In a base-3 numeral system, what is the representation of the decimal number 5?

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Question: In a base-3 numeral system, what is the representation of the decimal number 5? (2023)

Options:

Correct Answer: 21

Exam Year: 2023

Solution:

The decimal number 5 is represented as 21 in base-3 (2*3^1 + 1*3^0 = 5).

In a base-3 numeral system, what is the representation of the decimal number 5? (2023)

In a base-3 numeral system, what is the representation of the decimal number 5? (2023)

Step 1: Understand that we need to convert the decimal number 5 into base-3.
Step 2: Identify the largest power of 3 that is less than or equal to 5. The powers of 3 are 1 (3^0 = 1), 3 (3^1 = 3), and 9 (3^2 = 9). The largest power of 3 less than 5 is 3 (3^1).
Step 3: Determine how many times 3 fits into 5. 3 fits into 5 one time. So, we write down 1 for the coefficient of 3^1.
Step 4: Subtract the value of 3 from 5. 5 - 3 = 2.
Step 5: Now, we need to represent the remaining value (2) in base-3. The largest power of 3 less than or equal to 2 is 1 (3^0 = 1).
Step 6: Determine how many times 1 fits into 2. 1 fits into 2 two times. So, we write down 2 for the coefficient of 3^0.
Step 7: Combine the coefficients from steps 3 and 6. We have 1 for 3^1 and 2 for 3^0, which gives us 21 in base-3.

Base Conversion – Understanding how to convert a decimal number to a different base, specifically base-3 in this case.
Place Value System – Recognizing the significance of each digit's position in a numeral system, such as powers of the base.