Riemann prose, part 4


So far, I’ve been treating numbers as if they are on a single straight line. As we increment the counting numbers, we move farther away from 0 to the right, but we’re stuck on the same line. This isn’t very useful if we want to know how big a pan we need if we’re making a sheet cake. We need to introduce a second line at right angles (90 degrees) to the first one. If the first line is our “x-axis”, then the second will be the “y-axis”.


(Straight line, showing the zero crossing and the slope.)

Odds are that you saw the x-y axis in math class in school, and that it was used to graph equations like:

y = x + 1

This is a two-dimensional situation, and on an x-y graph, x is the part that changes from one point to another, and y is the result of the formula as x changes.

In this graph, x is infinitely continuous (no breaks or incremental steps) from negative infinity to positive infinity. There are two key elements in this formula:

Zero crossing
and the slope

If you have a straight line, you only need these two components to draw the line.

If y = 0, then x = -1, which gives us the zero-crossing.

Slope is the value multiplied against x. In my example, the slope is 1/1.

So, if we were to draw a line going through y=0 where x=-1, and go up one unit for every unit we go to the right, we’d have our graph without ever having to calculate any of the other points on the line.

One of the first things we look at with curves is where they cross y=0, and this is another important component of the Riemann Zeta function. But, more on that later.

Say we have a backyard, and we want to know how much sod to buy to re-sod it. What we could do is take a piece of graph paper and draw the yard on the x-y axes. It might look something like this:


(Backyard on graph paper.)

We start from the center of the yard and walk to the left, counting off feet, then turn left when we get to the edge and, starting from 0, count feet until we get to the far bottom corner. Maybe that was 50 feet left, and 20 down. We go back to the center and repeat, going to the left, but turning right at the edge and counting feet up from 0 again. So, 50 feet left and 20 feet up. Back to the center of the yard, and go right this time, giving us 50 feet right, 20 feet both up and down.

When I was a kid in school, we’d write these points out as (-50, -20), (-50, 20), (50, -20) and (50, 20), where the first value of the pair is x, and the second is y.

If we draw in the edges of the yard, we get a rectangle. Subtract the points from each other and we learn that our rectangle is 100 feet by 40 feet. To determine the amount of sod needed, we multiply the long side against the short side to get 4,000 square feet.

Odd… where did that “square” come from? I was using straight lines, and rectangles. Not no stupid squares…

 

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