# Quadrants of Coordinate Plane - I, II, III, IV  
[Geometry](/content/tag/geometry/index.html)

TL;DR  
The coordinate plane is divided by the x-axis and y-axis into four quadrants, numbered I, II, III, IV counterclockwise starting from the upper right. Each quadrant has a specific sign convention for (x,y): Quadrant I: both positive; II: x negative, y positive; III: both negative; IV: x positive, y negative.

## What Is a Quadrant?  
A **quadrant** is one of the four regions into which the **coordinate plane** is divided by the two coordinate axes — the horizontal **x-axis** and the vertical **y-axis**. The word _quadrant_ comes from the Latin _quadrans_, meaning _"a quarter"_ — and each quadrant is exactly one-quarter of the plane.

When the two axes intersect at right angles, they cut the plane into four equal infinite regions. Each region is labelled with a Roman numeral — **I, II, III, IV** — following a strict counterclockwise convention that traces back to René Descartes's 1637 _La Géométrie_.

A point in the coordinate plane is written as an **ordered pair** (x,y), where x is the horizontal distance from the y-axis and y is the vertical distance from the x-axis. The signs of x and y — positive or negative — determine which quadrant the point lies in.

## Four Quadrants in Coordinate Plane  
The four quadrants are arranged counterclockwise around the origin:

- **Quadrant I (upper right):** the region where x > 0 and y > 0. Every point here has both coordinates positive. Example: (3,5).
- **Quadrant II (upper left):** the region where x < 0 and y > 0. Negative x, positive y. Example: (−3,5).
- **Quadrant III (lower left):** the region where x < 0 and y < 0. Both coordinates negative. Example: (−3,−5).
- **Quadrant IV (lower right):** the region where x > 0 and y < 0. Positive x, negative y. Example: (3,−5).

The counterclockwise convention is not arbitrary — it matches the standard direction of _positive angle measurement_ in trigonometry.

## Sign Convention in Quadrants  
The sign of the x-coordinate and the y-coordinate uniquely determines a point's quadrant.

| Quadrant | x-coordinate | y-coordinate | Example point |  
| --- | --- | --- | --- |  
| **I** | + (positive) | + (positive) | (7,4) |  
| **II** | - (negative) | + (positive) | (−7,4) |  
| **III** | - (negative) | - (negative) | (−7,−4) |  
| **IV** | + (positive) | - (negative) | (7,−4) |

**Pattern:** starting from Q I with both signs positive, each subsequent quadrant flips one sign as you move counterclockwise: (+,+)→(−,+)→(−,−)→(+,−).

## What Is Origin?  
The **origin** is the point (0,0) — the single point where the x-axis and y-axis intersect.

Properties of the origin:

- Its coordinates are both zero: (0,0).
- It does **not** belong to any of the four quadrants.
- It is the reference point from which every other point's coordinates are measured.

## Plotting Points on Quadrants  
To plot a point (x,y) in the coordinate plane:

**Step 1.** Start at the origin (0,0).
**Step 2.** Read the **x-coordinate** and move horizontally — _right_ if x is positive, _left_ if x is negative.
**Step 3.** Read the **y-coordinate** and move vertically — _up_ if y is positive, _down_ if y is negative.
**Step 4.** Mark the point.

## Trigonometric Values in Different Quadrants  
The **ASTC rule** summarizes which trig functions are positive in each quadrant:

- **Q I — All:** all six trig functions are positive.
- **Q II — Sine:** sine is positive; the other four are negative.
- **Q III — Tangent:** tangent is positive; the other four are negative.
- **Q IV — Cosine:** cosine is positive; the other four are negative.

### Why this pattern.  
The signs of sin, cos, and tan follow the quadrant.

## Common Mistakes With Quadrants  
- Numbering quadrants clockwise instead of counterclockwise.
- Putting axis points in a quadrant.
- Confusing Q II with Q IV (or Q I with Q III).

## Key Takeaways  
- Four quadrants, numbered I, II, III, IV counterclockwise from the upper right.
- Sign conventions: Q I (+,+), Q II (−,+), Q III (−,−), Q IV (+,−).
- The origin (0,0) does not belong to any quadrant. 
- Axis points belong to no quadrant — only to the x-axis or y-axis.