32bit IEEE 754: Decimal ↗ Single Precision Floating Point Binary: 1 100 000 100 999 999 999 999 999 988 Convert the Number to 32 Bit Single Precision IEEE 754 Binary Floating Point Representation Standard, From a Base 10 Decimal System Number

Number 1 100 000 100 999 999 999 999 999 988(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;
  • 1 100 000 100 999 999 999 999 999 988 ÷ 2 = 550 000 050 499 999 999 999 999 994 + 0;
  • 550 000 050 499 999 999 999 999 994 ÷ 2 = 275 000 025 249 999 999 999 999 997 + 0;
  • 275 000 025 249 999 999 999 999 997 ÷ 2 = 137 500 012 624 999 999 999 999 998 + 1;
  • 137 500 012 624 999 999 999 999 998 ÷ 2 = 68 750 006 312 499 999 999 999 999 + 0;
  • 68 750 006 312 499 999 999 999 999 ÷ 2 = 34 375 003 156 249 999 999 999 999 + 1;
  • 34 375 003 156 249 999 999 999 999 ÷ 2 = 17 187 501 578 124 999 999 999 999 + 1;
  • 17 187 501 578 124 999 999 999 999 ÷ 2 = 8 593 750 789 062 499 999 999 999 + 1;
  • 8 593 750 789 062 499 999 999 999 ÷ 2 = 4 296 875 394 531 249 999 999 999 + 1;
  • 4 296 875 394 531 249 999 999 999 ÷ 2 = 2 148 437 697 265 624 999 999 999 + 1;
  • 2 148 437 697 265 624 999 999 999 ÷ 2 = 1 074 218 848 632 812 499 999 999 + 1;
  • 1 074 218 848 632 812 499 999 999 ÷ 2 = 537 109 424 316 406 249 999 999 + 1;
  • 537 109 424 316 406 249 999 999 ÷ 2 = 268 554 712 158 203 124 999 999 + 1;
  • 268 554 712 158 203 124 999 999 ÷ 2 = 134 277 356 079 101 562 499 999 + 1;
  • 134 277 356 079 101 562 499 999 ÷ 2 = 67 138 678 039 550 781 249 999 + 1;
  • 67 138 678 039 550 781 249 999 ÷ 2 = 33 569 339 019 775 390 624 999 + 1;
  • 33 569 339 019 775 390 624 999 ÷ 2 = 16 784 669 509 887 695 312 499 + 1;
  • 16 784 669 509 887 695 312 499 ÷ 2 = 8 392 334 754 943 847 656 249 + 1;
  • 8 392 334 754 943 847 656 249 ÷ 2 = 4 196 167 377 471 923 828 124 + 1;
  • 4 196 167 377 471 923 828 124 ÷ 2 = 2 098 083 688 735 961 914 062 + 0;
  • 2 098 083 688 735 961 914 062 ÷ 2 = 1 049 041 844 367 980 957 031 + 0;
  • 1 049 041 844 367 980 957 031 ÷ 2 = 524 520 922 183 990 478 515 + 1;
  • 524 520 922 183 990 478 515 ÷ 2 = 262 260 461 091 995 239 257 + 1;
  • 262 260 461 091 995 239 257 ÷ 2 = 131 130 230 545 997 619 628 + 1;
  • 131 130 230 545 997 619 628 ÷ 2 = 65 565 115 272 998 809 814 + 0;
  • 65 565 115 272 998 809 814 ÷ 2 = 32 782 557 636 499 404 907 + 0;
  • 32 782 557 636 499 404 907 ÷ 2 = 16 391 278 818 249 702 453 + 1;
  • 16 391 278 818 249 702 453 ÷ 2 = 8 195 639 409 124 851 226 + 1;
  • 8 195 639 409 124 851 226 ÷ 2 = 4 097 819 704 562 425 613 + 0;
  • 4 097 819 704 562 425 613 ÷ 2 = 2 048 909 852 281 212 806 + 1;
  • 2 048 909 852 281 212 806 ÷ 2 = 1 024 454 926 140 606 403 + 0;
  • 1 024 454 926 140 606 403 ÷ 2 = 512 227 463 070 303 201 + 1;
  • 512 227 463 070 303 201 ÷ 2 = 256 113 731 535 151 600 + 1;
  • 256 113 731 535 151 600 ÷ 2 = 128 056 865 767 575 800 + 0;
  • 128 056 865 767 575 800 ÷ 2 = 64 028 432 883 787 900 + 0;
  • 64 028 432 883 787 900 ÷ 2 = 32 014 216 441 893 950 + 0;
  • 32 014 216 441 893 950 ÷ 2 = 16 007 108 220 946 975 + 0;
  • 16 007 108 220 946 975 ÷ 2 = 8 003 554 110 473 487 + 1;
  • 8 003 554 110 473 487 ÷ 2 = 4 001 777 055 236 743 + 1;
  • 4 001 777 055 236 743 ÷ 2 = 2 000 888 527 618 371 + 1;
  • 2 000 888 527 618 371 ÷ 2 = 1 000 444 263 809 185 + 1;
  • 1 000 444 263 809 185 ÷ 2 = 500 222 131 904 592 + 1;
  • 500 222 131 904 592 ÷ 2 = 250 111 065 952 296 + 0;
  • 250 111 065 952 296 ÷ 2 = 125 055 532 976 148 + 0;
  • 125 055 532 976 148 ÷ 2 = 62 527 766 488 074 + 0;
  • 62 527 766 488 074 ÷ 2 = 31 263 883 244 037 + 0;
  • 31 263 883 244 037 ÷ 2 = 15 631 941 622 018 + 1;
  • 15 631 941 622 018 ÷ 2 = 7 815 970 811 009 + 0;
  • 7 815 970 811 009 ÷ 2 = 3 907 985 405 504 + 1;
  • 3 907 985 405 504 ÷ 2 = 1 953 992 702 752 + 0;
  • 1 953 992 702 752 ÷ 2 = 976 996 351 376 + 0;
  • 976 996 351 376 ÷ 2 = 488 498 175 688 + 0;
  • 488 498 175 688 ÷ 2 = 244 249 087 844 + 0;
  • 244 249 087 844 ÷ 2 = 122 124 543 922 + 0;
  • 122 124 543 922 ÷ 2 = 61 062 271 961 + 0;
  • 61 062 271 961 ÷ 2 = 30 531 135 980 + 1;
  • 30 531 135 980 ÷ 2 = 15 265 567 990 + 0;
  • 15 265 567 990 ÷ 2 = 7 632 783 995 + 0;
  • 7 632 783 995 ÷ 2 = 3 816 391 997 + 1;
  • 3 816 391 997 ÷ 2 = 1 908 195 998 + 1;
  • 1 908 195 998 ÷ 2 = 954 097 999 + 0;
  • 954 097 999 ÷ 2 = 477 048 999 + 1;
  • 477 048 999 ÷ 2 = 238 524 499 + 1;
  • 238 524 499 ÷ 2 = 119 262 249 + 1;
  • 119 262 249 ÷ 2 = 59 631 124 + 1;
  • 59 631 124 ÷ 2 = 29 815 562 + 0;
  • 29 815 562 ÷ 2 = 14 907 781 + 0;
  • 14 907 781 ÷ 2 = 7 453 890 + 1;
  • 7 453 890 ÷ 2 = 3 726 945 + 0;
  • 3 726 945 ÷ 2 = 1 863 472 + 1;
  • 1 863 472 ÷ 2 = 931 736 + 0;
  • 931 736 ÷ 2 = 465 868 + 0;
  • 465 868 ÷ 2 = 232 934 + 0;
  • 232 934 ÷ 2 = 116 467 + 0;
  • 116 467 ÷ 2 = 58 233 + 1;
  • 58 233 ÷ 2 = 29 116 + 1;
  • 29 116 ÷ 2 = 14 558 + 0;
  • 14 558 ÷ 2 = 7 279 + 0;
  • 7 279 ÷ 2 = 3 639 + 1;
  • 3 639 ÷ 2 = 1 819 + 1;
  • 1 819 ÷ 2 = 909 + 1;
  • 909 ÷ 2 = 454 + 1;
  • 454 ÷ 2 = 227 + 0;
  • 227 ÷ 2 = 113 + 1;
  • 113 ÷ 2 = 56 + 1;
  • 56 ÷ 2 = 28 + 0;
  • 28 ÷ 2 = 14 + 0;
  • 14 ÷ 2 = 7 + 0;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 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.


