Operation on Sexagesimal System

Mathematics operation like addition, subtraction, multiplication, and division can be applied in the sexagesimal system.

Example 1 (addition)

25º 38' 45" + 2º 12' 5" = ...º ...' ..."

solution:

grouping the similar unit.

• 25º + 2º = 27º
• 38' + 12' = 50'
• 45" + 5" = 50"

then 25º 38' 45" + 2º 12' 5" = 27º 50' 50"

Example 2 (addition)

55º 42' 35" + 14º 21' 33" = ...º ...' ..."

solution:

• 35" + 33" = 68" since the second unit is bigger than 60" then we must write the 68" into 1' 8". The 1' is added into the minute unit and the rest is 8".
• 1' + 42' + 21' = 64' similar with the first step the minute unit is bigger than 60' then we write it into 1º 4'. The 1º is added into the degree unit and the rest is 4'.
• 1º + 55º + 14º = 70º

then 55º 42' 35" + 14º 21' 33" = 70º 4' 8".

Example 3 (subtraction)

25º 38' 45" - 2º 12' 5" = ...º ...' ..."

solution:

Grouping the similar unit.

• 25º - 2º = 23º
• 38' - 12' = 26'
• 45" - 5" = 40"

then 25º 38' 45" - 2º 12' 5" = 23º 26' 40"

Example 4 (subtraction)
56º 45' 14" - 32º 23' 37" = ...º ...' ..."

There are two parts in the subtraction the front part is the minuend (56º 45' 14") and the end part is the subtrahend (32º 23' 37"). If the unit value of the subtrahend is bigger than the minuend, borrowing process can be used and the taken from the previous unit. The checking is started from second unit.

• Unit of second in the subtrahend (37") is bigger than the minuend (14") so we can borrow 1' (60") from minute unit and the minuend become 60" + 14" = 74". The subtraction in unit second become 74" - 37" = 37".
• The borrowing value from the previous process makes the value of the unit minute become 44' (borrowing: 45' - 1'). Since the value of the minuend is bigger than the subtrahend, the borrowing process is unnecessary used. The subtraction in minute unit is 44' - 23' = 21'. 
• The degree unit is similar with the minute unit (minuend > subtrahend) then the value of subtraction from the degree unit is 56º - 32º = 24º.

Then 56º 45' 14" - 32º 23' 37" = 24º 21' 37".

Example 5 (multiplication)
10º 11' 12" x 6 = ...º ...' ..."
Solution:
The first process is multiply each unit with the given number.

• 10º x 6 = 60º
• 11' x 6 = 66'
• 12" x 6 = 72"

then 10º 11' 12" x 6 = 60º 66' 72"
The second process is divided each unit with a value.

• 72"/60" = 1' + 12". There are two values 1' and 12". The value of 1' is added to minute unit, and the rest is 12".
• The minute unit become 66' + 1' = 67'. Then the value of 67'/60' = 1º + 7'. The value of 1º is added to degree unit and so the rest is 7'.
• We do not have to divide the degree unit like the previous process, just add the 1º + 60º  = 61º.

Then 10º 11' 12" x 6 = 61º 7' 12"

Example 6 (Division)
(12º 43' 15"): 5 = ...º ...' ..."
Solution:
Important note: 
a = b.n + r, 
example: 12 = 5.2 + 2
with a = 12, b = 2, n = 2, r = 2
n is multiplier and r is quotient.

Division can be solved by divided all the units by the given number. Look at the process below:

1. 12º = 5.2º + 2º. The quotient is 2º. This quotient is added to the minute unit. The value of conversion: 2º x 60' = 120'.
2. 120'+ 43' = 163'. The value written: 163' = 5.32' + 3'. The quotient is 3'. This quotient is added to second unit. The value of conversion: 3' x 60" = 180".
3. 180" + 15" = 195". The value written: 195" = 5.39" + 0".

Then (12º 43' 15"): 5 = 2º 32' 39".

-Vn-