What is the binary equivalent of 75 using the weights method?
Understand the Problem
The question is asking how to convert the decimal number 75 into its binary equivalent using the weights method. This method involves expressing the number in terms of powers of 2.
Answer
$1001011$
Answer for screen readers
The binary equivalent of the decimal number 75 is $1001011$.
Steps to Solve
- Identify the largest power of 2 less than 75
Start by listing the powers of 2 until you exceed 75. The powers of 2 are: $2^0 = 1$, $2^1 = 2$, $2^2 = 4$, $2^3 = 8$, $2^4 = 16$, $2^5 = 32$, $2^6 = 64$, and $2^7 = 128$.
The largest power of 2 less than 75 is $2^6 = 64$.
- Subtract and mark the power of 2 used
Subtract 64 from 75: $$ 75 - 64 = 11 $$ Now mark $2^6$ (which represents 64) as being used.
- Identify the largest power of 2 less than the remaining number (11)
Next, determine the largest power of 2 less than 11. The relevant powers of 2 are: $2^0 = 1$, $2^1 = 2$, $2^2 = 4$, and $2^3 = 8$.
The largest power of 2 less than 11 is $2^3 = 8$.
- Subtract and mark the power of 2 used
Subtract 8 from 11: $$ 11 - 8 = 3 $$ Now mark $2^3$ (which represents 8) as being used.
- Identify the largest power of 2 less than the remaining number (3)
For the remaining number (3), the largest power of 2 is $2^1 = 2$.
- Subtract and mark the power of 2 used
Subtract 2 from 3: $$ 3 - 2 = 1 $$ Now mark $2^1$ (which represents 2) as being used.
- Identify the largest power of 2 less than the remaining number (1)
The largest power of 2 less than 1 is $2^0 = 1$.
- Subtract and mark the power of 2 used
Subtract 1 from 1: $$ 1 - 1 = 0 $$ Mark $2^0$ (which represents 1) as being used.
- Compile the powers of 2 used
Now, we have used the powers: $2^6$, $2^3$, $2^1$, and $2^0$.
This gives us: $$ 75 = 2^6 + 2^3 + 2^1 + 2^0 $$
- Write the binary equivalent
The powers used correspond to the binary digits. Each power of 2 corresponds to a digit in binary, starting from the right for $2^0$. The complete binary representation is $1001011$.
The binary equivalent of the decimal number 75 is $1001011$.
More Information
Converting decimal numbers to binary is useful in computer science and digital electronics since computers operate using binary numbers.
Tips
One common mistake is to forget to include all the powers of 2 used when writing the binary equivalent. Make sure to account for each power of 2 that contributes to the decimal number.
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