The base eleven numeration system has the following eleven symbols: {0,1,2,3,4,5,6,7,8,9,A}, where A=10. Convert each of the following base 11 numerals into base ten numerals: 142 eleven = Aeieven ?= Aeleven ?= 1 Aeleven ?=
The correct answer and explanation is :
To convert numbers from base 11 to base 10, we use positional notation, where each digit’s value depends on its position and the base’s power. In base 11, the digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A}, with ‘A’ representing 10.
Conversion Steps:
- Identify each digit and its position: Starting from the right (least significant digit), positions are indexed from 0 upwards.
- Calculate each digit’s contribution: Multiply the digit by 11 raised to the power of its position index.
- Sum all contributions: The total is the equivalent base 10 value.
Examples:
- 142₁₁:
- Digits: 1 (position 2), 4 (position 1), 2 (position 0)
- Calculations:
- 1 × 11² = 1 × 121 = 121
- 4 × 11¹ = 4 × 11 = 44
- 2 × 11⁰ = 2 × 1 = 2
- Sum: 121 + 44 + 2 = 167
- Result: 142₁₁ = 167₁₀
- A1₁₁:
- Digits: A (position 1), 1 (position 0)
- Calculations:
- A = 10
- 10 × 11¹ = 10 × 11 = 110
- 1 × 11⁰ = 1 × 1 = 1
- Sum: 110 + 1 = 111
- Result: A1₁₁ = 111₁₀
- 1A1₁₁:
- Digits: 1 (position 2), A (position 1), 1 (position 0)
- Calculations:
- 1 × 11² = 1 × 121 = 121
- A = 10
- 10 × 11¹ = 10 × 11 = 110
- 1 × 11⁰ = 1 × 1 = 1
- Sum: 121 + 110 + 1 = 232
- Result: 1A1₁₁ = 232₁₀
Explanation:
In positional notation, the value of a digit is determined by its face value multiplied by the base raised to the power of its position index. For base 11, each position represents a power of 11:
- Position 0: 11⁰ = 1
- Position 1: 11¹ = 11
- Position 2: 11² = 121
- And so on.
By multiplying each digit by the corresponding power of 11 and summing the results, we obtain the number’s value in base 10.