The origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.
The origin is the starting point for every ordered pair: the first number says how far to go left or right from it, the second how far up or down.
Formal definition and notation
In a rectangular coordinate system, the origin is the intersection of the perpendicular axes, the point whose coordinates are all zero: (0, 0) in the plane and (0, 0, 0) in space. On a number line, the origin is the point for 0. Distances and directions of all other points are measured from it.
Symbol
Read it as
Meaning
O
O (the letter)
the usual label for the origin
(0, 0)
zero, zero
the coordinates of the origin in the plane
Key facts
The origin lies on both axes, so it belongs to no quadrant.
The distance from the origin to (x, y) is √x2 + y2: (3, 4) is 5 units away.
Zero is a number; the origin is a point, the place on a graph or number line that stands for zero.
Quick check: 3 questions
Question 1 of 3
What are the coordinates of the origin?
Show answer and explanation
Answer: D. The origin is where both axes cross, so both coordinates are 0. (0, 1) is on the y-axis, (1, 0) is on the x-axis, and (1, 1) is in Quadrant I.
Question 2 of 3
Which point is farthest from the origin?
Show answer and explanation
Answer: C.√16 + 16 = √32 ≈ 5.66. The others are exactly 5 away: √25 each. Adding coordinates or ignoring signs can make (3, 4) or (−4, −4) look closer than they are.
Question 3 of 3
Rotating (2, −7)180° about the origin gives:
Show answer and explanation
Answer: A. A half turn about the origin sends (x, y) to (−x, −y): (−2, 7). (2, 7) reflects across the x-axis, and the other two swap coordinates, which is a 90° turn or a reflection over y = x.
Origin: frequently asked questions
Why is the origin called the origin?
Origin means starting point. Every location on the coordinate plane is described by how far it is from this point: right or left, then up or down. Mathematicians chose (0, 0) as the reference, the same way a map might measure everything from one landmark. Moving the origin changes every point’s coordinates, but not the shapes.
Is the origin in a quadrant?
No. The four quadrants are the regions between the axes, and the origin sits on both axes at once. The same is true for every point on the x-axis or y-axis, such as (5, 0) or (0, −3). Only points with both coordinates nonzero lie in a quadrant, and their signs tell you which one.
Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6QuadrantA quadrant is one of the four regions that the x-axis and y-axis split the coordinate plane into. They are numbered I, II, III and IV counterclockwise, starting at the top right. The signs of a point’s coordinates tell its quadrant: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV.Geometry · Grade 5, Grade 6, Grade 7Y-InterceptThe y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).Algebra · Grade 8, Algebra 1Proportional RelationshipA proportional relationship is a relationship between two quantities in which one is always the same multiple of the other, so y = kx for a fixed number k. The ratio yx is the same for every pair, and the graph is a straight line through the origin. Example: at 3 dollars per pound, cost = 3 × pounds.Ratios and Proportions · Grade 7, Grade 8, Algebra 1RotationA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).Geometry · Grade 8, GeometrySubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7ZeroZero (0) is the number that represents none or no quantity. It is neither positive nor negative and sits in the middle of the number line, between the negative and positive numbers. Adding zero to a number does not change it, as in 9 + 0 = 9, so zero is called the additive identity.Number System · Grade 3, Grade 4, Grade 5NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7Distance FormulaThe distance formula finds the straight-line length between two points on the coordinate plane: d = √(x2 − x1)2 + (y2 − y1)2. It is the Pythagorean theorem in disguise: the horizontal and vertical differences are the legs of a right triangle, and the distance is the hypotenuse.Geometry · Grade 8, Algebra 1, Geometry