What does origin mean in math? Learn the definition of an origin in mathematics, where the origin is on a graph or coordinate plane, and see origin math examples.
Updated: 11/21/2023
Origin in Math | Definition, Graph & Examples
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Origin in Math: Multiple Choice Exercise
This activity will help you assess your knowledge of how origin in math is used.
Directions
For this activity, carefully read and select the best answer that completes each of the given statements. To do this, print or copy this page on blank paper and circle the letter of your answer.
Multiple Choice
1.) When you move towards the left of a point, x and y become __________.
A. points of origin
B. positive numbers
C. negative numbers
D. zero
2.) Which of the following statements is TRUE about plotting the point (4,7)?
A. You move 4 units to the left from the origin.
B. You move 7 units to the right from the origin.
C. You move 7 units to the left from the origin.
D. You move 4 units to the right from the origin.
3.) The y axis represents __________.
A. The number of units that you move to the left or right from the origin.
B. The number of units that you move up from the origin.
C. The number of units that you move up or down from the origin.
D. The number of units that you move down from the origin.
4.) How many units will you have to move up from the origin to plot the point (2,-5)
A. 0 unit
B. 2 units
C. -5 units
D. 5 units
5.) How many units will you need to move down the origin to get to the point (4,-7)?
A. -7 units
B. 7 units
C. 4 units
D. 0 unit
Answer Key
1.) C
2.) D
3.) C
4.) A
5.) A
Why is (0,0) called the origin?
The Cartesian plane is defined by two number lines, each with origin at 0, which intersect at the shared point (0,0). All points in the plane can be measured according to their distance from this central point.
What is the origin on a graph?
A graph in the two-dimensional coordinate plane has the point (0,0) as its origin. The origin is located at the intersection of the vertical and horizontal axes, and the distance to all points can be measured from this point.
What is an origin in math?
An origin is a single point of reference for a coordinate system, from which all other values can be measured. Its exact definition depends on the coordinates system in use and its dimension in particular.
Table of Contents
ShowThe term "origin" is regularly used in math classes and textbooks, but what does it mean?
Whenever we use numerical values to describe the physical world, including measuring lengths with a ruler or navigating using compass headings, measurements are always given relative to some agreed point of reference. Directions to head North and West mean moving in that direction starting from your current location. In mathematical terminology, this initial reference point is known as the origin.
Point of Origin Definition
The point of origin, or simply the origin for short, is the reference point in a coordinate system from which all other measurements are taken.
A coordinate system consists of one or more variables, each of which can be measured along a number line. In particular, the two-dimensional coordinate plane consists of two number lines, which, for example, could represent directions North and South and East and West. All other points on the plane can be treated as positions relative to a single point of reference called the origin, which is often labeled with the letter {eq}O {/eq}.
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Perhaps the most familiar use of coordinate planes is for graphing functions of a single variable. If {eq}y=f(x) {/eq}, we can represent each variable, {eq}x {/eq} and {eq}y {/eq} on its own number line. To understand how the origin is found in the resulting graph, we must first understand how the origin is located on a single number line, and how precisely two number lines are used to define the coordinate plane. Let's review these ideas first.
Origin on a Number Line
A number line is a straight line with graduated markings used to represent real numbers. One point on the line must represent the number zero. By default, positive numbers are counted toward the right of zero, and negative numbers are counted toward the left.
Zero plays the role of the origin on a number line because all numbers, both positive and negative, increase in absolute value (or magnitude, meaning size) the farther they are from zero. In this way, all numbers become measurements relative to a single origin.
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Constructing a Coordinate Plane Graph
A number line can be used to represent the values of a single measurement or variable. To represent two variables, we can create a coordinate plane by drawing two perpendicular number lines: a horizontal {eq}x {/eq}-axis and a vertical {eq}y {/eq}-axis. By referring to the axes, we can treat the two-dimensional plane as a grid in which any point can be identified by a coordinate pair {eq}(x, y) {/eq}. Positive and negative {eq}x {/eq} values are measured to the right and left respectively, and positive and negative {eq}y {/eq} values are measured up and down.
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The coordinate plane is also known as the Cartesian plane, after the French mathematician Rene Descartes, who first conceived the idea. {eq}x {/eq} and {eq}y {/eq} are referred to as Cartesian coordinates.
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The vertical and horizontal axes which create the Cartesian plane must intersect, and they are drawn so that the zero values of each coincide. The Cartesian coordinates of this single intersection point must then be {eq}(0, 0) {/eq}, and this point becomes the origin in the plane.
