-0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6(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
-0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6(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:

|-0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6| = 0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6


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: 0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6.

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) 0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6 × 2 = 0 + 0.000 000 000 000 000 000 054 365 636 560 000 002 801 277 499 2;
  • 2) 0.000 000 000 000 000 000 054 365 636 560 000 002 801 277 499 2 × 2 = 0 + 0.000 000 000 000 000 000 108 731 273 120 000 005 602 554 998 4;
  • 3) 0.000 000 000 000 000 000 108 731 273 120 000 005 602 554 998 4 × 2 = 0 + 0.000 000 000 000 000 000 217 462 546 240 000 011 205 109 996 8;
  • 4) 0.000 000 000 000 000 000 217 462 546 240 000 011 205 109 996 8 × 2 = 0 + 0.000 000 000 000 000 000 434 925 092 480 000 022 410 219 993 6;
  • 5) 0.000 000 000 000 000 000 434 925 092 480 000 022 410 219 993 6 × 2 = 0 + 0.000 000 000 000 000 000 869 850 184 960 000 044 820 439 987 2;
  • 6) 0.000 000 000 000 000 000 869 850 184 960 000 044 820 439 987 2 × 2 = 0 + 0.000 000 000 000 000 001 739 700 369 920 000 089 640 879 974 4;
  • 7) 0.000 000 000 000 000 001 739 700 369 920 000 089 640 879 974 4 × 2 = 0 + 0.000 000 000 000 000 003 479 400 739 840 000 179 281 759 948 8;
  • 8) 0.000 000 000 000 000 003 479 400 739 840 000 179 281 759 948 8 × 2 = 0 + 0.000 000 000 000 000 006 958 801 479 680 000 358 563 519 897 6;
  • 9) 0.000 000 000 000 000 006 958 801 479 680 000 358 563 519 897 6 × 2 = 0 + 0.000 000 000 000 000 013 917 602 959 360 000 717 127 039 795 2;
  • 10) 0.000 000 000 000 000 013 917 602 959 360 000 717 127 039 795 2 × 2 = 0 + 0.000 000 000 000 000 027 835 205 918 720 001 434 254 079 590 4;
  • 11) 0.000 000 000 000 000 027 835 205 918 720 001 434 254 079 590 4 × 2 = 0 + 0.000 000 000 000 000 055 670 411 837 440 002 868 508 159 180 8;
  • 12) 0.000 000 000 000 000 055 670 411 837 440 002 868 508 159 180 8 × 2 = 0 + 0.000 000 000 000 000 111 340 823 674 880 005 737 016 318 361 6;
  • 13) 0.000 000 000 000 000 111 340 823 674 880 005 737 016 318 361 6 × 2 = 0 + 0.000 000 000 000 000 222 681 647 349 760 011 474 032 636 723 2;
  • 14) 0.000 000 000 000 000 222 681 647 349 760 011 474 032 636 723 2 × 2 = 0 + 0.000 000 000 000 000 445 363 294 699 520 022 948 065 273 446 4;
  • 15) 0.000 000 000 000 000 445 363 294 699 520 022 948 065 273 446 4 × 2 = 0 + 0.000 000 000 000 000 890 726 589 399 040 045 896 130 546 892 8;
  • 16) 0.000 000 000 000 000 890 726 589 399 040 045 896 130 546 892 8 × 2 = 0 + 0.000 000 000 000 001 781 453 178 798 080 091 792 261 093 785 6;
  • 17) 0.000 000 000 000 001 781 453 178 798 080 091 792 261 093 785 6 × 2 = 0 + 0.000 000 000 000 003 562 906 357 596 160 183 584 522 187 571 2;
