Products, Powers and Roots
Last timeNumbering Things
The product sign is a loop that multiplies. Powers count the repeats, and the negative and fractional ones are the same idea pushed somewhere it still works.
Everything in the sum lesson applies here with one symbol changed, so this
lesson is short on the product sign and long on the marks above the line, which
is where the real reading difficulty sits.
The product sign
The capital pi is the sum sign with multiplication in place of addition.
The counter and its start are underneath, the stop is above and inclusive, and
the factor to the right is written once with the counter standing in for
whichever factor it is. A factor written to the left of the sign is outside the
product and applied once; written to the right it is inside and applied to
every term. All of that is carried over without change.
One difference is worth memorising. An empty sum is nought. An empty product is
one. Both conventions exist for the same reason, which is that each is the value
that leaves the surrounding arithmetic undisturbed: adding nought changes
nothing and multiplying by one changes nothing. A recursive definition that
starts from an empty product therefore starts from one, and a reader who
expects nought there will think the formula is broken.
The lesson stops here
6 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents