Question

write the first 10 counting numerals for each of the following bases.

a. base five
b. base eight

a. what are the first ten counting numerals in base five?
(use ascending order. use a comma to separate answers as needed.)

b. what are the first ten counting numerals in base eight?
(use ascending order. use a comma to separate answers as needed.)

Explanation:

Part a: Base Five

Step1: Understand Base Five Counting

In base five, digits range from 0 - 4, and we count up to 9 (decimal) and convert to base five. The first ten counting numerals (starting from 1) in base five:

• 1 (decimal 1) → $1_5$
• 2 (decimal 2) → $2_5$
• 3 (decimal 3) → $3_5$
• 4 (decimal 4) → $4_5$
• 5 (decimal 5) → $10_5$ (since $5 = 1\times5^1 + 0\times5^0$)
• 6 (decimal 6) → $11_5$ ( $6 = 1\times5 + 1$)
• 7 (decimal 7) → $12_5$ ( $7 = 1\times5 + 2$)
• 8 (decimal 8) → $13_5$ ( $8 = 1\times5 + 3$)
• 9 (decimal 9) → $14_5$ ( $9 = 1\times5 + 4$)
• 10 (decimal 10) → $20_5$ ( $10 = 2\times5 + 0$)

Step2: List in Ascending Order

The first ten counting numerals (1 to 10 in decimal) in base five are: 1, 2, 3, 4, 10, 11, 12, 13, 14, 20 (all in base five, written without subscript for simplicity here as per the question's comma-separated format).

Step1: Understand Base Eight Counting

In base eight, digits range from 0 - 7, and we count up to 9 (decimal) and convert to base eight. The first ten counting numerals (starting from 1) in base eight:

• 1 (decimal 1) → $1_8$
• 2 (decimal 2) → $2_8$
• 3 (decimal 3) → $3_8$
• 4 (decimal 4) → $4_8$
• 5 (decimal 5) → $5_8$
• 6 (decimal 6) → $6_8$
• 7 (decimal 7) → $7_8$
• 8 (decimal 8) → $10_8$ (since $8 = 1\times8^1 + 0\times8^0$)
• 9 (decimal 9) → $11_8$ ( $9 = 1\times8 + 1$)
• 10 (decimal 10) → $12_8$ ( $10 = 1\times8 + 2$)

Step2: List in Ascending Order

The first ten counting numerals (1 to 10 in decimal) in base eight are: 1, 2, 3, 4, 5, 6, 7, 10, 11, 12 (all in base eight, written without subscript for simplicity here as per the question's comma-separated format).

Answer:

(Part a):
1, 2, 3, 4, 10, 11, 12, 13, 14, 20

Part b: Base Eight