Find the numeral preceding and succeeding each of the following numerals. Use \( T \) and \( E \) as digits for 10 and 11 . a. \( T 50_{\text {eleven }} \) b. \( 100000_{\text {twelve }} \) c. \( 111_{\text {two }} \)

To find the numerals preceding and succeeding each of the given numerals, we first need to convert them to decimal (base 10) and then find the corresponding numerals in their respective bases. ### a. \( T 50_{\text{eleven}} \) 1. **Convert \( T 50_{11} \) to decimal:** - \( T \) represents 10 in base 11. - The numeral \( T 50_{11} \) can be expressed as: \[ T 50_{11} = 10 \cdot 11^2 + 5 \cdot 11^1 + 0 \cdot 11^0 = 10 \cdot 121 + 5 \cdot 11 + 0 = 1210 + 55 + 0 = 1265_{10} \] 2. **Find the preceding and succeeding numerals in base 11:** - Preceding: \( 1265 - 1 = 1264_{10} \) - Succeeding: \( 1265 + 1 = 1266_{10} \) 3. **Convert back to base 11:** - **For \( 1264_{10} \):** - \( 1264 \div 11 = 115 \) remainder \( 9 \) - \( 115 \div 11 = 10 \) remainder \( 5 \) - \( 10 \div 11 = 0 \) remainder \( 10 \) (which is \( T \)) - So, \( 1264_{10} = T 59_{11} \) - **For \( 1266_{10} \):** - \( 1266 \div 11 = 115 \) remainder \( 10 \) (which is \( T \)) - \( 115 \div 11 = 10 \) remainder \( 5 \) - \( 10 \div 11 = 0 \) remainder \( 10 \) (which is \( T \)) - So, \( 1266_{10} = T 5T_{11} \) Thus, the results for part (a) are: - Preceding: \( T 59_{11} \) - Succeeding: \( T 5T_{11} \) ### b. \( 100000_{\text{twelve}} \) 1. **Convert \( 100000_{12} \) to decimal:** \[ 100000_{12} = 1 \cdot 12^5 + 0 \cdot 12^4 + 0 \cdot 12^3 + 0 \cdot 12^2 + 0 \cdot 12^1 + 0 \cdot 12^0 = 1 \cdot 248832 = 248832_{10} \] 2. **Find the preceding and succeeding numerals in base 12:** - Preceding: \( 248832 - 1 = 248831_{10} \) - Succeeding: \( 248832 + 1 = 248833_{10} \) 3. **Convert back to base 12:** - **For \( 248831_{10} \):** - \( 248831 \div 12 = 20736 \) remainder \( 9 \) - \( 20736 \div 12 = 1728 \) remainder \( 0 \) - \( 1728 \div 12 = 144 \) remainder \( 0 \) - \( 144 \div 12 = 12 \) remainder \( 0 \) - \( 12 \div 12 = 1 \) remainder \( 0 \) - \( 1 \div 12 = 0 \) remainder \( 1 \) - So, \( 248831_{10} = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 = 100000_{12} - 1 =