The image shows the process of
converting a binary number to its decimal equivalent using the
doubling method, also known as the
"double-and-add" algorithm.
---
🔢 Given Binary Number:
$$
(10101101)_2
$$
We are converting this binary number to decimal, and the final result shown is:
$$
(173)_{10}
$$
Let’s go step-by-step through the method used in the image and explain how it works.
---
✔ Method: Double-and-Add (Left-to-Right)
This method processes the binary digits from
left to right, starting with an initial value of 0. For each digit:
1.
Double the current value.
2.
Add the current binary digit (0 or 1).
This mimics the positional value system of binary numbers but builds the decimal value iteratively.
---
🧮 Step-by-Step Breakdown:
We'll go digit by digit from left to right, showing the calculation at each step.
| Binary Digit | Current Value | Operation | New Value |
|--------------|----------------|-----------|-----------|
| 1 | 0 | (0 × 2) + 1 = 1 |
1 |
| 0 | 1 | (1 × 2) + 0 = 2 |
2 |
| 1 | 2 | (2 × 2) + 1 = 5 |
5 |
| 0 | 5 | (5 × 2) + 0 = 10 |
10 |
| 1 | 10 | (10 × 2) + 1 = 21 |
21 |
| 1 | 21 | (21 × 2) + 1 = 43 |
43 |
| 0 | 43 | (43 × 2) + 0 = 86 |
86 |
| 1 | 86 | (86 × 2) + 1 = 173 |
173 |
---
✔ Final Result:
$$
(10101101)_2 = (173)_{10}
$$
---
📌 Why This Works:
Each time we double the current value, we're shifting left in binary (equivalent to multiplying by 2), and then adding the next bit accounts for its contribution.
This is mathematically equivalent to computing:
$$
1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0
$$
$$
= 128 + 0 + 32 + 0 + 8 + 4 + 0 + 1 = 173
$$
✔ So both methods agree.
---
🎯 Summary:
- The
double-and-add method is an efficient way to convert binary to decimal.
- It processes bits from
left to right.
- At each step: `new_value = (current_value × 2) + current_bit`
- The final result is
173 in decimal.
✔️ Therefore, the solution is correct:
$$
\boxed{(10101101)_2 = (173)_{10}}
$$
Parent Tip: Review the logic above to help your child master the concept of binary to decimal worksheet.
