Convert 496.573 from base 10 to base 2
To convert 496.573 from base 10 to base 2, you need to separately convert the integer and fractional parts.
Integer Part (496):
To convert the integer part, continuously divide the number by 2 and write down the remainders until the quotient becomes 0. Then, the binary number is obtained by writing the remainders in reverse order.
Let's go step-by-step:
496 ÷ 2 = 248 remainder 0
248 ÷ 2 = 124 remainder 0
124 ÷ 2 = 62 remainder 0
62 ÷ 2 = 31 remainder 1
31 ÷ 2 = 15 remainder 1
15 ÷ 2 = 7 remainder 1
7 ÷ 2 = 3 remainder 1
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
In reverse order, the remainders are 111110000.
Therefore, the binary representation of the integer part 496 in base 10 is 111110000 in base 2.
Fractional Part (0.573):
To convert the fractional part, continuously multiply the number by 2 and write down the integer part until the fractional part becomes 0. Then, the binary number is obtained by writing the integer parts in order.
Let's go step-by-step:
0.573 × 2 = 1.146 → 1
0.146 × 2 = 0.292 → 0
0.292 × 2 = 0.584 → 0
0.584 × 2 = 1.168 → 1
0.168 × 2 = 0.336 → 0
0.336 × 2 = 0.672 → 0
0.672 × 2 = 1.344 → 1
0.344 × 2 = 0.688 → 0
In order, the integer parts are 10010010.
Therefore, the binary representation of the fractional part 0.573 in base 10 is 10010010 in base 2.
Putting both parts together, we get:
496.573 in base 10 is equal to 111110000.10010010 in base 2.
To convert a number from base 10 to base 2 (binary), you can use a method called "long division."
Step 1: Start by dividing the base 10 number (496.573) by 2.
496 ÷ 2 = 248 remainder 0
Step 2: Write down the remainder (0) on the right side of the division line.
Step 3: Continue dividing the quotient from the previous step (248) by 2.
248 ÷ 2 = 124 remainder 0
Step 4: Again, write down the remainder (0) on the right side.
Step 5: Repeat the process by dividing the new quotient (124) by 2.
124 ÷ 2 = 62 remainder 0
Step 6: Write down the remainder (0).
Step 7: Divide 62 by 2.
62 ÷ 2 = 31 remainder 0
Step 8: Continue writing down the remainders.
Step 9: Divide 31 by 2.
31 ÷ 2 = 15 remainder 1
Step 10: Write down the remainder (1).
Step 11: Divide 15 by 2.
15 ÷ 2 = 7 remainder 1
Step 12: Write down the remainder (1).
Step 13: Divide 7 by 2.
7 ÷ 2 = 3 remainder 1
Step 14: Write down the remainder (1).
Step 15: Finally, divide 3 by 2.
3 ÷ 2 = 1 remainder 1
Step 16: Write down the remainder (1).
Step 17: Since the last quotient is 1, write it down.
Step 18: Write down all the remainders in reverse order, starting from the bottom.
The remainders from Step 16 to Step 1 are: 111110000.
Therefore, the base 10 number 496.573 can be expressed as 111110000 in base 2 (binary).
To convert a number from base 10 to base 2 (binary), you can use the following steps:
Step 1: Divide the number by 2
Step 2: Keep track of the remainder (0 or 1)
Step 3: Repeat steps 1 and 2 with the quotient obtained in step 1 until the quotient becomes 0
Step 4: Write down the remainders from the last step (in reverse order) to get the binary representation
Let's apply these steps to convert 496.573 from base 10 to base 2:
Step 1: Divide 496 (integer part) by 2: 496 ÷ 2 = 248, remainder 0
Step 2: Divide 248 by 2: 248 ÷ 2 = 124, remainder 0
Step 3: Divide 124 by 2: 124 ÷ 2 = 62, remainder 0
Step 4: Divide 62 by 2: 62 ÷ 2 = 31, remainder 1
Step 5: Divide 31 by 2: 31 ÷ 2 = 15, remainder 1
Step 6: Divide 15 by 2: 15 ÷ 2 = 7, remainder 1
Step 7: Divide 7 by 2: 7 ÷ 2 = 3, remainder 1
Step 8: Divide 3 by 2: 3 ÷ 2 = 1, remainder 1
Step 9: Divide 1 by 2: 1 ÷ 2 = 0, remainder 1
Now, let's write down the remainders we obtained in reverse order: 111110000
Therefore, the binary representation of 496 in base 10 is 111110000. Now, let's deal with the fractional part.
Step 1: Multiply the fractional part (0.573) by 2
Step 2: Keep track of the whole number part of the result as the binary digit
Step 3: Repeat steps 1 and 2 with the fractional part until you obtain the desired precision or the fractional part becomes 0
Let's apply these steps to convert the fractional part of 573 (0.573) from base 10 to base 2:
Step 1: Multiply 0.573 by 2: 0.573 * 2 = 1.146
Step 2: Write down the whole number part (1) as the first binary digit
Step 3: Multiply the fractional part (0.146) by 2: 0.146 * 2 = 0.292
Step 4: Write down the whole number part (0) as the next binary digit
Step 5: Repeat step 3 with the new fractional part (0.292 * 2 = 0.584)
Step 6: Write down the whole number part (0) as the next binary digit
Step 7: Repeat step 3 with the new fractional part (0.584 * 2 = 1.168)
Step 8: Write down the whole number part (1) as the next binary digit
Step 9: Repeat step 3 with the new fractional part (0.168 * 2 = 0.336)
Step 10: Write down the whole number part (0) as the next binary digit
Repeat this process until you achieve the desired precision or the fractional part becomes zero. In this case, the pattern repeats after a certain point, so let's stop at 10 binary digits.
Thus, the binary representation of 0.573 in base 10 is approximately 0.1001000111. Combining the integer and fractional parts, the final binary representation is:
496.573 (base 10) = 111110000.1001000111 (base 2)