Background: In 2000 BC, the Babylonians were using a base 60 positional number system (much like our base 10 system) before Roman numerals were even invented!
Below are the symbols the Babylonians used in their system:
Example: Converting a Babylonian base 60 number to our base 10 system goes like this:
Another Example: Converting a base 10 number to base 60 is the same process we used to find the base 8 representation in POW 2. For example, let's convert the number $11115$ to base $60.$
We find the highest power of $60$ that divides into $11115$ which is $60^2=3600.$ Then we divide: $3600$ divides into $11115$ 3 times with a remainder of 315. Thus, $11115=3 \cdot 3600+315$.
Now we divide the remainder by the highest power of $60$ which is simply $60^1=60$. Since $60$ divides into $315$ 5 times with a remainder of 15, we have $315=5 \cdot 60+15$. We now have our base 60 representation of our number: $$11115=3 \cdot 3600 + 5 \cdot 60 + 15=(3,5,15)_{60}.$$ Using the chart above, we may now write the number in Babylonian symbols:
Directions for POW 3: Your problem of the week has two parts. Using the above examples as guides to thinking (and writing up your solution), perform the following computations:
a) Convert the following Babylonian numeral to our base $10$ system.
b) Convert the base $10$ numeral $9801$ to base $60$ using Babylonian symbols.