Answer :
Certainly! Let's convert the base 10 number [tex]\(9_{10}\)[/tex] into its equivalent in base 2.
1. Understand the Conversion Process:
- To convert a number from base 10 to base 2, we need to express the number as a sum of powers of 2.
2. Identify the Powers of 2:
- We start by identifying the largest power of 2 less than or equal to 9. The powers of 2 are: [tex]\(2^0 = 1\)[/tex], [tex]\(2^1 = 2\)[/tex], [tex]\(2^2 = 4\)[/tex], [tex]\(2^3 = 8\)[/tex], [tex]\(2^4 = 16\)[/tex], and so on.
- The largest power of 2 that is less than or equal to 9 is [tex]\(2^3 = 8\)[/tex].
3. Subtract and Continue:
- Subtract this largest power of 2 from 9:
[tex]\[ 9 - 8 = 1 \][/tex]
- Now, we need to express the remainder (1) as a sum of powers of 2. The remainder 1 itself is a power of 2, specifically [tex]\(2^0 = 1\)[/tex].
4. Combine the Powers:
- We combine the powers of 2 used:
[tex]\[ 9 = 2^3 + 2^0 \][/tex]
5. Construct the Binary Representation:
- In binary notation, [tex]\(2^3\)[/tex] is represented as a "1" in the [tex]\(2^3\)[/tex] place (i.e., the fourth position from the right).
- [tex]\(2^0\)[/tex] is represented as a "1" in the [tex]\(2^0\)[/tex] place (i.e., the first position from the right).
- All other positional values are zero.
- Therefore, you arrange these as follows:
[tex]\[ 1001_2 \][/tex]
So, converting [tex]\(9_{10}\)[/tex] to base 2 gives [tex]\(1001_2\)[/tex]. Hence, the correct answer is:
b. [tex]\(1001_2\)[/tex].
1. Understand the Conversion Process:
- To convert a number from base 10 to base 2, we need to express the number as a sum of powers of 2.
2. Identify the Powers of 2:
- We start by identifying the largest power of 2 less than or equal to 9. The powers of 2 are: [tex]\(2^0 = 1\)[/tex], [tex]\(2^1 = 2\)[/tex], [tex]\(2^2 = 4\)[/tex], [tex]\(2^3 = 8\)[/tex], [tex]\(2^4 = 16\)[/tex], and so on.
- The largest power of 2 that is less than or equal to 9 is [tex]\(2^3 = 8\)[/tex].
3. Subtract and Continue:
- Subtract this largest power of 2 from 9:
[tex]\[ 9 - 8 = 1 \][/tex]
- Now, we need to express the remainder (1) as a sum of powers of 2. The remainder 1 itself is a power of 2, specifically [tex]\(2^0 = 1\)[/tex].
4. Combine the Powers:
- We combine the powers of 2 used:
[tex]\[ 9 = 2^3 + 2^0 \][/tex]
5. Construct the Binary Representation:
- In binary notation, [tex]\(2^3\)[/tex] is represented as a "1" in the [tex]\(2^3\)[/tex] place (i.e., the fourth position from the right).
- [tex]\(2^0\)[/tex] is represented as a "1" in the [tex]\(2^0\)[/tex] place (i.e., the first position from the right).
- All other positional values are zero.
- Therefore, you arrange these as follows:
[tex]\[ 1001_2 \][/tex]
So, converting [tex]\(9_{10}\)[/tex] to base 2 gives [tex]\(1001_2\)[/tex]. Hence, the correct answer is:
b. [tex]\(1001_2\)[/tex].
