32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: 11 111 000 011 111 100 111 100 001 110 998 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

Number 11 111 000 011 111 100 111 100 001 110 998(10) converted and written in 32 bit single precision IEEE 754 binary floating point representation (1 bit for sign, 8 bits for exponent, 23 bits for mantissa)

1. 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;
  • 11 111 000 011 111 100 111 100 001 110 998 ÷ 2 = 5 555 500 005 555 550 055 550 000 555 499 + 0;
  • 5 555 500 005 555 550 055 550 000 555 499 ÷ 2 = 2 777 750 002 777 775 027 775 000 277 749 + 1;
  • 2 777 750 002 777 775 027 775 000 277 749 ÷ 2 = 1 388 875 001 388 887 513 887 500 138 874 + 1;
  • 1 388 875 001 388 887 513 887 500 138 874 ÷ 2 = 694 437 500 694 443 756 943 750 069 437 + 0;
  • 694 437 500 694 443 756 943 750 069 437 ÷ 2 = 347 218 750 347 221 878 471 875 034 718 + 1;
  • 347 218 750 347 221 878 471 875 034 718 ÷ 2 = 173 609 375 173 610 939 235 937 517 359 + 0;
  • 173 609 375 173 610 939 235 937 517 359 ÷ 2 = 86 804 687 586 805 469 617 968 758 679 + 1;
  • 86 804 687 586 805 469 617 968 758 679 ÷ 2 = 43 402 343 793 402 734 808 984 379 339 + 1;
  • 43 402 343 793 402 734 808 984 379 339 ÷ 2 = 21 701 171 896 701 367 404 492 189 669 + 1;
  • 21 701 171 896 701 367 404 492 189 669 ÷ 2 = 10 850 585 948 350 683 702 246 094 834 + 1;
  • 10 850 585 948 350 683 702 246 094 834 ÷ 2 = 5 425 292 974 175 341 851 123 047 417 + 0;
  • 5 425 292 974 175 341 851 123 047 417 ÷ 2 = 2 712 646 487 087 670 925 561 523 708 + 1;
  • 2 712 646 487 087 670 925 561 523 708 ÷ 2 = 1 356 323 243 543 835 462 780 761 854 + 0;
  • 1 356 323 243 543 835 462 780 761 854 ÷ 2 = 678 161 621 771 917 731 390 380 927 + 0;
  • 678 161 621 771 917 731 390 380 927 ÷ 2 = 339 080 810 885 958 865 695 190 463 + 1;
  • 339 080 810 885 958 865 695 190 463 ÷ 2 = 169 540 405 442 979 432 847 595 231 + 1;
  • 169 540 405 442 979 432 847 595 231 ÷ 2 = 84 770 202 721 489 716 423 797 615 + 1;
  • 84 770 202 721 489 716 423 797 615 ÷ 2 = 42 385 101 360 744 858 211 898 807 + 1;
  • 42 385 101 360 744 858 211 898 807 ÷ 2 = 21 192 550 680 372 429 105 949 403 + 1;
  • 21 192 550 680 372 429 105 949 403 ÷ 2 = 10 596 275 340 186 214 552 974 701 + 1;
  • 10 596 275 340 186 214 552 974 701 ÷ 2 = 5 298 137 670 093 107 276 487 350 + 1;
  • 5 298 137 670 093 107 276 487 350 ÷ 2 = 2 649 068 835 046 553 638 243 675 + 0;
  • 2 649 068 835 046 553 638 243 675 ÷ 2 = 1 324 534 417 523 276 819 121 837 + 1;
  • 1 324 534 417 523 276 819 121 837 ÷ 2 = 662 267 208 761 638 409 560 918 + 1;
  • 662 267 208 761 638 409 560 918 ÷ 2 = 331 133 604 380 819 204 780 459 + 0;
  • 331 133 604 380 819 204 780 459 ÷ 2 = 165 566 802 190 409 602 390 229 + 1;
  • 165 566 802 190 409 602 390 229 ÷ 2 = 82 783 401 095 204 801 195 114 + 1;
  • 82 783 401 095 204 801 195 114 ÷ 2 = 41 391 700 547 602 400 597 557 + 0;
  • 41 391 700 547 602 400 597 557 ÷ 2 = 20 695 850 273 801 200 298 778 + 1;
  • 20 695 850 273 801 200 298 778 ÷ 2 = 10 347 925 136 900 600 149 389 + 0;
  • 10 347 925 136 900 600 149 389 ÷ 2 = 5 173 962 568 450 300 074 694 + 1;
  • 5 173 962 568 450 300 074 694 ÷ 2 = 2 586 981 284 225 150 037 347 + 0;
