Answer :
Sure, let's convert the decimal number 122 to its binary form step by step:
1. Find the largest power of 2 less than or equal to 122:
- 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], [tex]\(2^5=32\)[/tex], [tex]\(2^6=64\)[/tex], [tex]\(2^7=128\)[/tex].
- The largest power of 2 less than or equal to 122 is [tex]\(2^6 = 64\)[/tex].
2. Start with 122 and subtract 64:
- [tex]\(122 - 64 = 58\)[/tex]
3. Find the largest power of 2 less than or equal to 58:
- Powers of 2 less than or equal to 58 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], [tex]\(2^5=32\)[/tex], [tex]\(2^6=64\)[/tex]
- The largest power of 2 less than or equal to 58 is [tex]\(2^5 = 32\)[/tex].
4. Subtract 32 from 58:
- [tex]\(58 - 32 = 26\)[/tex]
5. Find the largest power of 2 less than or equal to 26:
- Powers of 2 less than or equal to 26 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], [tex]\(2^5=32\)[/tex]
- The largest power of 2 less than or equal to 26 is [tex]\(2^4 = 16\)[/tex].
6. Subtract 16 from 26:
- [tex]\(26 - 16 = 10\)[/tex]
7. Find the largest power of 2 less than or equal to 10:
- Powers of 2 less than or equal to 10 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].
- The largest power of 2 less than or equal to 10 is [tex]\(2^3 = 8\)[/tex].
8. Subtract 8 from 10:
- [tex]\(10 - 8 = 2\)[/tex]
9. Find the largest power of 2 less than or equal to 2:
- Powers of 2 less than or equal to 2 are [tex]\(2^0=1\)[/tex], [tex]\(2^1=2\)[/tex]
- The largest power of 2 less than or equal to 2 is [tex]\(2^1 = 2\)[/tex].
10. Subtract 2 from 2:
- [tex]\(2 - 2 = 0\)[/tex]
Now, we have identified the powers of 2 which sum up to 122: [tex]\(2^6\)[/tex], [tex]\(2^5\)[/tex], [tex]\(2^4\)[/tex], [tex]\(2^3\)[/tex], and [tex]\(2^1\)[/tex].
We write a '1' in each position corresponding to these powers of 2 (from [tex]\(2^6\)[/tex] to [tex]\(2^0\)[/tex]) and '0' in other positions.
So, we get the binary form of 122:
- [tex]\(2^6\)[/tex] (64) : 1
- [tex]\(2^5\)[/tex] (32) : 1
- [tex]\(2^4\)[/tex] (16) : 1
- [tex]\(2^3\)[/tex] (8) : 1
- [tex]\(2^2\)[/tex] (4) : 0
- [tex]\(2^1\)[/tex] (2) : 1
- [tex]\(2^0\)[/tex] (1) : 0
The binary representation of 122 is:
[tex]\[ 1111010 \][/tex]
So, the binary form of 122 is [tex]\(1111010\)[/tex].
1. Find the largest power of 2 less than or equal to 122:
- 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], [tex]\(2^5=32\)[/tex], [tex]\(2^6=64\)[/tex], [tex]\(2^7=128\)[/tex].
- The largest power of 2 less than or equal to 122 is [tex]\(2^6 = 64\)[/tex].
2. Start with 122 and subtract 64:
- [tex]\(122 - 64 = 58\)[/tex]
3. Find the largest power of 2 less than or equal to 58:
- Powers of 2 less than or equal to 58 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], [tex]\(2^5=32\)[/tex], [tex]\(2^6=64\)[/tex]
- The largest power of 2 less than or equal to 58 is [tex]\(2^5 = 32\)[/tex].
4. Subtract 32 from 58:
- [tex]\(58 - 32 = 26\)[/tex]
5. Find the largest power of 2 less than or equal to 26:
- Powers of 2 less than or equal to 26 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], [tex]\(2^5=32\)[/tex]
- The largest power of 2 less than or equal to 26 is [tex]\(2^4 = 16\)[/tex].
6. Subtract 16 from 26:
- [tex]\(26 - 16 = 10\)[/tex]
7. Find the largest power of 2 less than or equal to 10:
- Powers of 2 less than or equal to 10 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].
- The largest power of 2 less than or equal to 10 is [tex]\(2^3 = 8\)[/tex].
8. Subtract 8 from 10:
- [tex]\(10 - 8 = 2\)[/tex]
9. Find the largest power of 2 less than or equal to 2:
- Powers of 2 less than or equal to 2 are [tex]\(2^0=1\)[/tex], [tex]\(2^1=2\)[/tex]
- The largest power of 2 less than or equal to 2 is [tex]\(2^1 = 2\)[/tex].
10. Subtract 2 from 2:
- [tex]\(2 - 2 = 0\)[/tex]
Now, we have identified the powers of 2 which sum up to 122: [tex]\(2^6\)[/tex], [tex]\(2^5\)[/tex], [tex]\(2^4\)[/tex], [tex]\(2^3\)[/tex], and [tex]\(2^1\)[/tex].
We write a '1' in each position corresponding to these powers of 2 (from [tex]\(2^6\)[/tex] to [tex]\(2^0\)[/tex]) and '0' in other positions.
So, we get the binary form of 122:
- [tex]\(2^6\)[/tex] (64) : 1
- [tex]\(2^5\)[/tex] (32) : 1
- [tex]\(2^4\)[/tex] (16) : 1
- [tex]\(2^3\)[/tex] (8) : 1
- [tex]\(2^2\)[/tex] (4) : 0
- [tex]\(2^1\)[/tex] (2) : 1
- [tex]\(2^0\)[/tex] (1) : 0
The binary representation of 122 is:
[tex]\[ 1111010 \][/tex]
So, the binary form of 122 is [tex]\(1111010\)[/tex].
