-0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4 Converted to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal -0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4(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
-0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4(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:

|-0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4| = 0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4


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: 0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4.

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) 0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4 × 2 = 0 + 0.000 000 000 000 000 000 054 365 636 560 000 002 801 277 488 8;
  • 2) 0.000 000 000 000 000 000 054 365 636 560 000 002 801 277 488 8 × 2 = 0 + 0.000 000 000 000 000 000 108 731 273 120 000 005 602 554 977 6;
  • 3) 0.000 000 000 000 000 000 108 731 273 120 000 005 602 554 977 6 × 2 = 0 + 0.000 000 000 000 000 000 217 462 546 240 000 011 205 109 955 2;
  • 4) 0.000 000 000 000 000 000 217 462 546 240 000 011 205 109 955 2 × 2 = 0 + 0.000 000 000 000 000 000 434 925 092 480 000 022 410 219 910 4;
  • 5) 0.000 000 000 000 000 000 434 925 092 480 000 022 410 219 910 4 × 2 = 0 + 0.000 000 000 000 000 000 869 850 184 960 000 044 820 439 820 8;
  • 6) 0.000 000 000 000 000 000 869 850 184 960 000 044 820 439 820 8 × 2 = 0 + 0.000 000 000 000 000 001 739 700 369 920 000 089 640 879 641 6;
  • 7) 0.000 000 000 000 000 001 739 700 369 920 000 089 640 879 641 6 × 2 = 0 + 0.000 000 000 000 000 003 479 400 739 840 000 179 281 759 283 2;
  • 8) 0.000 000 000 000 000 003 479 400 739 840 000 179 281 759 283 2 × 2 = 0 + 0.000 000 000 000 000 006 958 801 479 680 000 358 563 518 566 4;
  • 9) 0.000 000 000 000 000 006 958 801 479 680 000 358 563 518 566 4 × 2 = 0 + 0.000 000 000 000 000 013 917 602 959 360 000 717 127 037 132 8;
  • 10) 0.000 000 000 000 000 013 917 602 959 360 000 717 127 037 132 8 × 2 = 0 + 0.000 000 000 000 000 027 835 205 918 720 001 434 254 074 265 6;
  • 11) 0.000 000 000 000 000 027 835 205 918 720 001 434 254 074 265 6 × 2 = 0 + 0.000 000 000 000 000 055 670 411 837 440 002 868 508 148 531 2;
  • 12) 0.000 000 000 000 000 055 670 411 837 440 002 868 508 148 531 2 × 2 = 0 + 0.000 000 000 000 000 111 340 823 674 880 005 737 016 297 062 4;
  • 13) 0.000 000 000 000 000 111 340 823 674 880 005 737 016 297 062 4 × 2 = 0 + 0.000 000 000 000 000 222 681 647 349 760 011 474 032 594 124 8;
  • 14) 0.000 000 000 000 000 222 681 647 349 760 011 474 032 594 124 8 × 2 = 0 + 0.000 000 000 000 000 445 363 294 699 520 022 948 065 188 249 6;
  • 15) 0.000 000 000 000 000 445 363 294 699 520 022 948 065 188 249 6 × 2 = 0 + 0.000 000 000 000 000 890 726 589 399 040 045 896 130 376 499 2;
  • 16) 0.000 000 000 000 000 890 726 589 399 040 045 896 130 376 499 2 × 2 = 0 + 0.000 000 000 000 001 781 453 178 798 080 091 792 260 752 998 4;
  • 17) 0.000 000 000 000 001 781 453 178 798 080 091 792 260 752 998 4 × 2 = 0 + 0.000 000 000 000 003 562 906 357 596 160 183 584 521 505 996 8;
  • 18) 0.000 000 000 000 003 562 906 357 596 160 183 584 521 505 996 8 × 2 = 0 + 0.000 000 000 000 007 125 812 715 192 320 367 169 043 011 993 6;
  • 19) 0.000 000 000 000 007 125 812 715 192 320 367 169 043 011 993 6 × 2 = 0 + 0.000 000 000 000 014 251 625 430 384 640 734 338 086 023 987 2;
  • 20) 0.000 000 000 000 014 251 625 430 384 640 734 338 086 023 987 2 × 2 = 0 + 0.000 000 000 000 028 503 250 860 769 281 468 676 172 047 974 4;
