Each line is a line with numbers on it, so to define a point in the PLANE, we just read of the corresponding points on each line. For example I pick a point in the plane

Then I draw a line from the horizontal line all the way to P, and I draw another line from the vertical line to the point P. Now I measure the lengths of these lines, which is in essence the same as measuring the distance UP to the point P from the horizontal line, and the distance to the RIGHT from the vertical line. We know how to do that by now!! Say the distance to the right is 4 units (the units could be anything, m, cm, mm), and 3 units UP. SO the point is defined by the COORDINATE (4,3) in the PLANE!!! We will write this as P(4,3) to denote that P is the point with coordinate (4,3)

But what is (4,3)?? Is it different than (3,4)?? Sure looks like the interval from 4 to 3 !!! Well Let's put it formally.

By an ordered pair of real numbers we mean two real numbers in an assigned order. Thus, there is a "first number" and a ,” second number”. The symbol ( a, b) is used to denote the ordered pair of real numbers in which a is the first number and b is the second number. Because order matters, the ordered pairs (4, 3) and (3,4) are regarded to be different. AND, although the interval from 4 to 3 is written as (4,3) as well, the CONTEXT of the discussion will make it clear whether the interval is being discussed or the ordered pair. This is unfortunately the case in most of math, since if we didn’t repeat most of the notation, then just the notation will be hard to handle, let alone the concepts!!!

You must be wondering why we took the vertical line as a point of reference to count units to the right, and the horizontal line to count units up??? Well, the reason is that I want to have a point of reference. Just like on the real number line where we had the 0 as the ORIGIN, similarly, in this case we will take the COORDINATE or the POINT (0,0) as our reference or the ORIGIN.

Notice that when I measured the units UP and DOWN in the above example, I was actually measuring from the point (0,0). Convince yourself of this.

It is hard to keep saying vertical line and horizontal line. So we introduce terminology here. You may have heard this before. We will call the Vertical line the Y-AXIS, and the Horizontal line the X-AXIS. Then in the POINT (4,3), 4 will be called the X-COORDINATE, and 3 will be called the Y-COORDINATE. Sometimes we will say x=4, y=3, for short.

BUT HERE IS AN IMPORTANT QUESTION???? We saw how to assign ordered pairs to the points in the plane. Is this a one-to-one assignment?? That is, are we sure that we won’t get the same ordered pair assigned to a different point in the plane.

Well we are sure that we WON’T. The reason is the way we draw perpendicular lines from the coordinate lines to the POINT. The intersection of any such two perpendicular lines has to be a unique point!!!!! Conversely, if we start with an ordered pair of real numbers (a, b) and construct lines perpendicular to the x-axis and y-axis that pass through the points with coordinates a and b, respectively, then these lines intersect at a unique point P in the plane whose coordinates are (a, b). Thus, we have a one-to-one correspondence between ordered pairs of real numbers and points in a coordinate plane. I don’t think you need a long discussion about this, its fairly straight forward.

To plot a point P(a, b) (Remember that this means the point P with coordinates (a,b) ) means to locate the point with coordinates (a, b) in a coordinate plane. For example, In the figure below we have plotted the points P(2,5), Q(-4,3), R(-5,-2), and S(4,-3).Now this idea will enable us to visualise algebraic equations as geometric curves and, conversely, to represent geometric curves by algebraic equations. Remember we said this in the first lecture!?? This is in essence what the famous French mathematician Descartes developed.

Ordered Pair
Each line is a line with numbers on it, so to define a point in the PLANE, we just read of the corresponding points on each line. For example I pick a point in the plane Then I draw a line from the horizontal line all the way to P, and I draw another line from the vertical line to the point P. Now I measure the lengths of these lines, which is in essence the same as measuring the distance UP to the point P from the horizontal line, and the distance to the RIGHT from the vertical line. We know how to do that by now!! Say the distance to the right is 4 units (the units could be anything, m, cm, mm), and 3 units UP. SO the point is defined by the COORDINATE (4,3) in the PLANE!!! We will write this as P(4,3) to denote that P is the point with coordinate (4,3) But what is (4,3)?? Is it different than (3,4)?? Sure looks like the interval from 4 to 3 !!! Well Let's put it formally. By an ordered pair of real numbers we mean two real numbers in an assigned order. Thus, there is a "first number" and a ,” second number”. The symbol ( a, b) is used to denote the ordered pair of real numbers in which a is the first number and b is the second number. Because order matters, the ordered pairs (4, 3) and (3,4) are regarded to be different. AND, although the interval from 4 to 3 is written as (4,3) as well, the CONTEXT of the discussion will make it clear whether the interval is being discussed or the ordered pair. This is unfortunately the case in most of math, since if we didn’t repeat most of the notation, then just the notation will be hard to handle, let alone the concepts!!! You must be wondering why we took the vertical line as a point of reference to count units to the right, and the horizontal line to count units up??? Well, the reason is that I want to have a point of reference. Just like on the real number line where we had the 0 as the ORIGIN, similarly, in this case we will take the COORDINATE or the POINT (0,0) as our reference or the ORIGIN. Notice that when I measured the units UP and DOWN in the above example, I was actually measuring from the point (0,0). Convince yourself of this. It is hard to keep saying vertical line and horizontal line. So we introduce terminology here. You may have heard this before. We will call the Vertical line the Y-AXIS, and the Horizontal line the X-AXIS. Then in the POINT (4,3), 4 will be called the X-COORDINATE, and 3 will be called the Y-COORDINATE. Sometimes we will say x=4, y=3, for short. BUT HERE IS AN IMPORTANT QUESTION???? We saw how to assign ordered pairs to the points in the plane. Is this a one-to-one assignment?? That is, are we sure that we won’t get the same ordered pair assigned to a different point in the plane. BUT HERE IS AN IMPORTANT QUESTION???? We saw how to assign ordered pairs to the points in the plane. Is this a one-to-one assignment?? That is, are we sure that we won’t get the same ordered pair assigned to a different point in the plane. Well we are sure that we WON’T. The reason is the way we draw perpendicular lines from the coordinate lines to the POINT. The intersection of any such two perpendicular lines has to be a unique point!!!!! Conversely, if we start with an ordered pair of real numbers (a, b) and construct lines perpendicular to the x-axis and y-axis that pass through the points with coordinates a and b, respectively, then these lines intersect at a unique point P in the plane whose coordinates are (a, b). Thus, we have a one-to-one correspondence between ordered pairs of real numbers and points in a coordinate plane. I don’t think you need a long discussion about this, its fairly straight forward. To plot a point P(a, b) (Remember that this means the point P with coordinates (a,b) ) means to locate the point with coordinates (a, b) in a coordinate plane. For example, In the figure below we have plotted the points P(2,5), Q(-4,3), R(-5,-2), and S(4,-3).Now this idea will enable us to visualise algebraic equations as geometric curves and, conversely, to represent geometric curves by algebraic equations. Remember we said this in the first lecture!?? This is in essence what the famous French mathematician Descartes developed.
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