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Lets consider a few examples of graphs in the Cartesian plane and how they related to the origin {eq}(0,0) {/eq}. Let's start by looking at the graph of a linear function:
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Any point on the line is a coordinate pair {eq}(x,y) {/eq} that satisfies the equation. The easiest points to identify are the intercepts, where the line crosses the coordinate axes. The {eq}x {/eq}-intercept is {eq}(6,0) {/eq} and the {eq}y {/eq}-intercept is {eq}(0,3) {/eq}. How far away from the origin is each of these points?
Because the intercepts fall exactly on an axis, we can measure their distance to the origin at {eq}(0,0) {/eq} just like counting on a number line. The {eq}x {/eq}-intercept is 6 units away from the origin, and the {eq}y {/eq}-intercept is 3 units away.
If we look at other points on the line, you can probably see whether they are closer or farther from the origin, but finding the exact distance is not nearly so easy. To find the distance from the origin to any point in the plane requires using the formula for the hypotenuse of a right triangle, but that's a topic for another lesson.
Let's look at another example: the unit circle, which has radius 1 and is centered at the origin.
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The circle intersects each axis twice, at {eq}x=\pm 1 {/eq} and {eq}y = \pm 1 {/eq}. Referring to the scale on each axis, the distance between the origin {eq}(0,0) {/eq} and all four intercepts is exactly 1.
All points on a circle are, by definition, the same distance away from the center of the circle. Can you visualize how distance would be measured to other points on the circle, and that they also measure 1 unit away from the origin?
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In this lesson, we've learned how, in math, the point of origin, or simply origin, is a reference point from which all other measurements are taken.
A single measurement or variable can be represented on a number line. Positive and negative values are measured in relation to their origin at the number zero.
By drawing two perpendicular lines we can define a two-dimensional coordinate plane, or Cartesian plane, which can be used for graphing functions. Every point in the plane can be identified by its Cartesian coordinates {eq}(x,y) {/eq}, measured against the coordinate axes.
In the plane, the horizontal and vertical axes intersect at the point {eq}(0,0) {/eq}, which is the origin of a graph in the coordinate plane. Distances from points on the axes to the origin can be measured on their respective number lines, but calculating the distance to points off-axis requires more advanced calculations.
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Video Transcript
Origin in Math Definition
The banana you had for lunch probably originated in Costa Rica. The bus you took to school may have originated at a bus station.
An origin is a beginning or starting point, and, in mathematics, the origin can also be thought of as a starting point. The coordinates for every other point are based on how far that point is from the origin. At the origin, both x and y are equal to zero, and the x-axis and the y-axis intersect.
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Math Origin Example
Imagine that you're a pirate and you have buried your loot on a small island in the Pacific. Being one of those careful pirates who thinks ahead, you create a map so that you can find the treasure again later.
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This map isn't very useful, of course, if it doesn't give you any idea of the distance or direction to the treasure from some other useful place, like your hideout.
So, in order to keep track of where the treasure is buried relative to your hideout, you create a grid on your map.
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Each line on the grid represents 100 steps. By counting lines between the big X and your hideout (the triangle), you know how far to travel in both the up/down and right/left directions.
You find that your hideout and the treasure are three blue lines (300 steps) apart in the right/left direction. Your hideout and the treasure are also two green lines (200 steps) apart in the up/down direction. Of course, you would probably really travel diagonally straight between the two places, but it's much easier to describe the directions and distances by pretending the travel would happen along the lines on the grid.
Now, it would be much easier to describe travel between the hideout and the treasure if you numbered the green and blue lines. It would also be easier if you chose one of the two locations as the starting point.
For example, you might choose the hideout as the starting point, or origin. That way, you can say, ''Go three lines (or 300 feet) to the right.'' Your directions to your fellow pirates would become much clearer. In this case, it doesn't really matter if you call the hideout the starting point (or originating point or origin) and describe travel to the treasure, or vice versa. It's simply important that you're clear and consistent. If the hideout is the origin, the map looks like this:
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When you're at the hideout, you have moved zero footsteps in any direction. So you're at the point (0,0). To travel to the treasure, you go three lines to the right and two lines up. You end up at the point (3,2).
The numbers on the grid simply represent distances from the origin. Positive numbers indicate moving to the right or upward. Negative numbers indicate moving to the left or downward. The point (-1,-2) tells you where you are on the map (in the middle of the ocean). However, it also tells you how far you are vertically and horizontally from the hideout, as in the origin.
Lesson Summary
Let's review. In mathematics, an origin is a starting point on a grid that's the point (0,0), where the x-axis and y-axis intercept. The origin is used to determine the coordinates for every other point on the graph. In a treasure map, for example, you would determine the location of your buried treasure according to its distance from your hideout, the origin.
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Origin in Math | Definition, Graph & Examples
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