Cartesian system

Two number lines, one origin. The Cartesian plane is built from two number lines placed at right angles: a horizontal x-axis and a vertical y-axis. They cross at a single point called the origin, labelled $O(0, 0)$, which is the zero of both scales simultaneously. Because the two lines are number lines, both carry a direction of increase (rightward and upward) and a direction of decrease (leftward and downward), and both are marked off in equal units.

Ordered pairs are addresses. Every point $P$ in the plane is described by exactly one ordered pair of real numbers, written $P(x, y)$:

where: $x$ (the abscissa) is the point's signed horizontal distance from the y-axis — positive if $P$ is to the right, negative if to the left — and $y$ (the ordinate) is its signed vertical distance from the x-axis — positive if $P$ is above, negative if below. "Signed distance" is doing real work here: it is not just how far the point is from the axis, it is how far and in which direction, which is exactly why a single pair of numbers is enough to fix the point uniquely. The word "ordered" is not decoration. $(2, 5)$ and $(5, 2)$ name two different points — one reached by going $2$ right then $5$ up, the other by going $5$ right then $2$ up — so swapping the two numbers swaps the point. This matches the everyday convention "along the corridor, then up the stairs": read the x-coordinate first (how far along), then the y-coordinate (how far up).

Four quadrants, one sign rule. The two axes cut the plane into four regions called quadrants, numbered I to IV going anticlockwise starting from the top-right region. Each quadrant has a fixed sign pattern for $(x, y)$:

A point that lies exactly on an axis is a special case and does not belong to any quadrant: any point with $y = 0$ sits on the x-axis, and any point with $x = 0$ sits on the y-axis. The origin itself, $(0,0)$, is the one point common to both axes.

Dependent and independent variables. Coordinate geometry becomes useful the moment two real quantities are linked: for example, the number of pencils bought ($x$) and the total cost of buying them ($y$). Here $x$ is the independent variable, chosen freely, and $y$ is the dependent variable, whose value follows from $x$. Tabulating a few values of $x$ against the corresponding $y$ produces a set of ordered pairs, which can then be plotted on the Cartesian plane exactly as any other point is plotted — this is the bridge between arithmetic tables and geometric pictures that the rest of the chapter builds on.

The Cartesian system needs no formula because it is a definition, not a derived result — but it is worth seeing why two perpendicular number lines are enough to locate every point in a flat plane, and why nothing less would do. A single number line can only locate points along one direction: given a number line, "$3$" names one specific point and no other. A flat plane, however, has two independent directions of freedom — you can move a point left-right and, separately, up-down, and neither motion undoes the other. To pin down a point in a space with two independent directions of freedom, you need exactly two independent numbers, one for each direction. That is precisely what the x-axis and y-axis supply: the x-axis measures how far along the horizontal direction, the y-axis measures how far along the vertical direction, and because the two axes are perpendicular, moving along one never changes your position along the other — the two measurements genuinely do not interfere. This is also why the axes must be number lines with a common origin and a common unit. The shared origin $O(0,0)$ gives every point a single reference to measure from, so "distance from the y-axis" and "distance from the x-axis" are always measured from the same starting point for every point in the plane. Without a shared origin, two people could disagree about where "zero" is, and the same point would get two different addresses — the whole system would stop being a reliable, unambiguous grid. The sign convention (right and up positive, left and down negative) is what lets one ordered pair distinguish all four directions around the origin, which is exactly why the four quadrants fall out automatically once the sign rule is fixed.

Plot the points $A(4, 3)$, $B(-2, 5)$, $C(-3, -4)$, $D(5, -2)$ on a Cartesian plane, state the quadrant each lies in, and then plot a fifth point $E(0, -6)$ and describe its position.

Step 1. Draw the x-axis and y-axis intersecting at $O(0,0)$, marking equal units along each so that distances can be read directly off the grid.

Step 2. Locate $A(4, 3)$: move $4$ units right of $O$, then $3$ units up. Both coordinates are positive, so by the sign rule $A$ lies in Quadrant I.

Step 3. Locate $B(-2, 5)$: move $2$ units left, then $5$ units up. Here $x$ is negative and $y$ is positive, so $B$ lies in Quadrant II.

Step 4. Locate $C(-3, -4)$: move $3$ units left, then $4$ units down. Both coordinates are negative, so $C$ lies in Quadrant III.

Step 5. Locate $D(5, -2)$: move $5$ units right, then $2$ units down. Here $x$ is positive and $y$ is negative, so $D$ lies in Quadrant IV.

Step 6. Locate $E(0, -6)$: the abscissa is $0$, so $E$ does not move left or right of the origin at all — it sits directly on the y-axis, $6$ units below $O$. Since $x = 0$, $E$ lies on the y-axis, not inside any quadrant. The five points, taken together, land one in each quadrant plus one on an axis, giving a complete picture of every sign case: $A$ in Quadrant I, $B$ in Quadrant II, $C$ in Quadrant III, $D$ in Quadrant IV, and $E$ on the y-axis.

Competitive stretch

A shopkeeper sells pencils at ₹2 each, so $1$ pencil costs ₹2, $2$ pencils cost ₹4, $3$ pencils cost ₹6, and so on, giving $y = 2x$. Which quadrant do all the plotted points $(x, y)$ for this relation lie in, and why can $x$ never be negative here? Since the number of pencils bought is never negative and the cost is never negative, both coordinates stay positive for every meaningful value, so every point lies in Quadrant I — a reminder that the mathematics of the plane allows all four quadrants, but the real-world context of a problem can restrict which ones are actually possible.

• Reversing the coordinate order. Writing $(3, 5)$ for a point that is really $5$ right and $3$ up (which is $(5, 3)$). Always read $x$ first, then $y$ — horizontal move before vertical move.
• Numbering the quadrants clockwise, or starting from the top-left. Quadrants go I, II, III, IV strictly anticlockwise, starting from the top-right region where both coordinates are positive.
• Forcing an axis point into a quadrant. A point like $(0, 6)$ or $(-5, 0)$ lies on an axis, not inside any quadrant — check for a zero coordinate before assigning a quadrant.
• Mixing up independent and dependent variables. In a real situation, the quantity chosen freely (often time, or a count of items) is independent ($x$); the quantity that results from it (often a cost, a distance travelled, or a total) is dependent ($y$).

Ask the tutor: If plotting points or naming quadrants still feels shaky, ask the AI tutor to give you a fresh batch of coordinates to plot and check your answers one by one.