  • 18) 0.000 000 000 000 003 562 906 357 596 160 183 584 522 187 571 2 × 2 = 0 + 0.000 000 000 000 007 125 812 715 192 320 367 169 044 375 142 4;
  • 19) 0.000 000 000 000 007 125 812 715 192 320 367 169 044 375 142 4 × 2 = 0 + 0.000 000 000 000 014 251 625 430 384 640 734 338 088 750 284 8;
  • 20) 0.000 000 000 000 014 251 625 430 384 640 734 338 088 750 284 8 × 2 = 0 + 0.000 000 000 000 028 503 250 860 769 281 468 676 177 500 569 6;
  • 21) 0.000 000 000 000 028 503 250 860 769 281 468 676 177 500 569 6 × 2 = 0 + 0.000 000 000 000 057 006 501 721 538 562 937 352 355 001 139 2;
  • 22) 0.000 000 000 000 057 006 501 721 538 562 937 352 355 001 139 2 × 2 = 0 + 0.000 000 000 000 114 013 003 443 077 125 874 704 710 002 278 4;
  • 23) 0.000 000 000 000 114 013 003 443 077 125 874 704 710 002 278 4 × 2 = 0 + 0.000 000 000 000 228 026 006 886 154 251 749 409 420 004 556 8;
  • 24) 0.000 000 000 000 228 026 006 886 154 251 749 409 420 004 556 8 × 2 = 0 + 0.000 000 000 000 456 052 013 772 308 503 498 818 840 009 113 6;
  • 25) 0.000 000 000 000 456 052 013 772 308 503 498 818 840 009 113 6 × 2 = 0 + 0.000 000 000 000 912 104 027 544 617 006 997 637 680 018 227 2;
  • 26) 0.000 000 000 000 912 104 027 544 617 006 997 637 680 018 227 2 × 2 = 0 + 0.000 000 000 001 824 208 055 089 234 013 995 275 360 036 454 4;
  • 27) 0.000 000 000 001 824 208 055 089 234 013 995 275 360 036 454 4 × 2 = 0 + 0.000 000 000 003 648 416 110 178 468 027 990 550 720 072 908 8;
  • 28) 0.000 000 000 003 648 416 110 178 468 027 990 550 720 072 908 8 × 2 = 0 + 0.000 000 000 007 296 832 220 356 936 055 981 101 440 145 817 6;
  • 29) 0.000 000 000 007 296 832 220 356 936 055 981 101 440 145 817 6 × 2 = 0 + 0.000 000 000 014 593 664 440 713 872 111 962 202 880 291 635 2;
  • 30) 0.000 000 000 014 593 664 440 713 872 111 962 202 880 291 635 2 × 2 = 0 + 0.000 000 000 029 187 328 881 427 744 223 924 405 760 583 270 4;
  • 31) 0.000 000 000 029 187 328 881 427 744 223 924 405 760 583 270 4 × 2 = 0 + 0.000 000 000 058 374 657 762 855 488 447 848 811 521 166 540 8;
  • 32) 0.000 000 000 058 374 657 762 855 488 447 848 811 521 166 540 8 × 2 = 0 + 0.000 000 000 116 749 315 525 710 976 895 697 623 042 333 081 6;
  • 33) 0.000 000 000 116 749 315 525 710 976 895 697 623 042 333 081 6 × 2 = 0 + 0.000 000 000 233 498 631 051 421 953 791 395 246 084 666 163 2;
  • 34) 0.000 000 000 233 498 631 051 421 953 791 395 246 084 666 163 2 × 2 = 0 + 0.000 000 000 466 997 262 102 843 907 582 790 492 169 332 326 4;
  • 35) 0.000 000 000 466 997 262 102 843 907 582 790 492 169 332 326 4 × 2 = 0 + 0.000 000 000 933 994 524 205 687 815 165 580 984 338 664 652 8;
  • 36) 0.000 000 000 933 994 524 205 687 815 165 580 984 338 664 652 8 × 2 = 0 + 0.000 000 001 867 989 048 411 375 630 331 161 968 677 329 305 6;
  • 37) 0.000 000 001 867 989 048 411 375 630 331 161 968 677 329 305 6 × 2 = 0 + 0.000 000 003 735 978 096 822 751 260 662 323 937 354 658 611 2;
