Base-seven Multiplication

I woke this morning thinking about base-seven numbers. Realizing that the standard base-ten multiplication process must work as well in base seven, I had to try it before getting out of bed.

Base-seven numbers use powers of seven rather than powers of ten, so the digits turn over after 6. Counting in base seven goes like this:


two stars

two starsstar

two starstwo stars

two starstwo starsstar

two starstwo starstwo stars

10 seven stars

11 seven starsstar

12 seven starstwo stars

13  seven starstwo starsstar

14  seven starstwo starstwo stars

15  seven starstwo starstwo starsstar

16  seven starstwo starstwo starstwo stars

20  seven stars seven stars


Seizing the idea of multipying base-seven numbers, which I discovered in my morning thinking, I immediately set to work out an example of it, like [warning: Homeric simile approaching] a traveller who, when he is unpacking his bags at home and discovers among his packed belongings a wonderful cup, which one of his hosts must have secretly given him, but which has become tarnished on the return journey, immediately sets to polishing a portion of it so that he can see how splendid his gift is. In much the same way, I worked through my example of base-seven multiplication while still lying in bed, until I began to see the full beauty of this gift given to me in the dream world.

In base ten, 13 × 5 = 65.

The same product, in base seven, is 16 × 5:


First, 6 × 5 = 42 (i.e., 4 sevens and 2 ones).

(The 6-times table in base seven is like the 9-times table in base ten: 6 × 2 = 15; 6 × 3 = 24, and so on, up to 6 × 6 = 51.)

Then 5 × 1 + 4 = 12 (i.e., 1 seven and 2 ones).

So the product is 122 (i.e., 1 × 7² + 2 × 7 + 2 × 1).

Going back to base ten, 49 + 14 + 2 = 65.

Satisfied, I got up and had a shower.

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