Below is the base 60 system used by Babylonians? Using the the translation table translate the following: B->Decimal a) 2, 25 b) 3, 36; 22, 32, 45 Decimal->B a) 1245 b) 65 c) 147 Compare to decimal and give some comments?

Sep 13, 2016

Explanation:

We normally use decimal system which is Base 10 system and hence uses 10 symbols viz., $1 , 2.3 .4 .5 .6 .7 .8 .9 .0$.

Similarly, hexadecimal system, which is Base 16 system use 16 symbols viz., $1 , 2.3 .4 .5 .6 .7 .8 .9 .0 , A , B , C , D , E , F$.

In any such system, if base is $B$, a number ${X}_{4} {X}_{3} {X}_{2} {X}_{1}$, with four symbols is described as

${X}_{4} {X}_{3} {X}_{2} {X}_{1} = {X}_{4} \times {B}^{3} + {X}_{3} \times {B}^{2} + {X}_{2} \times B + {X}_{1}$

In such systems every number has a place value, which is very important and the a digit on the left hand side is $B$ times the value of a similar digit on its immediate right.

Note in such systems, we need the same number of digits as is the Base. However in Babylonian system, we do not have $60$ symbols and hence it does not fall in the same genre as decimal or hexadecimal systems. For example in the given Figure 5 (in the question) $16 , 24 , 26 , 43 , 44 , 51$ have been described using two symbols. This clearly breaks down the the rules relating to base system.

Hence Babylonian system can lead to multiple values when converted to decimal or any other base system.

If above constraints are ignored and one focuses primary on the question, the following are the decimal values of given Babylonian numbers

$B a b y l o n i a n \left(2 , 25\right) = 2 \times 60 + 25 = 145$
$B a b y l o n i a n \left(3 , 36\right) = 3 \times 60 + 36 = 216$
$B a b y l o n i a n \left(22 , 32 , 45\right) = 22 \times {60}^{2} + 32 \times 60 + 45 = 22 \times 3600 + 32 \times 60 + 45 = 81165$

And Decimal $1245 = B a b y l o n i a n \left(20 , 45\right)$ as $1245 = 20 \times 60 + 45$
Decimal $65 = B a b y l o n i a n \left(1 , 5\right)$ as $65 = 1 \times 60 + 5$ and
Decimal $147 = B a b y l o n i a n \left(2 , 27\right)$ as $147 = 2 \times 60 + 27$.