Algebraic geometry and analytic geometry

Algebraic geometry is about shapes and spaces, but instead of using numbers to describe them, we use mathematical expressions made up of letters and symbols.

Let's pretend we are drawing a circle on a piece of paper. We could use algebra to describe this circle by saying that all the points on the paper that are the same distance from the center point is our circle. We could use a formula like (x-a)^2 + (y-b)^2 = r^2, where x and y are the coordinates of any point on the paper, a and b are the coordinates of the center point, and r is the radius of the circle. This formula tells us that any point on the paper that satisfies this formula is on our circle.

Now, let's talk about analytic geometry. In analytic geometry, we give shapes and spaces numerical coordinates. We can plot points on a graph, and each point has a unique (x, y) coordinate. By connecting these points with lines, we can create shapes.

For example, we could plot a point on a graph at (2,3), and another point at (4,5). We could then connect these two points with a line. This line has a mathematical equation, y = (2/3)x + (5/3). This equation tells us that for any value of x, we can determine the corresponding y value that will be on this line.

In summary, algebraic geometry uses algebra to describe shapes, while analytic geometry uses numerical coordinates to plot points and create shapes.