Conversion to sexagesimal

The question of converting $0.125$ into the babylonian sexagesimal expression $0;7,30$ is answered thusly:

First, write $0.125$ as a fraction: $0.125=\frac{1}{8}$

Now Babylonian sexagesimal numbers have the form


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where $d_{0}$ is the whole number portion of the number $a$, and the other integers are the numerators of the fractions of the powers of $60.$

For $a=1/8$, it is clear that $d_{0}=0.$ The numerator $d_{1}$ is determined as the largest integer such that
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In this case $d_{1}=7.$ (Clearly, the next integer gives MATH is negative.)

Now subtract to get
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In general we now perform the same arithmetic with respect to the denominator $60^{2}$. Thus we solve for the largest integer
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Since MATH we conclude that $d_{2}=30.$ Also, there is no excess from the subtraction; so the process stops.

Example

Here's another example. Find the sexagesimal representation of $9\frac{5}{18}$

Solution

First $d_{0}=9.$ Now subtract MATH. Solve for the largest integer $d_{1}$ such that
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or $50-3d_{1}\geq 0$. We see that $d_{1}=16$. The difference MATH is therefore $\dfrac{2}{180}$. Since MATH $\ $we see that $d_{2}=40$.Thus we have exhausted powers of sixty needed. It follows that
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