  • 38) 0.000 000 003 735 978 096 822 751 260 662 323 937 354 658 611 2 × 2 = 0 + 0.000 000 007 471 956 193 645 502 521 324 647 874 709 317 222 4;
  • 39) 0.000 000 007 471 956 193 645 502 521 324 647 874 709 317 222 4 × 2 = 0 + 0.000 000 014 943 912 387 291 005 042 649 295 749 418 634 444 8;
  • 40) 0.000 000 014 943 912 387 291 005 042 649 295 749 418 634 444 8 × 2 = 0 + 0.000 000 029 887 824 774 582 010 085 298 591 498 837 268 889 6;
  • 41) 0.000 000 029 887 824 774 582 010 085 298 591 498 837 268 889 6 × 2 = 0 + 0.000 000 059 775 649 549 164 020 170 597 182 997 674 537 779 2;
  • 42) 0.000 000 059 775 649 549 164 020 170 597 182 997 674 537 779 2 × 2 = 0 + 0.000 000 119 551 299 098 328 040 341 194 365 995 349 075 558 4;
  • 43) 0.000 000 119 551 299 098 328 040 341 194 365 995 349 075 558 4 × 2 = 0 + 0.000 000 239 102 598 196 656 080 682 388 731 990 698 151 116 8;
  • 44) 0.000 000 239 102 598 196 656 080 682 388 731 990 698 151 116 8 × 2 = 0 + 0.000 000 478 205 196 393 312 161 364 777 463 981 396 302 233 6;
  • 45) 0.000 000 478 205 196 393 312 161 364 777 463 981 396 302 233 6 × 2 = 0 + 0.000 000 956 410 392 786 624 322 729 554 927 962 792 604 467 2;
  • 46) 0.000 000 956 410 392 786 624 322 729 554 927 962 792 604 467 2 × 2 = 0 + 0.000 001 912 820 785 573 248 645 459 109 855 925 585 208 934 4;
  • 47) 0.000 001 912 820 785 573 248 645 459 109 855 925 585 208 934 4 × 2 = 0 + 0.000 003 825 641 571 146 497 290 918 219 711 851 170 417 868 8;
  • 48) 0.000 003 825 641 571 146 497 290 918 219 711 851 170 417 868 8 × 2 = 0 + 0.000 007 651 283 142 292 994 581 836 439 423 702 340 835 737 6;
  • 49) 0.000 007 651 283 142 292 994 581 836 439 423 702 340 835 737 6 × 2 = 0 + 0.000 015 302 566 284 585 989 163 672 878 847 404 681 671 475 2;
  • 50) 0.000 015 302 566 284 585 989 163 672 878 847 404 681 671 475 2 × 2 = 0 + 0.000 030 605 132 569 171 978 327 345 757 694 809 363 342 950 4;
  • 51) 0.000 030 605 132 569 171 978 327 345 757 694 809 363 342 950 4 × 2 = 0 + 0.000 061 210 265 138 343 956 654 691 515 389 618 726 685 900 8;
  • 52) 0.000 061 210 265 138 343 956 654 691 515 389 618 726 685 900 8 × 2 = 0 + 0.000 122 420 530 276 687 913 309 383 030 779 237 453 371 801 6;
  • 53) 0.000 122 420 530 276 687 913 309 383 030 779 237 453 371 801 6 × 2 = 0 + 0.000 244 841 060 553 375 826 618 766 061 558 474 906 743 603 2;
  • 54) 0.000 244 841 060 553 375 826 618 766 061 558 474 906 743 603 2 × 2 = 0 + 0.000 489 682 121 106 751 653 237 532 123 116 949 813 487 206 4;
  • 55) 0.000 489 682 121 106 751 653 237 532 123 116 949 813 487 206 4 × 2 = 0 + 0.000 979 364 242 213 503 306 475 064 246 233 899 626 974 412 8;
  • 56) 0.000 979 364 242 213 503 306 475 064 246 233 899 626 974 412 8 × 2 = 0 + 0.001 958 728 484 427 006 612 950 128 492 467 799 253 948 825 6;
  • 57) 0.001 958 728 484 427 006 612 950 128 492 467 799 253 948 825 6 × 2 = 0 + 0.003 917 456 968 854 013 225 900 256 984 935 598 507 897 651 2;
  • 58) 0.003 917 456 968 854 013 225 900 256 984 935 598 507 897 651 2 × 2 = 0 + 0.007 834 913 937 708 026 451 800 513 969 871 197 015 795 302 4;
