Why a complete circle has 360^o ?

History

The circle was divided into 360 degrees during the reign of Nebuchadnezzar (605-562 BC) in the Chaldean dynasty in Babylon. The Chaldeans calculated that a complete year numbered 360 days by observation and inference.

The basis of angular measure for the mathematicians of Babylon was the angle at each of the corners of an equilateral triangle. They did not have decimal fractions and thus found it difficult to deal with remainders when doing division. So they agreed to divide the corner of an equilateral triangle into 60 degrees. This is because 60 could be divided by 2, 3, 4, 5 and 6 without remainder. Each degree was divided into 60 minutes and each minute into 60 seconds. If the angles at the corners of six equilateral triangles are placed together they form the angle formed by a complete circle. It is for this reason that there are six times 60 degrees of arc in the complete circle.

How to count the number of factors?

We begin with the Fundamental Theorem of Arithmetic. Any positive integer can be represented in exactly one way as a product of primes. Therefore:

8 = 2³

12 = 2² ´ 3

Now, define: d(n) is the number of positive divisors of n, including 1 and n itself.

Since the factors of 8 are 1, 2, 4, 8, we write d(8) = 4.

Since the factors of 12 are 1, 2, 3, 4, 6, 12, we write d(12) = 6.

If you write the number N = p^aq^br^cs^d where p, q, r, s are prime numbers,

then d(n) = (a + 1)(b + 1)(c + 1)(d + 1).

8 = 2³, d(8) = 3 + 1 = 4

12 = 2² ´ 3 d(12) = (2 + 1)(1 + 1) = 6

240 = 2⁴ ´ 3 ´ 5 d(240) = (4 + 1)(1 + 1) (1 + 1) = 20

Why some units are not decimals?

10 = 2 ´ 5 d(10) = (1 + 1) (1 + 1) = 4

12 = 2² ´ 3 d(12) = (2 + 1)(1 + 1) = 6

16 = 2⁴ d(16) = 4 + 1 = 5

20 = 2² ´ 5 d(20) = (2 + 1)(1 + 1) = 6

Therefore the number of factors of 12, 16 and 20 are bigger than 10.

That is why we have : 1 foot = 12 inches

1 pound = 16 ounces

£ 1 = 20 shillings

in English and European units.

Graph

You can plot the number of factors, d(n), against n. The red dots show the numbers with

“first appearance of bigger d(n)”.

Table

Number n	Number of factors d(n) with “first appearance”
1	1
2	2
4	3
6	4
12	6
24	8
36	9
48	10
60	12
120	16
180	18
240	20
360	24
720	30

As you can see, the number 360 has 24 factors. You have to wait until the number 720, which has more factors than 360!