  • 2 586 981 284 225 150 037 347 ÷ 2 = 1 293 490 642 112 575 018 673 + 1;
  • 1 293 490 642 112 575 018 673 ÷ 2 = 646 745 321 056 287 509 336 + 1;
  • 646 745 321 056 287 509 336 ÷ 2 = 323 372 660 528 143 754 668 + 0;
  • 323 372 660 528 143 754 668 ÷ 2 = 161 686 330 264 071 877 334 + 0;
  • 161 686 330 264 071 877 334 ÷ 2 = 80 843 165 132 035 938 667 + 0;
  • 80 843 165 132 035 938 667 ÷ 2 = 40 421 582 566 017 969 333 + 1;
  • 40 421 582 566 017 969 333 ÷ 2 = 20 210 791 283 008 984 666 + 1;
  • 20 210 791 283 008 984 666 ÷ 2 = 10 105 395 641 504 492 333 + 0;
  • 10 105 395 641 504 492 333 ÷ 2 = 5 052 697 820 752 246 166 + 1;
  • 5 052 697 820 752 246 166 ÷ 2 = 2 526 348 910 376 123 083 + 0;
  • 2 526 348 910 376 123 083 ÷ 2 = 1 263 174 455 188 061 541 + 1;
  • 1 263 174 455 188 061 541 ÷ 2 = 631 587 227 594 030 770 + 1;
  • 631 587 227 594 030 770 ÷ 2 = 315 793 613 797 015 385 + 0;
  • 315 793 613 797 015 385 ÷ 2 = 157 896 806 898 507 692 + 1;
  • 157 896 806 898 507 692 ÷ 2 = 78 948 403 449 253 846 + 0;
  • 78 948 403 449 253 846 ÷ 2 = 39 474 201 724 626 923 + 0;
  • 39 474 201 724 626 923 ÷ 2 = 19 737 100 862 313 461 + 1;
  • 19 737 100 862 313 461 ÷ 2 = 9 868 550 431 156 730 + 1;
  • 9 868 550 431 156 730 ÷ 2 = 4 934 275 215 578 365 + 0;
  • 4 934 275 215 578 365 ÷ 2 = 2 467 137 607 789 182 + 1;
  • 2 467 137 607 789 182 ÷ 2 = 1 233 568 803 894 591 + 0;
  • 1 233 568 803 894 591 ÷ 2 = 616 784 401 947 295 + 1;
  • 616 784 401 947 295 ÷ 2 = 308 392 200 973 647 + 1;
  • 308 392 200 973 647 ÷ 2 = 154 196 100 486 823 + 1;
  • 154 196 100 486 823 ÷ 2 = 77 098 050 243 411 + 1;
  • 77 098 050 243 411 ÷ 2 = 38 549 025 121 705 + 1;
  • 38 549 025 121 705 ÷ 2 = 19 274 512 560 852 + 1;
  • 19 274 512 560 852 ÷ 2 = 9 637 256 280 426 + 0;
  • 9 637 256 280 426 ÷ 2 = 4 818 628 140 213 + 0;
  • 4 818 628 140 213 ÷ 2 = 2 409 314 070 106 + 1;
  • 2 409 314 070 106 ÷ 2 = 1 204 657 035 053 + 0;
  • 1 204 657 035 053 ÷ 2 = 602 328 517 526 + 1;
  • 602 328 517 526 ÷ 2 = 301 164 258 763 + 0;
  • 301 164 258 763 ÷ 2 = 150 582 129 381 + 1;
  • 150 582 129 381 ÷ 2 = 75 291 064 690 + 1;
  • 75 291 064 690 ÷ 2 = 37 645 532 345 + 0;
  • 37 645 532 345 ÷ 2 = 18 822 766 172 + 1;
  • 18 822 766 172 ÷ 2 = 9 411 383 086 + 0;
  • 9 411 383 086 ÷ 2 = 4 705 691 543 + 0;
  • 4 705 691 543 ÷ 2 = 2 352 845 771 + 1;
  • 2 352 845 771 ÷ 2 = 1 176 422 885 + 1;
  • 1 176 422 885 ÷ 2 = 588 211 442 + 1;
  • 588 211 442 ÷ 2 = 294 105 721 + 0;
  • 294 105 721 ÷ 2 = 147 052 860 + 1;
  • 147 052 860 ÷ 2 = 73 526 430 + 0;
  • 73 526 430 ÷ 2 = 36 763 215 + 0;
  • 36 763 215 ÷ 2 = 18 381 607 + 1;
  • 18 381 607 ÷ 2 = 9 190 803 + 1;
  • 9 190 803 ÷ 2 = 4 595 401 + 1;
  • 4 595 401 ÷ 2 = 2 297 700 + 1;
  • 2 297 700 ÷ 2 = 1 148 850 + 0;
  • 1 148 850 ÷ 2 = 574 425 + 0;
  • 574 425 ÷ 2 = 287 212 + 1;
  • 287 212 ÷ 2 = 143 606 + 0;
  • 143 606 ÷ 2 = 71 803 + 0;
  • 71 803 ÷ 2 = 35 901 + 1;
  • 35 901 ÷ 2 = 17 950 + 1;
  • 17 950 ÷ 2 = 8 975 + 0;
  • 8 975 ÷ 2 = 4 487 + 1;
  • 4 487 ÷ 2 = 2 243 + 1;
  • 2 243 ÷ 2 = 1 121 + 1;
  • 1 121 ÷ 2 = 560 + 1;
  • 560 ÷ 2 = 280 + 0;
  • 280 ÷ 2 = 140 + 0;
  • 140 ÷ 2 = 70 + 0;
  • 70 ÷ 2 = 35 + 0;
  • 35 ÷ 2 = 17 + 1;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number.

