If a language of natural numbers has binary vocabulary of 0 and 1, then which one of the following strings represents the natural number 7 ?

Options

A. 11
B. 101
C. 110
D. 111

Answer

To find the binary representation of the natural number 7, we can follow these steps:

### Step 1: Understand the binary system
The binary system uses two digits: 0 and 1. Each digit represents a power of 2, starting from the rightmost digit (which is 2^0).

### Step 2: Determine the powers of 2 needed for 7
We need to express the number 7 as a sum of powers of 2:
- The powers of 2 we will consider are:
- 2^2 = 4
- 2^1 = 2
- 2^0 = 1

### Step 3: Find the combination of powers of 2 that sum to 7
We can find the combination of these powers:
- 4 (2^2) + 2 (2^1) + 1 (2^0) = 7

### Step 4: Write the binary representation
Since we are using 3 bits (for 2^2, 2^1, and 2^0):
- We need to place a '1' for each power of 2 that we are using and a '0' for those we are not.
- For 7, we use all three powers:
- 2^2 (4) → 1
- 2^1 (2) → 1
- 2^0 (1) → 1
- Therefore, the binary representation of 7 is 111.

### Step 5: Match with the given options
Now, we need to check which option corresponds to the binary representation 111. According to the question, it matches with Option D.

### Final Answer:
The binary representation of the natural number 7 is 111, which corresponds to Option D.

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