Cartesian coordinates can be used to pinpoint where we are on a map or graph.
Using Cartesian Coordinates we mark a point on a graph by how far along and how far up it is:
|The point (12,5) is 12 units along, and 5 units up.|
X and Y Axis
|The left-right (horizontal) direction is commonly called X.|
|The up-down (vertical) direction is commonly called Y.|
|Put them together on a graph ...|
... and we are ready to go
Where they cross over is the "0" point,
Axis: The reference line from which distances are measured.
The plural of Axis is Axes, and is pronounced ax-eez
Point (6,4) is
6 units across (in the x direction), and
4 units up (in the y direction)
So (6,4) means:
Go along 6 and then go up 4 then "plot the dot".
And you can remember which axis is which by:
x is A CROSS, so x is ACROSS the page.
Play With It !
Now is a good time to play with
to see for yourself how it all works.
Like 2 Number Lines Put Together
It is like we put two Number Lines together, one going left-right, and the other going down-up.
As x increases, the point moves further right.
As y increases, the point moves further up.
The coordinates are always written in a certain order:
- the horizontal distance first,
- then the vertical distance.
This is called an "ordered pair" (a pair of numbers in a special order)
And usually the numbers are separated by a comma, and parentheses are put around the whole thing like this:
Example: (3,2) means 3 units to the right, and 2 units up
Example: (0,5) means 0 units to the right, and 5 units up.
In other words, only 5 units up.
The point (0,0) is given the special name "The Origin", and is sometimes given the letter "O".
Abscissa and Ordinate
You may hear the words "Abscissa" and "Ordinate" ... they are just the x and y values:
- Abscissa: the horizontal ("x") value in a pair of coordinates: how far along the point is
- Ordinate: the vertical ("y") value in a pair of coordinates: how far up or down the point is
"Cartesian" ... ?
They are called Cartesian because the idea was developed by the mathematician and philosopher Rene Descartes who was also known as Cartesius.
He is also famous for saying "I think, therefore I am".
What About Negative Values of X and Y?
Just like with the Number Line, we can also have negative values.
Negative: start at zero and head in the opposite direction:
- Negative x goes to the left
- Negative y goes down
So, for a negative number:
When we include negative values, the x and y axes divide the space up into 4 pieces:
Quadrants I, II, III and IV
(They are numbered in a counterclockwise direction)
In Quadrant I both x and y are positive, but ...
- in Quadrant II x is negative (y is still positive),
- in Quadrant III both x and y are negative, and
- in Quadrant IV x is positive again, while y is negative.
Example: The point "A" (3,2) is 3 units along, and 2 units up.
Both x and y are positive, so that point is in "Quadrant I"
Example: The point "C" (-2,-1) is 2 units along in the negative direction, and 1 unit down (i.e. negative direction).
Both x and y are negative, so that point is in "Quadrant III"
Dimensions: 1, 2, 3 and more ...
Think about this:
The Number Line can only go:
so any position needs just one number
Cartesian coordinates can go:
so any position needs two numbers
How do we locate a spot in the real world (such as the tip of your nose)? We need to know:
that is three numbers, or 3 dimensions!
Cartesian coordinates can be used for locating points in 3 dimensions as in this example:
Here the point (2, 4, 5) is shown in
three-dimensional Cartesian coordinates.
In fact, this idea can be continued into four dimensions and more - I just can't work out how to illustrate that for you!