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:
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.