Sexagesimal System

Sexagesimal System is a numeral system used for measuring time, angles and geographical coordinates based on the number 60. It is one of the ancient numeral systems. In the sexagesimal numeral system, a unit of the concerned physical quantity is divided into 60 equal divisions. The reason to use the number 60 is that it has multiple divisors, making it convenient to create divisions of the physical quantity such as hour, minute, degree, etc.

In modern world, the sexagesimal system is mostly replaced by the decimal system as it is more convenient and practical. But, it is still prevalent in measurements involving angles, time, longitude and latitude.

In this article, we will learn about the sexagesimal system, reasons to use 60 as the base number, examples where it is used, practice questions and related frequently asked questions.

Sexagesimal systems are defined as,

Sexagesimal system is a numerical system based on the number 60, i.e. it involves expressing numbers in terms of multiples and powers of 60.

Sexagesimal system is also known as the sexagenary system or base 60.

Actual reason to use 60 as the base number in the sexagesimal system is uncertain. However, it is said that it is used as it has multiple factors or divisors. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. Thus, it is a highly composite number.

Also, its factors contain prime numbers that are 2, 3 and 5. The sexagesimal system was predominantly used before the introduction of computers. However, due to the compatibility of computer systems with binary and decimal systems only, the use of the sexagesimal system has pretty much declined in the modern world.

Sexagesimal system was used in a variety of measurements but it has been replaced by decimal system in modern world. Here, we will discuss the regions where it is still prevalent that are angle measurements, time measurements and representation of geographical coordinates.

Angle Measurement

Angles are measured in terms of degrees, minutes and seconds. A complete circle angle is comprised of an angle of 360 degrees (sixth multiple of 60). Also, one degree is divided into 60 minutes and one minute is divided into 60 seconds. It is represented mathematically as follows:

• 1 complete circle = 360 degrees
• 1 degree = 60 minutes
• 1 minute = 60 seconds

A degree is symbolized as (°), a minute as (‘), and a second as (“). For example,

• 30 degrees would be written as 30°.
• 30 degrees and 15 minutes would be written as 30° 15′.
• 30 degrees, 15 minutes, and 20 seconds would be written as 30° 15′ 20″.

This division is helpful when precise angle measurements are required such as in determining projectile of rockets and missiles, navigation, etc.

Time Measurement

Sexagesimal system is predominantly used in time measurement from long ago. A complete day is divided into 24 hours (a multiple of 6). One hour is divided into 60 minutes and one minute into 60 seconds. Mathematically, it is represented as follows:

• 1 complete day = 24 hours
• 1 hour = 60 minutes
• 1 minute = 60 seconds

Thus, time in a day can be expressed in form of hours (h), minutes (m), and seconds (s). It is written in the format of hours:minutes:seconds. For example:

• 6 hours, 30 minutes, and 45 seconds would be written as 6:30:45.
• 5 hours, 15 minutes, and 20 seconds would be written as 5:15:20.
• 12 hours, 10 minutes, and 20 seconds would be written as 12:10:20.

Geographical Coordinates

Sexagesimal system is also used to express location of a point on earth in terms of geographical coordinates, i.e. latitude and longitude. Latitude represents how far north or south a point is from the equator whereas Longitude represents how far east or west a point is from the Prime Meridian. Prime Meridian is the line that passes through the Royal Observatory in Greenwich, London, United Kingdom. Degrees, Minutes, and Seconds are used to precisely express the location of a point on the earth. It is explained briefly as follows:

• Degrees (°): Latitude between poles is divided into 90 degrees. Positive values represent points north of the equator, while negative values represent points south of the equator. While, Longitude is measured in degrees ranging from 0° at the Prime Meridian to 180° eastward and westward.

• Minutes (′): Each degree is divided into 60 minutes.

• Seconds (″): Each minute is further divided into 60 seconds.

Using the sexagismal system, a geographical coordinate is typically expressed as,

• Latitude: degrees° minutes′ seconds′′

• Longitude: degrees° minutes′ seconds′′

Direction can be expressed using positive-negative signs or by writing the first letter of the direction at the end of latitude and longitude, i.e. N for north, E for east, etc. For example, location of Qutub Minar can be represented as follows using sexagesimal system for geographical coordinates:

• Qutub Minar, New Delhi: Latitude: 28° 31′ 27″ N, Longitude: 77° 11′ 9″ E

In conclusion, sexagesimal system is a numeral system with base number as sixty (60) which was used predomnantly by ancient civilizations for counting and measurements but due to advancement of computing technologies which majorly work on binary (base 2) and decimal (base 10) numeral systems, its use has been limited to time measurements, angle measurements, and geographical coordinates.

In this system, one unit of a physical quantity is divide into 60 or its multiples. For instance, in angular measurements a complete circle is divide into 360 degrees and in time measurements, one hour is divided into 60 minutes which is further divided into 60 seconds.

Q1. Convert the following angle measure from decimal degrees to degrees, minutes, and seconds using the sexagesimal system: 68.25°.

Q2. Convert the following angle measure from degrees, minutes, and seconds to decimal degrees: 30° 15′ 20″.

Q3. How many seconds are there in 1 day?

Q4. Determine the number of hours in 7200 seconds.

Q5. Convert 1440 seconds into minutes.

What is Sexagesimal System?

Sexagesimal system is a numeral system with the number sixty as base. It is widely used in measuring angles, time, and geographical coordinates.

Why 60 is used as a base number in sexagesimal system?

Actual reason to use 60 as base number is uncertain but approximations are made that it used as it is a highly composite number which facilitates its division in various factors easier.

What are some other names for sexagesimal system?

Sexagesimal System is also known as the Sexagenary System or the base 60 numeral system.

Why the use of sexagesimal system decreased nowadays?

While sexagesimal system remains in use for certain applications such as time and angle measurements, but other numeral systems such as the decimal system (base-10) and binary system (base-2) are more prevalent nowadays due to their compatibility with digital computing.

Give example of where sexagesimal system is used today?

Most common example of sexagesimal system is found in time measurement, where a complete day is divided into 24 hours, one hour into 60 minutes and one minute into 60 seconds.

How do you convert degrees to minutes and minutes to seconds?

Given conversions achieved using the given relation:

• 1 degree = 60 minutes
• 1 minute = 60 seconds