To convert the binary number [tex]\(101010\)[/tex] to its decimal equivalent, follow these steps:
1. Understand Positions: Each digit in a binary number represents a power of 2, starting from the rightmost digit which is [tex]\(2^0\)[/tex].
2. Write the Powers of 2: Write down the power of 2 corresponding to each position in the binary number [tex]\(101010\)[/tex]:
[tex]\[
\begin{array}{cccccc}
2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0 \\
\end{array}
\][/tex]
3. Assign Binary Digits: Place each corresponding binary digit below its power of 2:
[tex]\[
\begin{array}{cccccc}
1 & 0 & 1 & 0 & 1 & 0 \\
2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0 \\
\end{array}
\][/tex]
4. Multiply Each Digit by Its Corresponding Power of 2: Perform the multiplication:
[tex]\[
1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0
\][/tex]
5. Calculate Each Term:
[tex]\[
1 \cdot 32 = 32 \\
0 \cdot 16 = 0 \\
1 \cdot 8 = 8 \\
0 \cdot 4 = 0 \\
1 \cdot 2 = 2 \\
0 \cdot 1 = 0 \\
\][/tex]
6. Sum All Products: Add all the terms together:
[tex]\[
32 + 0 + 8 + 0 + 2 + 0 = 42
\][/tex]
Therefore, the decimal equivalent of the binary number [tex]\(101010\)[/tex] is [tex]\(42\)[/tex].
