To convert the decimal number [tex]\( 122_{10} \)[/tex] to its binary representation and to understand the result step-by-step, follow this process:
1. Division by 2 Method:
- We repeatedly divide the decimal number by 2 and record the quotient and the remainder.
- The remainder will be either 0 or 1 and it forms the binary digits.
- The binary number is formed by the remainders read from bottom to top.
2. Conversion Process:
- Start with the decimal number 122 and divide it by 2:
[tex]\( 122 \div 2 = 61 \)[/tex] with a remainder of 0.
- Take the quotient from the previous division (61) and divide it by 2:
[tex]\( 61 \div 2 = 30 \)[/tex] with a remainder of 1.
- Take the quotient from the previous division (30) and divide it by 2:
[tex]\( 30 \div 2 = 15 \)[/tex] with a remainder of 0.
- Take the quotient from the previous division (15) and divide it by 2:
[tex]\( 15 \div 2 = 7 \)[/tex] with a remainder of 1.
- Take the quotient from the previous division (7) and divide it by 2:
[tex]\( 7 \div 2 = 3 \)[/tex] with a remainder of 1.
- Take the quotient from the previous division (3) and divide it by 2:
[tex]\( 3 \div 2 = 1 \)[/tex] with a remainder of 1.
- Take the quotient from the previous division (1) and divide it by 2:
[tex]\( 1 \div 2 = 0 \)[/tex] with a remainder of 1.
3. Compiling the Binary Number:
- We compile the binary number from the remainders, from bottom to top: [tex]\( 1111010 \)[/tex].
4. Conclusion:
- Therefore, the binary representation of [tex]\( 122_{10} \)[/tex] is [tex]\( 1111010_2 \)[/tex].
