# Difficulty understanding how Babylonian reciprocals work

According to the reciprocal tables for Babylonian's base $$60$$ system, dividing by $$2$$ is like multiplying like $$30$$. Dividing by $$3$$ is like multiplying by $$20$$. Dividing by $$4$$ is like multiplying by $$15$$. Dividing by $$k$$ is like multiplying by $$60/k$$.

I don't understand why the reciprocal works like this and not like $$1/k$$. You might say "It multiplies by $$60/k$$ because it's base $$60$$" but this doesn't make sense to me, it's not like our base-$$10$$ reciprocals look like $$10/k$$.

What's going on?

• Note that $\frac n3=\frac {20n}{60}$, for example Nov 11 '18 at 8:30
• @MarkBennet I don't understand how it would get used though. Say I wanted to divide $8$ in half. How does multiplying by $30$ help me get $4$? Nov 11 '18 at 14:43
• $30\times 8=240 =40_{60}$ (ie translated to base $60$) and there is your $4$ Nov 11 '18 at 14:58
• @MarkBennet Oh, wow! But then why wouldn't they say it's $40$ as opposed to $4$? They had symbols for both $4$ and $40$ separately. How would they do the multiplication itself? The symbol for $8$ times the symbol for $30$ (both had their own symbols in base $60$) somehow giving them just.... $4$ Nov 11 '18 at 15:42
• Well, when I multiply by $25$ by dividing by $4$ I do need to make sure I get the decimal point in the right place. Nov 11 '18 at 16:33

Our base-$$10$$ reciprocals do look like $$10/k$$! Dividing by $$2$$ is like multiplying by $$5$$ (and then shifting the decimal point) and vice versa. Indeed, I regularly divide by $$5$$ by doubling the number and shifting the decimal point.

$$10$$ has fewer divisors than $$60$$, so this "trick" (if you like) doesn't have as many applications—basically only this one.

• For example I don't see how dividing by $2$ is like multiplying by $30$. For example if I wanted to divide $8$ in half to get $4$ it's somehow the same as taking $8$ times $30$? Nov 11 '18 at 14:33
• 8 times 30 is 240, that is 4*60 + 0
– mau
Nov 18 '18 at 19:15

Babylonians - at least in the tablets we have found - do not have the concept of order of magnitude: probably they calculated it in other ways. This means that - as far as the tablets are concerned - 1, 60, 3600=60*60, 1/60, 1/3600 are all represented as 1.

Therefore, if you have a number $$n$$ and you want to divide it by 4 (say), if the number is greater than 4 there is no problem; 9 / 4 = 2;15 (the semicolon has the same use as the comma in our system). For 3/4, they would have 0;45; but since they could not accept a 0 by itself, they multiplied the result for 60 obtaining 45, which is the same as 3*15. In this way, "divide by 4" is the same as "multiply by 15".

• A note that the semicolon notation is not Babylonian, but, I think, from the 20th century by Neugebauer (e.g., see www-groups.dcs.st-and.ac.uk/history/HistTopics/…). Nov 19 '18 at 2:11
• yes, this is only a practical way for us to separate numbers.
– mau
Nov 20 '18 at 8:03

In our number system, $$\dfrac 15 = \dfrac{2}{10}$$. So,

to divide $$678$$ by $$5$$,

first divide by $$10$$, getting $$67.8$$,

and then multiply by $$2$$, getting $$135.6$$.

In a base $$60$$ number system, $$\dfrac 15 = \dfrac{12}{60}$$. So

to divide $$[6,7,8]_{60}$$ by $$5$$,

first divide by $$[1,0]_{60}$$, getting $$[6,7 . 8]_{60}$$

then multiply by $$12_{60}$$, getting $$[72, 84. 96]_{60} = [72, 85. 36]_{60} =[73, 25. 36]_{60}==[1,13, 25. 36]_{60}$$