  • 59) 0.007 834 913 937 708 026 451 800 513 969 871 197 015 795 302 4 × 2 = 0 + 0.015 669 827 875 416 052 903 601 027 939 742 394 031 590 604 8;
  • 60) 0.015 669 827 875 416 052 903 601 027 939 742 394 031 590 604 8 × 2 = 0 + 0.031 339 655 750 832 105 807 202 055 879 484 788 063 181 209 6;
  • 61) 0.031 339 655 750 832 105 807 202 055 879 484 788 063 181 209 6 × 2 = 0 + 0.062 679 311 501 664 211 614 404 111 758 969 576 126 362 419 2;
  • 62) 0.062 679 311 501 664 211 614 404 111 758 969 576 126 362 419 2 × 2 = 0 + 0.125 358 623 003 328 423 228 808 223 517 939 152 252 724 838 4;
  • 63) 0.125 358 623 003 328 423 228 808 223 517 939 152 252 724 838 4 × 2 = 0 + 0.250 717 246 006 656 846 457 616 447 035 878 304 505 449 676 8;
  • 64) 0.250 717 246 006 656 846 457 616 447 035 878 304 505 449 676 8 × 2 = 0 + 0.501 434 492 013 313 692 915 232 894 071 756 609 010 899 353 6;
  • 65) 0.501 434 492 013 313 692 915 232 894 071 756 609 010 899 353 6 × 2 = 1 + 0.002 868 984 026 627 385 830 465 788 143 513 218 021 798 707 2;
  • 66) 0.002 868 984 026 627 385 830 465 788 143 513 218 021 798 707 2 × 2 = 0 + 0.005 737 968 053 254 771 660 931 576 287 026 436 043 597 414 4;
  • 67) 0.005 737 968 053 254 771 660 931 576 287 026 436 043 597 414 4 × 2 = 0 + 0.011 475 936 106 509 543 321 863 152 574 052 872 087 194 828 8;
  • 68) 0.011 475 936 106 509 543 321 863 152 574 052 872 087 194 828 8 × 2 = 0 + 0.022 951 872 213 019 086 643 726 305 148 105 744 174 389 657 6;
  • 69) 0.022 951 872 213 019 086 643 726 305 148 105 744 174 389 657 6 × 2 = 0 + 0.045 903 744 426 038 173 287 452 610 296 211 488 348 779 315 2;
  • 70) 0.045 903 744 426 038 173 287 452 610 296 211 488 348 779 315 2 × 2 = 0 + 0.091 807 488 852 076 346 574 905 220 592 422 976 697 558 630 4;
  • 71) 0.091 807 488 852 076 346 574 905 220 592 422 976 697 558 630 4 × 2 = 0 + 0.183 614 977 704 152 693 149 810 441 184 845 953 395 117 260 8;
  • 72) 0.183 614 977 704 152 693 149 810 441 184 845 953 395 117 260 8 × 2 = 0 + 0.367 229 955 408 305 386 299 620 882 369 691 906 790 234 521 6;
  • 73) 0.367 229 955 408 305 386 299 620 882 369 691 906 790 234 521 6 × 2 = 0 + 0.734 459 910 816 610 772 599 241 764 739 383 813 580 469 043 2;
  • 74) 0.734 459 910 816 610 772 599 241 764 739 383 813 580 469 043 2 × 2 = 1 + 0.468 919 821 633 221 545 198 483 529 478 767 627 160 938 086 4;
  • 75) 0.468 919 821 633 221 545 198 483 529 478 767 627 160 938 086 4 × 2 = 0 + 0.937 839 643 266 443 090 396 967 058 957 535 254 321 876 172 8;
  • 76) 0.937 839 643 266 443 090 396 967 058 957 535 254 321 876 172 8 × 2 = 1 + 0.875 679 286 532 886 180 793 934 117 915 070 508 643 752 345 6;
  • 77) 0.875 679 286 532 886 180 793 934 117 915 070 508 643 752 345 6 × 2 = 1 + 0.751 358 573 065 772 361 587 868 235 830 141 017 287 504 691 2;
  • 78) 0.751 358 573 065 772 361 587 868 235 830 141 017 287 504 691 2 × 2 = 1 + 0.502 717 146 131 544 723 175 736 471 660 282 034 575 009 382 4;
