Introduction
While we typically use the decimal system (Base 10) for counting, our measurement of time and angles relies on an ancient system called Sexagesimal (Base 60). Developed by the Sumerians and Babylonians thousands of years ago, this system divides units into increments of 60. Whether you are calculating the duration of a flight or the precise coordinates of a map, understanding how to navigate Base 60 is a vital skill in both GCSE and A-Level Mathematics.
Theory
The Sexagesimal system is defined by the use of 60 as its base. In this system, each larger unit is composed of 60 smaller sub-units.
Notation and Units
In mathematics, we primarily use this for Time and Angular Measurement. The notation is as follows:
- Degrees or Hours (° or h): The primary unit.
- Minutes ('): There are 60 minutes in one degree or hour.
- Seconds (''): There are 60 seconds in one minute.
Conversion Fundamentals
To convert between units, we use the following ratios:
| From Unit | To Unit | Operation |
|---|---|---|
| Degrees | Minutes | Multiply by 60 |
| Minutes | Seconds | Multiply by 60 |
| Seconds | Minutes | Divide by 60 |
| Minutes | Degrees | Divide by 60 |
| Degrees | Seconds | Multiply by 3600 |
Operations in Base 60
When adding or subtracting sexagesimal numbers, we must remember the Regrouping Rule:
- Addition: If the sum of seconds or minutes reaches 60 or more, you must carry over to the next higher unit.
- Subtraction: If the value being subtracted is larger than the starting value in a column, you must borrow 60 from the next higher unit.
Worked Example
Convert 14.38° into degrees, minutes, and seconds (DMS).
Step 1: Identify the whole degrees - The whole number part is the degrees. 14°
Step 2: Convert the decimal remainder to minutes - Take the decimal part (0.38) and multiply it by 60:
The whole number part is the minutes: 22'.
Step 3: Convert the remaining decimal to seconds - Take the new decimal part (0.8) and multiply it by 60.
Final Answer: 14° 22' 48''
Practice Questions & Solutions
Convert the following time into total seconds:
Convert hours to seconds:
Convert minutes to seconds:
Add all values together:
Convert the following measurement into degrees, minutes, and seconds:
The whole degrees:
Multiply the decimal by 60 for minutes:
Multiply the remaining decimal by 60 for seconds:
Final result:
Perform the following addition:
Sum the columns:
Adjust the seconds (subtract 60, carry 1):
Adjust the minutes (subtract 60, carry 1):
Convert the total seconds into the sexagesimal form:
Divide by 60 to find minutes:
Divide minutes by 60 to find degrees:
Final result:
Subtract the following:
Borrow 1 minute for the seconds:
Borrow 1 degree for the minutes:
Subtract remaining values:
Summarise with AI:
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Time Is Amazing It’s More Than One