147 408 888 505 249 488 306 312 566 538 24 × 2 = 0 + 0.294 817 777 010 498 976 612 625 133 076 48;
  • 91) 0.294 817 777 010 498 976 612 625 133 076 48 × 2 = 0 + 0.589 635 554 020 997 953 225 250 266 152 96;
  • 92) 0.589 635 554 020 997 953 225 250 266 152 96 × 2 = 1 + 0.179 271 108 041 995 906 450 500 532 305 92;
  • 93) 0.179 271 108 041 995 906 450 500 532 305 92 × 2 = 0 + 0.358 542 216 083 991 812 901 001 064 611 84;
  • 94) 0.358 542 216 083 991 812 901 001 064 611 84 × 2 = 0 + 0.717 084 432 167 983 625 802 002 129 223 68;
  • 95) 0.717 084 432 167 983 625 802 002 129 223 68 × 2 = 1 + 0.434 168 864 335 967 251 604 004 258 447 36;
  • 96) 0.434 168 864 335 967 251 604 004 258 447 36 × 2 = 0 + 0.868 337 728 671 934 503 208 008 516 894 72;
  • 97) 0.868 337 728 671 934 503 208 008 516 894 72 × 2 = 1 + 0.736 675 457 343 869 006 416 017 033 789 44;
  • 98) 0.736 675 457 343 869 006 416 017 033 789 44 × 2 = 1 + 0.473 350 914 687 738 012 832 034 067 578 88;
  • 99) 0.473 350 914 687 738 012 832 034 067 578 88 × 2 = 0 + 0.946 701 829 375 476 025 664 068 135 157 76;
  • 100) 0.946 701 829 375 476 025 664 068 135 157 76 × 2 = 1 + 0.893 403 658 750 952 051 328 136 270 315 52;
  • 101) 0.893 403 658 750 952 051 328 136 270 315 52 × 2 = 1 + 0.786 807 317 501 904 102 656 272 540 631 04;
  • 102) 0.786 807 317 501 904 102 656 272 540 631 04 × 2 = 1 + 0.573 614 635 003 808 205 312 545 081 262 08;
  • 103) 0.573 614 635 003 808 205 312 545 081 262 08 × 2 = 1 + 0.147 229 270 007 616 410 625 090 162 524 16;
  • 104) 0.147 229 270 007 616 410 625 090 162 524 16 × 2 = 0 + 0.294 458 540 015 232 821 250 180 325 048 32;
  • 105) 0.294 458 540 015 232 821 250 180 325 048 32 × 2 = 0 + 0.588 917 080 030 465 642 500 360 650 096 64;
  • 106) 0.588 917 080 030 465 642 500 360 650 096 64 × 2 = 1 + 0.177 834 160 060 931 285 000 721 300 193 28;
  • 107) 0.177 834 160 060 931 285 000 721 300 193 28 × 2 = 0 + 0.355 668 320 121 862 570 001 442 600 386 56;

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:


0.000 000 000 000 000 000 000 000 085 864 27(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 1001 0010 1101 1110 010(2)

6. Positive number before normalization:

0.000 000 000 000 000 000 000 000 085 864 27(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 1001 0010 1101 1110 010(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 84 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 000 000 085 864 27(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 1001 0010 1101 1110 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1010 1001 0010 1101 1110 010(2) × 20 =

1.1010 1001 0010 1101 1110 010(2) × 2-84


8. 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 1 (a negative number)

Exponent (unadjusted): -84

Mantissa (not normalized):
1.1010 1001 0010 1101 1110 010


9. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-84 + 2(8-1) - 1 =

(-84 + 127)(10) =

43(10)


10. 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;
  • 43 ÷ 2 = 21 + 1;
  • 21 ÷ 2 = 10 + 1;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 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) =

43(10) =

0010 1011(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 23 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 101 0100 1001 0110 1111 0010 =

101 0100 1001 0110 1111 0010


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

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

Exponent (8 bits) =
0010 1011

Mantissa (23 bits) =
101 0100 1001 0110 1111 0010


Decimal number -0.000 000 000 000 000 000 000 000 085 864 27 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0010 1011 - 101 0100 1001 0110 1111 0010

Share

» Convert decimal -0.000 000 000 000 000 000 000 000 085 864 3 to 32 bit single precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 08, 2026 [August]: Decimal Numbers Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: August - by our visitors

» Improve your social skills: Reputation: Bridge Between Company's Actions & Market Value. Gemini Notebook YouTube Video of 6.55 minutes

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