  • 79) 0.502 717 146 131 544 723 175 736 471 660 282 034 575 009 382 4 × 2 = 1 + 0.005 434 292 263 089 446 351 472 943 320 564 069 150 018 764 8;
  • 80) 0.005 434 292 263 089 446 351 472 943 320 564 069 150 018 764 8 × 2 = 0 + 0.010 868 584 526 178 892 702 945 886 641 128 138 300 037 529 6;
  • 81) 0.010 868 584 526 178 892 702 945 886 641 128 138 300 037 529 6 × 2 = 0 + 0.021 737 169 052 357 785 405 891 773 282 256 276 600 075 059 2;
  • 82) 0.021 737 169 052 357 785 405 891 773 282 256 276 600 075 059 2 × 2 = 0 + 0.043 474 338 104 715 570 811 783 546 564 512 553 200 150 118 4;
  • 83) 0.043 474 338 104 715 570 811 783 546 564 512 553 200 150 118 4 × 2 = 0 + 0.086 948 676 209 431 141 623 567 093 129 025 106 400 300 236 8;
  • 84) 0.086 948 676 209 431 141 623 567 093 129 025 106 400 300 236 8 × 2 = 0 + 0.173 897 352 418 862 283 247 134 186 258 050 212 800 600 473 6;
  • 85) 0.173 897 352 418 862 283 247 134 186 258 050 212 800 600 473 6 × 2 = 0 + 0.347 794 704 837 724 566 494 268 372 516 100 425 601 200 947 2;
  • 86) 0.347 794 704 837 724 566 494 268 372 516 100 425 601 200 947 2 × 2 = 0 + 0.695 589 409 675 449 132 988 536 745 032 200 851 202 401 894 4;
  • 87) 0.695 589 409 675 449 132 988 536 745 032 200 851 202 401 894 4 × 2 = 1 + 0.391 178 819 350 898 265 977 073 490 064 401 702 404 803 788 8;
  • 88) 0.391 178 819 350 898 265 977 073 490 064 401 702 404 803 788 8 × 2 = 0 + 0.782 357 638 701 796 531 954 146 980 128 803 404 809 607 577 6;

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:


0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 0000 0101 1110 0000 0010(2)

6. Positive number before normalization:

0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 0000 0101 1110 0000 0010(2)

7. Normalize the binary representation of the number.

Shift the decimal mark 65 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 0000 0101 1110 0000 0010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 0000 0101 1110 0000 0010(2) × 20 =

1.0000 0000 1011 1100 0000 010(2) × 2-65


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): -65

Mantissa (not normalized):
1.0000 0000 1011 1100 0000 010


9. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =

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

-65 + 2(8-1) - 1 =

(-65 + 127)(10) =

62(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;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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) =

62(10) =

0011 1110(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. 000 0000 0101 1110 0000 0010 =

000 0000 0101 1110 0000 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) =
0011 1110

Mantissa (23 bits) =
000 0000 0101 1110 0000 0010


Decimal number -0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 749 6 converted to 32 bit single precision IEEE 754 binary floating point representation:

1 - 0011 1110 - 000 0000 0101 1110 0000 0010

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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