The following is a base 60 (sexagesimal) number written in modern form: Find its value in the base-10 system. Round your answer to three decimal places, if necessary. (15,18,46; 31, 49) 60 Answer: In base-10, (15,18,46; 31,49_ (Round to 3 places) Recall that the notation (12, 34)60 means you have the number 12,34 in base 60_
The Correct Answer and Explanation is:
To convert the base-60 (sexagesimal) number (15,18,46; 31,49)\₆₀ into base-10, we break it into two parts: the whole number part and the fractional part.
Step 1: Understand the notation
In base-60, the notation (15,18,46; 31,49)\₆₀ means:
- Whole number part = 15 × 60² + 18 × 60¹ + 46 × 60⁰
- Fractional part = 31 × 60⁻¹ + 49 × 60⁻²
Step 2: Convert the whole number part
- 15 × 60² = 15 × 3600 = 54000
- 18 × 60¹ = 18 × 60 = 1080
- 46 × 1 = 46
Adding them together:
54000 + 1080 + 46 = 55126
Step 3: Convert the fractional part
- 31 × (1/60) = 31 ÷ 60 = 0.516666…
- 49 × (1/3600) = 49 ÷ 3600 = 0.013611…
Adding them together:
0.516666… + 0.013611… = 0.530278
Step 4: Add both parts
55126 + 0.530278 = 55126.530
Final Answer:
In base-10, (15,18,46; 31,49)\₆₀ = 55126.530
Explanation
The base-60 number system, also called sexagesimal, was used by ancient Mesopotamians and is still used today in measuring time and angles. It operates similarly to our familiar base-10 system but uses powers of 60 instead of powers of 10. Each position in a sexagesimal number represents a multiple of a power of 60.
In the number (15,18,46; 31,49)\₆₀, the digits to the left of the semicolon represent the whole number part, and those to the right represent the fractional part. To convert this to base-10, we multiply each digit by 60 raised to a specific power, starting with the leftmost digit. For example, the leftmost 15 is in the “60 squared” place, so we multiply 15 by 3600. The next digit, 18, is multiplied by 60. The last digit of the whole part, 46, is in the ones place.
The fractional part uses negative powers of 60. The first digit after the semicolon, 31, is in the 1/60 place, and the next digit, 49, is in the 1/3600 place. Each is converted to decimal and added to the whole number.
Finally, we add the decimal equivalents of both the whole and fractional parts to get the base-10 value. Rounding to three decimal places gives the final result: 55126.530.
