Sexagesimal System / Degree Measure

The origin of the counting system is irreparable in nature and precedes the invention of writing. They were associated with the ancient Sumerians, whose language contained the only known system of sexagesimal counting. The Sumerians are also likely to be the inventors of the arithmetic system, originating from a counter system (also known as a small clay object, token or stone) that represents a unit of  1, 10, 60, 60×10, 60 2 and 603. This offset system can be traced back to about a year ago. The method of numerical notation in 3500 BCE clearly arose from the system of counters by adapting it to the format. From the Sumerians, it was passed down to the Babylonians and is still used in modern times.

Sexagesimal system was an ancient system of enumeration, arithmetic, and numerical notation that used powers of 60. The wreckage of the old system survives in the form of dysgenesis by dividing the time into 60 minutes and the minutes into 60 seconds.

What is the sexagesimal system?

Understanding the ancient form of counting can be tedious if not understood well. Described below are the concepts of this system.

The sexagesimal system of counting is also known as sexagenary or base 60.

 Reason to use 60 as the base number

• A good reason to explain this is using 60 as the base makes it easier to divide as 60 has multiple divisors.
• 60 is a highly composite number with 12 factors—1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. Among these, 2, 3 and 5 are prime numbers.

• The sexagesimal system measurement of an angle is done in degrees, minutes and seconds.
• Full rotation represents 360°. In this system, the right angle is divided into 90  equal parts, each of which is called a degree (1°). Degrees are divided into equal parts of 60, each of which is called a sexagesimal minute (1`), and the minute is further divided into equal parts of 60, each with a sexagesimal second (1”). In summary, 

1 right angle = 90 degrees (or 90 degrees) 

1 degree (or 1°) = 60 minutes (or 60`) 

1 minute (or 1`) = 60 seconds (or 60”).

• For numbers from 1 to 59, different combinations of numbers for 1 and 10 were used.
• To express greater value, the concept of place value was adopted by the Babylonians. For example, 60,70 and so on were represented with similar symbols. In fact, mathematics can represent powers of 60. 
• The context determined whether power inclusion and effect were meant to be included.
• By the third century BCE, the Babylonians seemed to have developed a placeholder symbol that acted as zero, but its exact meaning and usage are still unknown. 
• Also, there was no sign to divide a number into whole and decimal (like a modern decimal point). Therefore, the three digits are 3 7 3 , (i.e., 3 + 7✕60 + 30✕602)or multiples of these numbers to the power of 60.

Angles and degree measure

• When two lines are joined together, an angle is formed. 
• The point where they meet is the vertex or the node.
• An angle is represented as ㄥand are symbolised as θ, α, β, etc. in gradient, degree and radian.

Sexagesimal system examples

Here are two sexagesimal system examples.

1. Using the thumb to point to each of the three finger bones of each finger, a human can count up to 12 fingers with one hand. Traditional counting systems still work this way and help explain the existence of 12 and 60 based number systems alongside those based on 10, 20, and 5. With one hand, count the number of times a person hits 12 with the first hand. Five fingers count five sets of 12 or 60.
2. The degree of angle is expressed in radians and probably originated from the Babylonians who used the radix 60 (sexagesimal) notation. Their calendar had a total of 360 days. Therefore, they assumed all angles of 360°. First, they tried to subdivide the perfect angle into angles using the angles of an equilateral triangle. Then they divided 60° by 60 according to the number system (radix 60) and defined this as 1°. It is also called the arc degree.

Conclusion

Even though modern counting techniques have been developed, there are a number of similarities between the counting methods of the present time and the sexagesimal method. Sexagesimal examples and Babylonian notation uses the radix 60 (sexagesimal) instead of 10. It is not too difficult to decode, as it uses the weighted number system as we do. For us, the number 4 means 4, 40/10, 400/100, or 4/1,  which is based on where the number appears.