10.01 The number plane

The grid reference system commonly used in street directories and topographic maps makes use of ordered pairs of numbers, or sometimes letter-number pairs, to describe different regions within an area.

In mathematics, the number plane is used to describe the location of actual points, not regions, in a two-dimensional space. These points are labelled by an ordered pair of numbers of the form \left(x,\,y\right), called coordinates.

We can think of the number plane as being built from two separate number lines. The first number line we lay down horizontally on the plane. This line is called the x-axis, and for every point in the plane we can talk about its x-coordinate, which is its horizontal position along this axis.

The second number line is placed in a vertical direction, perpendicular to the first number line. This vertical line is called the y-axis, and every point in the plane has a y-coordinate, which is its vertical distance along this axis.

The point at which the x-axis and the y-axis intercept is called the origin. The coordinates of the origin are \left(0,0\right).

The coordinates of a point are given with respect to the origin. In the image below, we can see that the cat is 6 units to the right of the origin, and 2 units above the origin. So we can say the cat has the coordinates (6,\,2). The x-coordinate is 6 and the y-coordinate is 2.

Notice that it would be incorrect to say the cat has the coordinates (2,\,6). These coordinates refer to the point 2 units to the right of the origin, and 6 units above the origin. The order of the numbers is important.

Coordinates are always written with brackets in the form (a,\,b) where the first number, a, is the x-coordinate and the second number, b, is the y-coordinate.

The expression "a,\,b" is a list of two numbers, a and b, and it does not convey the same information as the ordered pair of coordinates (a,\,b).

Examples

Example 1

Here is a number plane.

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What object has coordinates \left(6,2\right)?

Worked Solution

Create a strategy

Use the numbers on the axes in locating the coordinates.

Apply the idea

Start at (0,\,0). Move 6 spaces to the right, then 2 spaces up.

The object with coordinates (6,\,2) is a bike.

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b

Write the coordinates of the rabbit.

Worked Solution

Create a strategy

Count the number of horizontal and vertical units required to move away from the origin and determine the location of the rabbit.

Apply the idea

The rabbit is located 9 spaces to the right, then 8 spaces up. So, the coordinates are (9,\,8).

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Example 2

Plot the point \left(6,3\right) onto the number plane.

Worked Solution

Create a strategy

Use the numbers on the axes to move on the coordinate plane.

Apply the idea

Start at (0,\,0). Plot the point 6 spaces to the right, then 3 spaces up.

This will be the point on the plane described by (6,\,3).

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Example 3

Write the coordinates of the point that is 5 units to the right of \left(9,\,6\right).

Worked Solution

Create a strategy

Plot the given coordinates, then move horizontally by the required units.

Apply the idea

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Plot (9,\,6) on the number plane and move 5 units to the right.

So, the new coordinates are (14,\,6).

Reflect and check

Another way to find the coordinates of the new point is by realising that moving right will increase the x-coordinate so we need to add 5 to the x-coordinate.

\displaystyle \text{New coordinates}	\displaystyle =	\displaystyle (9+5,6)	Add 5 to 9
	\displaystyle =	\displaystyle (14,6)	Evaluate

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