  • 21) 0.000 000 000 000 028 503 250 860 769 281 468 676 172 047 974 4 × 2 = 0 + 0.000 000 000 000 057 006 501 721 538 562 937 352 344 095 948 8;
  • 22) 0.000 000 000 000 057 006 501 721 538 562 937 352 344 095 948 8 × 2 = 0 + 0.000 000 000 000 114 013 003 443 077 125 874 704 688 191 897 6;
  • 23) 0.000 000 000 000 114 013 003 443 077 125 874 704 688 191 897 6 × 2 = 0 + 0.000 000 000 000 228 026 006 886 154 251 749 409 376 383 795 2;
  • 24) 0.000 000 000 000 228 026 006 886 154 251 749 409 376 383 795 2 × 2 = 0 + 0.000 000 000 000 456 052 013 772 308 503 498 818 752 767 590 4;
  • 25) 0.000 000 000 000 456 052 013 772 308 503 498 818 752 767 590 4 × 2 = 0 + 0.000 000 000 000 912 104 027 544 617 006 997 637 505 535 180 8;
  • 26) 0.000 000 000 000 912 104 027 544 617 006 997 637 505 535 180 8 × 2 = 0 + 0.000 000 000 001 824 208 055 089 234 013 995 275 011 070 361 6;
  • 27) 0.000 000 000 001 824 208 055 089 234 013 995 275 011 070 361 6 × 2 = 0 + 0.000 000 000 003 648 416 110 178 468 027 990 550 022 140 723 2;
  • 28) 0.000 000 000 003 648 416 110 178 468 027 990 550 022 140 723 2 × 2 = 0 + 0.000 000 000 007 296 832 220 356 936 055 981 100 044 281 446 4;
  • 29) 0.000 000 000 007 296 832 220 356 936 055 981 100 044 281 446 4 × 2 = 0 + 0.000 000 000 014 593 664 440 713 872 111 962 200 088 562 892 8;
  • 30) 0.000 000 000 014 593 664 440 713 872 111 962 200 088 562 892 8 × 2 = 0 + 0.000 000 000 029 187 328 881 427 744 223 924 400 177 125 785 6;
  • 31) 0.000 000 000 029 187 328 881 427 744 223 924 400 177 125 785 6 × 2 = 0 + 0.000 000 000 058 374 657 762 855 488 447 848 800 354 251 571 2;
  • 32) 0.000 000 000 058 374 657 762 855 488 447 848 800 354 251 571 2 × 2 = 0 + 0.000 000 000 116 749 315 525 710 976 895 697 600 708 503 142 4;
  • 33) 0.000 000 000 116 749 315 525 710 976 895 697 600 708 503 142 4 × 2 = 0 + 0.000 000 000 233 498 631 051 421 953 791 395 201 417 006 284 8;
  • 34) 0.000 000 000 233 498 631 051 421 953 791 395 201 417 006 284 8 × 2 = 0 + 0.000 000 000 466 997 262 102 843 907 582 790 402 834 012 569 6;
  • 35) 0.000 000 000 466 997 262 102 843 907 582 790 402 834 012 569 6 × 2 = 0 + 0.000 000 000 933 994 524 205 687 815 165 580 805 668 025 139 2;
  • 36) 0.000 000 000 933 994 524 205 687 815 165 580 805 668 025 139 2 × 2 = 0 + 0.000 000 001 867 989 048 411 375 630 331 161 611 336 050 278 4;
  • 37) 0.000 000 001 867 989 048 411 375 630 331 161 611 336 050 278 4 × 2 = 0 + 0.000 000 003 735 978 096 822 751 260 662 323 222 672 100 556 8;
  • 38) 0.000 000 003 735 978 096 822 751 260 662 323 222 672 100 556 8 × 2 = 0 + 0.000 000 007 471 956 193 645 502 521 324 646 445 344 201 113 6;
  • 39) 0.000 000 007 471 956 193 645 502 521 324 646 445 344 201 113 6 × 2 = 0 + 0.000 000 014 943 912 387 291 005 042 649 292 890 688 402 227 2;
  • 40) 0.000 000 014 943 912 387 291 005 042 649 292 890 688 402 227 2 × 2 = 0 + 0.000 000 029 887 824 774 582 010 085 298 585 781 376 804 454 4;
  • 41) 0.000 000 029 887 824 774 582 010 085 298 585 781 376 804 454 4 × 2 = 0 + 0.000 000 059 775 649 549 164 020 170 597 171 562 753 608 908 8;
  • 42) 0.000 000 059 775 649 549 164 020 170 597 171 562 753 608 908 8 × 2 = 0 + 0.000 000 119 551 299 098 328 040 341 194 343 125 507 217 817 6;
  • 43) 0.000 000 119 551 299 098 328 040 341 194 343 125 507 217 817 6 × 2 = 0 + 0.000 000 239 102 598 196 656 080 682 388 686 251 014 435 635 2;
  • 44) 0.000 000 239 102 598 196 656 080 682 388 686 251 014 435 635 2 × 2 = 0 + 0.000 000 478 205 196 393 312 161 364 777 372 502 028 871 270 4;
