Small Business Doctor

Ancient Israelites did not do arithmetic like we do arithmetic in America today. Multiplication like we do it today did not arrive in Europe from India where it originated until the 13th century with Fibonacci.

So, how would an ancient Israelite do arithmetic like multiplication? They almost certainly did multiplication like the ancient Egyptians did it. This method is also known as the peasant method, and is still done by some people today who lack formal math education.

Multiplication in this method is based on successive additions and doubling. It also uses powers of 2. Powers of 2 are 1, 2, 4, 8, 16, 32, 64, and so on.

To demonstrate this, consider 25 X 100. For many of us, we could look at this and say the answer is 2,500. The ancients could not do this. Instead, they would do this.

Ancient Multiplication

1. Decompose one of the two numbers. To decompose 25, we ask:
   • What is the largest power of 2 that is less than or equal to 25?
     • The answer is 16.
   • Subtract this power of 2 from 25.
     • 25 – 16 = 9.
   • What is the largest power of 2 that is less than or equal to 9?
     • The answer is 8.
   • Subtract this power of 2 from 9.
     • 9 – 8 = 1.
   • What is the largest power of 2 that is less than or equal to 1?
     • The answer is 1.
   • Subtract this power of 2 from 1.
     • 1 – 1 = 0.
   • 25 is the sum of 16 + 8 + 1.
2. Construct a table of powers of 2 times the other number (100) from 1 through the largest power of 2 from step 1, which is 16. They did this by doubling.
   • Begin with 1.
     • Begin with the 100.
   • 2 is the next power of 2.
     • Double 100 to get 200.
   • 4 is the next power of 2.
     • Double 200 to get 400.
   • 8 is the next power of 2.
     • Double 400 to get 800.
   • 16 is the next power of 2.
     • Double 800 to get 1,600.
3. Add only those rows from step 2 that contain the results from step 1.
   • Use only 1, 8, and 16.
   • These are 100, 800 and 1,600
   • 100 + 800 + 1,600 = 2,500

Likely Implications from Multiplying Like This

As you can see, this is exceptionally tedious, and there’s no realistic way to reliably calculate this in your head. As a result, everyone would be much more likely to think and remember counts in terms of X sets of Y rather than in terms of the final product Z.

This would be like saying “I have 8 dozen sheep,” instead of saying “I have 96 sheep.” To go from 8 X 12 = 96, which in words is 8 of the dozen unit size is equal to 96, would take all of those intermediate steps above. So, ancients would have been much more likely to say “8 dozen” instead of “96.”

This way of counting would lead to having certain standardized unit sizes that could be readily identified, because you’d be counting in terms of those sets. In America, this would be akin to a dozen. From a dozen we could have a half dozen, which is 6, or even a gross, which is a dozen dozens.

Also note that even though 96 is a compact and efficient way of conveying the number of sheep I have, it is also a summary of information. If a honey farm received 1 order of 96 jars of honey, that is very different than receiving 8 orders of 12 jars of honey, which is also different than 96 orders of 1 jar of honey. If I only receive 1 order, then I know I have 1 very large buyer, which is risky for a business. If I have 8 orders of 12 jars each, then I know I likely have resellers like stores buying my honey. If I have 96 single jar orders, then I know I likely have individual consumers buying my honey. Potentially important information like this is lost when I only use “96” instead of “8 dozen.”

Counting instead of multiplying

Instead of using any arithmetic, why would they not simply count their 96 sheep? If we primarily think about counting in terms of a relatively small number of stationary objects, this would make sense. Someone could go through an encampment and count the number of tents. But sheep and people tend to move around while you’re trying to count them. So, there are logistical constraints to counting.

Another difficulty with counting a large number of objects outside in the real world is that as the numbers get larger and the time spent counting increases, it becomes increasingly likely that you would miscount or lose your place in the counting. So, the likelihood of errors increases alongside the duration of counting.

It takes only a minute to count to a hundred, but it took Jeremy Harper about 3 months of 16-hour days to count to a million. His overall average was only counting about 700 times in an hour. There are diminishing returns to counting.

For example, if we use his documented average, it would take a little more than 571.4 person hours to count to the 400,000 used in many translations of Judges 20:2. That’s nearly 24 days of nonstop 24-hours a day counting. Even if they split this task up among hundreds of counters, why would they do this massive counting effort on the eve of battle?

Fourthly, it is actually much more efficient when counting a large number of items to first divide them up into smaller sets and count those. This is often referred to by math educators as counting collections.