1 100 000 100 999 999 999 999 999 988(10) =


11 1000 1101 1110 0110 0001 0100 1111 0110 0100 0000 1010 0001 1111 0000 1101 0110 0111 0011 1111 1111 1111 0100(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 89 positions to the left, so that only one non zero digit remains to the left of it:


1 100 000 100 999 999 999 999 999 988(10) =


11 1000 1101 1110 0110 0001 0100 1111 0110 0100 0000 1010 0001 1111 0000 1101 0110 0111 0011 1111 1111 1111 0100(2) =


11 1000 1101 1110 0110 0001 0100 1111 0110 0100 0000 1010 0001 1111 0000 1101 0110 0111 0011 1111 1111 1111 0100(2) × 20 =


1.1100 0110 1111 0011 0000 1010 0111 1011 0010 0000 0101 0000 1111 1000 0110 1011 0011 1001 1111 1111 1111 1010 0(2) × 289


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): 89


Mantissa (not normalized):
1.1100 0110 1111 0011 0000 1010 0111 1011 0010 0000 0101 0000 1111 1000 0110 1011 0011 1001 1111 1111 1111 1010 0


5. Adjust the exponent.

Use the 8 bit excess/bias notation:


Exponent (adjusted) =


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


89 + 2(8-1) - 1 =


(89 + 127)(10) =


216(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;
  • 216 ÷ 2 = 108 + 0;
  • 108 ÷ 2 = 54 + 0;
  • 54 ÷ 2 = 27 + 0;
  • 27 ÷ 2 = 13 + 1;
  • 13 ÷ 2 = 6 + 1;
  • 6 ÷ 2 = 3 + 0;
  • 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) =


216(10) =


1101 1000(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. 110 0011 0111 1001 1000 0101 00 1111 0110 0100 0000 1010 0001 1111 0000 1101 0110 0111 0011 1111 1111 1111 0100 =


110 0011 0111 1001 1000 0101


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) =
1101 1000


Mantissa (23 bits) =
110 0011 0111 1001 1000 0101


The base ten decimal number 1 100 000 100 999 999 999 999 999 988 converted and written in 32 bit single precision IEEE 754 binary floating point representation:
0 - 1101 1000 - 110 0011 0111 1001 1000 0101

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