  • 45) 0.000 000 478 205 196 393 312 161 364 777 372 502 028 871 270 4 × 2 = 0 + 0.000 000 956 410 392 786 624 322 729 554 745 004 057 742 540 8;
  • 46) 0.000 000 956 410 392 786 624 322 729 554 745 004 057 742 540 8 × 2 = 0 + 0.000 001 912 820 785 573 248 645 459 109 490 008 115 485 081 6;
  • 47) 0.000 001 912 820 785 573 248 645 459 109 490 008 115 485 081 6 × 2 = 0 + 0.000 003 825 641 571 146 497 290 918 218 980 016 230 970 163 2;
  • 48) 0.000 003 825 641 571 146 497 290 918 218 980 016 230 970 163 2 × 2 = 0 + 0.000 007 651 283 142 292 994 581 836 437 960 032 461 940 326 4;
  • 49) 0.000 007 651 283 142 292 994 581 836 437 960 032 461 940 326 4 × 2 = 0 + 0.000 015 302 566 284 585 989 163 672 875 920 064 923 880 652 8;
  • 50) 0.000 015 302 566 284 585 989 163 672 875 920 064 923 880 652 8 × 2 = 0 + 0.000 030 605 132 569 171 978 327 345 751 840 129 847 761 305 6;
  • 51) 0.000 030 605 132 569 171 978 327 345 751 840 129 847 761 305 6 × 2 = 0 + 0.000 061 210 265 138 343 956 654 691 503 680 259 695 522 611 2;
  • 52) 0.000 061 210 265 138 343 956 654 691 503 680 259 695 522 611 2 × 2 = 0 + 0.000 122 420 530 276 687 913 309 383 007 360 519 391 045 222 4;
  • 53) 0.000 122 420 530 276 687 913 309 383 007 360 519 391 045 222 4 × 2 = 0 + 0.000 244 841 060 553 375 826 618 766 014 721 038 782 090 444 8;
  • 54) 0.000 244 841 060 553 375 826 618 766 014 721 038 782 090 444 8 × 2 = 0 + 0.000 489 682 121 106 751 653 237 532 029 442 077 564 180 889 6;
  • 55) 0.000 489 682 121 106 751 653 237 532 029 442 077 564 180 889 6 × 2 = 0 + 0.000 979 364 242 213 503 306 475 064 058 884 155 128 361 779 2;
  • 56) 0.000 979 364 242 213 503 306 475 064 058 884 155 128 361 779 2 × 2 = 0 + 0.001 958 728 484 427 006 612 950 128 117 768 310 256 723 558 4;
  • 57) 0.001 958 728 484 427 006 612 950 128 117 768 310 256 723 558 4 × 2 = 0 + 0.003 917 456 968 854 013 225 900 256 235 536 620 513 447 116 8;
  • 58) 0.003 917 456 968 854 013 225 900 256 235 536 620 513 447 116 8 × 2 = 0 + 0.007 834 913 937 708 026 451 800 512 471 073 241 026 894 233 6;
  • 59) 0.007 834 913 937 708 026 451 800 512 471 073 241 026 894 233 6 × 2 = 0 + 0.015 669 827 875 416 052 903 601 024 942 146 482 053 788 467 2;
  • 60) 0.015 669 827 875 416 052 903 601 024 942 146 482 053 788 467 2 × 2 = 0 + 0.031 339 655 750 832 105 807 202 049 884 292 964 107 576 934 4;
  • 61) 0.031 339 655 750 832 105 807 202 049 884 292 964 107 576 934 4 × 2 = 0 + 0.062 679 311 501 664 211 614 404 099 768 585 928 215 153 868 8;
  • 62) 0.062 679 311 501 664 211 614 404 099 768 585 928 215 153 868 8 × 2 = 0 + 0.125 358 623 003 328 423 228 808 199 537 171 856 430 307 737 6;
  • 63) 0.125 358 623 003 328 423 228 808 199 537 171 856 430 307 737 6 × 2 = 0 + 0.250 717 246 006 656 846 457 616 399 074 343 712 860 615 475 2;
  • 64) 0.250 717 246 006 656 846 457 616 399 074 343 712 860 615 475 2 × 2 = 0 + 0.501 434 492 013 313 692 915 232 798 148 687 425 721 230 950 4;
  • 65) 0.501 434 492 013 313 692 915 232 798 148 687 425 721 230 950 4 × 2 = 1 + 0.002 868 984 026 627 385 830 465 596 297 374 851 442 461 900 8;
  • 66) 0.002 868 984 026 627 385 830 465 596 297 374 851 442 461 900 8 × 2 = 0 + 0.005 737 968 053 254 771 660 931 192 594 749 702 884 923 801 6;
  • 67) 0.005 737 968 053 254 771 660 931 192 594 749 702 884 923 801 6 × 2 = 0 + 0.011 475 936 106 509 543 321 862 385 189 499 405 769 847 603 2;
  • 68) 0.011 475 936 106 509 543 321 862 385 189 499 405 769 847 603 2 × 2 = 0 + 0.022 951 872 213 019 086 643 724 770 378 998 811 539 695 206 4;