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


11 111 000 011 111 100 111 100 001 110 998(10) =


1000 1100 0011 1101 1001 0011 1100 1011 1001 0110 1010 0111 1110 1011 0010 1101 0110 0011 0101 0110 1101 1111 1100 1011 1101 0110(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 103 positions to the left, so that only one non zero digit remains to the left of it:


11 111 000 011 111 100 111 100 001 110 998(10) =


1000 1100 0011 1101 1001 0011 1100 1011 1001 0110 1010 0111 1110 1011 0010 1101 0110 0011 0101 0110 1101 1111 1100 1011 1101 0110(2) =


1000 1100 0011 1101 1001 0011 1100 1011 1001 0110 1010 0111 1110 1011 0010 1101 0110 0011 0101 0110 1101 1111 1100 1011 1101 0110(2) × 20 =


1.0001 1000 0111 1011 0010 0111 1001 0111 0010 1101 0100 1111 1101 0110 0101 1010 1100 0110 1010 1101 1011 1111 1001 0111 1010 110(2) × 2103


4. 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): 103


Mantissa (not normalized):
1.0001 1000 0111 1011 0010 0111 1001 0111 0010 1101 0100 1111 1101 0110 0101 1010 1100 0110 1010 1101 1011 1111 1001 0111 1010 110


5. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =


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


103 + 2(8-1) - 1 =


(103 + 127)(10) =


230(10)


6. 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;
  • 230 ÷ 2 = 115 + 0;
  • 115 ÷ 2 = 57 + 1;
  • 57 ÷ 2 = 28 + 1;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =


230(10) =


1110 0110(2)


8. 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =


1. 000 1100 0011 1101 1001 0011 1100 1011 1001 0110 1010 0111 1110 1011 0010 1101 0110 0011 0101 0110 1101 1111 1100 1011 1101 0110 =


000 1100 0011 1101 1001 0011


9. 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) =
1110 0110


Mantissa (23 bits) =
000 1100 0011 1101 1001 0011


The base ten decimal number 11 111 000 011 111 100 111 100 001 110 998 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 1110 0110 - 000 1100 0011 1101 1001 0011

The latest decimal numbers converted from base ten to 32 bit single precision IEEE 754 floating point binary standard representation