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Quizzes
Conversion From Binary To Decimal Numbers (Expansion Method)
Topic Content:
- Conversion From Binary To Decimal Numbers (Expansion Method)
- Locating Place Values
- Worked Examples
The expansion and place value method could be employed to do this type of conversion. This method could equally be used to convert from any other base number such as base 4, base 8, etc to decimal.
We use the expansion method to convert from binary back to decimal. In doing this, you have to expand the old number base.
Worked Example 1:
Convert 1012 to decimal
Solution
Step 1:
Locate Place Values.
Power Expansion entails multiplying out the number base using their base values.
- First, write the place values starting from the right-hand side.
- Write each digit under its place value.
- Multiply each digit by its base raised to the corresponding place value.
(i.e. baseplace value, Note: for this question, it will be 2place value) - Add up the products. The answer will be the decimal number in base ten.
Students often find it difficult to locate the place values. The easiest way is to start from the last digit to the first digit. The last Digit always starts with a 0, the digit that follows a 1, and so on. (0, 1 2, 3, etc)
Step 2:
Then write down the base raised to the place value.
| Place value | 22 | 21 | 20 |
| Binary Number | 1 | 0 | 1 |
Step 3:
Multiply each digit by its base raised to the corresponding place value.
1012 = (1 × 22) + (0 × 21) + (1 × 20)
1012 = (1 × 4) + (0 × 1) + (1 × 1)
1012 = 4 + 0 + 1
= 510
Answer: 1012 = 510
Remember: that any number raised to the power 0 is 1, Except 0. e.g. 1000 = 1
Worked Example 2:
Convert 101112 to decimal
Solution
Step 1:
Then write down the base raised to the place value.
| Place value | 24 | 23 | 22 | 21 | 20 |
| Binary Number | 1 | 0 | 1 | 1 | 1 |
Step 2:
Multiply each digit by its base raised to the corresponding place value.
101112 = (1 × 24) + (0 × 23) + (1 × 22) + (1 × 21) + (1 × 20)
1012 = (1 × 16) + (0 × 8) + (1 × 4) + (1 × 2) + (1 × 1)
1012 = 16 + 0 + 4 + 2 + 1
= 2310
Answer: 101112 = 2310
Worked Example 3:
Convert 00101 to decimal
Solution
Step 1:
Then write down the base raised to the place value.
| Place value | 24 | 23 | 22 | 21 | 20 |
| Binary Number | 0 | 0 | 1 | 0 | 1 |
Step 2:
Multiply each digit by its base raised to the corresponding place value.
001012 = (0 × 24) + (0 × 23) + (1 × 22) + (0 × 21) + (1 × 20)
001012 = (0 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (1 × 1)
001012 = 0 + 0 + 4 + 0 + 1
= 510
Answer: 001012 = 510
Note: The first 0 at the beginning of any number has no effect on the actual number (i.e. 00101 is the same as 101)
Class Work:
Convert the following numbers from binary to decimal.
(a) 101002
(b) 100002
(c) 110012