To convert the binary number 1000101 to its decimal equivalent, you calculate the sum of each digit multiplied by its corresponding power of 2. The decimal equivalent of 1000101 binary is 69.
Understanding Binary and Decimal Systems
The binary system (base-2) uses only two digits: 0 and 1. It's the fundamental language of computers. The decimal system (base-10), which we use daily, utilizes ten digits (0-9). Converting from binary to decimal involves understanding the place value of each binary digit (bit).
Each position in a binary number represents a power of 2, starting from $2^0$ (which is 1) for the rightmost digit, $2^1$ (2) for the next, $2^2$ (4) for the one after that, and so on, increasing as you move left.
Step-by-Step Conversion of 1000101
Here’s how to convert 1000101 from binary to decimal:
-
Identify Place Values: Write down the binary number and assign the corresponding power of 2 to each digit, starting from the rightmost digit as $2^0$.
- For 1000101:
- Rightmost
1is at position $2^0$ (1) - Next
0is at position $2^1$ (2) - Next
1is at position $2^2$ (4) - Next
0is at position $2^3$ (8) - Next
0is at position $2^4$ (16) - Next
0is at position $2^5$ (32) - Leftmost
1is at position $2^6$ (64)
- Rightmost
- For 1000101:
-
Multiply and Sum: Multiply each binary digit by its corresponding place value (power of 2). Then, sum up the results.
Binary Digit Place Value (Power of 2) Calculation Decimal Value 1 $2^6$ (64) $1 \times 64$ 64 0 $2^5$ (32) $0 \times 32$ 0 0 $2^4$ (16) $0 \times 16$ 0 0 $2^3$ (8) $0 \times 8$ 0 1 $2^2$ (4) $1 \times 4$ 4 0 $2^1$ (2) $0 \times 2$ 0 1 $2^0$ (1) $1 \times 1$ 1 Total 69 -
Final Result: Add the decimal values from the calculation.
$64 + 0 + 0 + 0 + 4 + 0 + 1 = 69$
Therefore, the binary number 1000101 is equal to 69 in decimal.
Practical Application
Understanding binary-to-decimal conversion is fundamental in computer science and digital electronics. It allows us to interpret the numbers that computers process, bridging the gap between machine-level operations and human-readable values. This conversion process is used extensively in networking, data storage, and programming.