  • 69) 0.022 951 872 213 019 086 643 724 770 378 998 811 539 695 206 4 × 2 = 0 + 0.045 903 744 426 038 173 287 449 540 757 997 623 079 390 412 8;
  • 70) 0.045 903 744 426 038 173 287 449 540 757 997 623 079 390 412 8 × 2 = 0 + 0.091 807 488 852 076 346 574 899 081 515 995 246 158 780 825 6;
  • 71) 0.091 807 488 852 076 346 574 899 081 515 995 246 158 780 825 6 × 2 = 0 + 0.183 614 977 704 152 693 149 798 163 031 990 492 317 561 651 2;
  • 72) 0.183 614 977 704 152 693 149 798 163 031 990 492 317 561 651 2 × 2 = 0 + 0.367 229 955 408 305 386 299 596 326 063 980 984 635 123 302 4;
  • 73) 0.367 229 955 408 305 386 299 596 326 063 980 984 635 123 302 4 × 2 = 0 + 0.734 459 910 816 610 772 599 192 652 127 961 969 270 246 604 8;
  • 74) 0.734 459 910 816 610 772 599 192 652 127 961 969 270 246 604 8 × 2 = 1 + 0.468 919 821 633 221 545 198 385 304 255 923 938 540 493 209 6;
  • 75) 0.468 919 821 633 221 545 198 385 304 255 923 938 540 493 209 6 × 2 = 0 + 0.937 839 643 266 443 090 396 770 608 511 847 877 080 986 419 2;
  • 76) 0.937 839 643 266 443 090 396 770 608 511 847 877 080 986 419 2 × 2 = 1 + 0.875 679 286 532 886 180 793 541 217 023 695 754 161 972 838 4;
  • 77) 0.875 679 286 532 886 180 793 541 217 023 695 754 161 972 838 4 × 2 = 1 + 0.751 358 573 065 772 361 587 082 434 047 391 508 323 945 676 8;
  • 78) 0.751 358 573 065 772 361 587 082 434 047 391 508 323 945 676 8 × 2 = 1 + 0.502 717 146 131 544 723 174 164 868 094 783 016 647 891 353 6;
  • 79) 0.502 717 146 131 544 723 174 164 868 094 783 016 647 891 353 6 × 2 = 1 + 0.005 434 292 263 089 446 348 329 736 189 566 033 295 782 707 2;
  • 80) 0.005 434 292 263 089 446 348 329 736 189 566 033 295 782 707 2 × 2 = 0 + 0.010 868 584 526 178 892 696 659 472 379 132 066 591 565 414 4;
  • 81) 0.010 868 584 526 178 892 696 659 472 379 132 066 591 565 414 4 × 2 = 0 + 0.021 737 169 052 357 785 393 318 944 758 264 133 183 130 828 8;
  • 82) 0.021 737 169 052 357 785 393 318 944 758 264 133 183 130 828 8 × 2 = 0 + 0.043 474 338 104 715 570 786 637 889 516 528 266 366 261 657 6;
  • 83) 0.043 474 338 104 715 570 786 637 889 516 528 266 366 261 657 6 × 2 = 0 + 0.086 948 676 209 431 141 573 275 779 033 056 532 732 523 315 2;
  • 84) 0.086 948 676 209 431 141 573 275 779 033 056 532 732 523 315 2 × 2 = 0 + 0.173 897 352 418 862 283 146 551 558 066 113 065 465 046 630 4;
  • 85) 0.173 897 352 418 862 283 146 551 558 066 113 065 465 046 630 4 × 2 = 0 + 0.347 794 704 837 724 566 293 103 116 132 226 130 930 093 260 8;
  • 86) 0.347 794 704 837 724 566 293 103 116 132 226 130 930 093 260 8 × 2 = 0 + 0.695 589 409 675 449 132 586 206 232 264 452 261 860 186 521 6;
  • 87) 0.695 589 409 675 449 132 586 206 232 264 452 261 860 186 521 6 × 2 = 1 + 0.391 178 819 350 898 265 172 412 464 528 904 523 720 373 043 2;
  • 88) 0.391 178 819 350 898 265 172 412 464 528 904 523 720 373 043 2 × 2 = 0 + 0.782 357 638 701 796 530 344 824 929 057 809 047 440 746 086 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:


0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4(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:

0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4(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:


0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4(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 -0.000 000 000 000 000 000 027 182 818 280 000 001 400 638 744 4 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