Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.
Definition of the Ruler Postulate
The ruler postulate tells us how to find the distance between two points. First, we have to match each of the two points to a number. We call these numbers the coordinates of the points. Then, we subtract the coordinates from each other. If the resulting difference is negative, we take the absolute value of the result because the distance is always a positive number. We can see this process in the following formula:
Distance is equal to the absolute value of C1 and C2 where C1 and C2 stand for the coordinates of the two points. It does not matter in which order you put the coordinates into the formula.
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The Ruler Postulate on a Ruler
Believe it or not, the ruler postulate actually has to do with using a ruler. The ruler postulate doesn't care whether you're using a ruler with inches or centimeters (or some other measuring system you want to create). When you use the ruler postulate on a ruler, you are not using the ruler in the normal way that you usually measure between two points. Usually, we line up the zero with one point and then see where the other point falls on the ruler. With the ruler postulate, we are just using the ruler to give each point a coordinate. So, we line up the ruler along all of the points at one time and assign each point the number on the ruler that it falls on. Just make sure the ruler is long enough to include all of the points that you are working with so that each point ends up with a coordinate.
Let's start with a couple of examples with points on a ruler. Here, we have a diagram of a ruler with points R, U, L, and E. Each point is matched with a coordinate: R is matched with 1, U is matched with 2, L is matched to 4, and E is matched to 5.5.
Now, let's find the distance between point R and point L. We will fill in the ruler postulate formula with the coordinates for R and L and simplify:
So, the distance from R to L is 3 units. Notice that I didn't say inches or millimeters. We're not using the ruler to measure length. Distance here is about the number of units between coordinates - no matter what those units are.
Let's find the distance between another pair of points on the ruler: point E and point U. We will fill in the ruler postulate formula with the coordinates for E and U and simplify:
So, the distance from E to U is 3.5 units.
The ruler postulate can also be used to find distances on a number line. We will look at a couple of examples of that in the next section.
The Ruler Postulate on a Number Line
When you need to find the distance between two points on a number line, you are going to use the same formula we just used on the ruler. The only thing that may be different between the number line and the ruler is that a number line contains negative and positive numbers, so be careful with your negatives! If you tend to have trouble subtracting negative numbers, you can always double-check the result you get from the ruler postulate by counting the spaces on the number line between the two points.
Let's try a couple of examples using points A, B, C, and D on the number line:
Each point is matched with a coordinate: A is matched to -6, B is matched to -4, C is matched to -1, and D is matched to 3.
Now, let's find the distance between point C and point A. We will fill in the ruler postulate formula with the coordinates for C and A and simplify. Make sure you are careful about your negatives as you simplify.
So, the distance from C to A is 5 units.
Let's find the distance between another pair of points on the number line, point B and point D. We will fill in the ruler postulate formula with the coordinates for B and D and simplify:
So, the distance from B to D is 7 units.
The ruler postulate is used to find the distance between two points. The formula for the ruler postulate is:
where C1 and C2 stand for the coordinates of the two points.
The coordinates of the points are the number values assigned to each point, either on a ruler or a number line. When using the ruler postulate with a number line, be especially careful in your calculations with negative numbers.
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Ruler Postulate: Definition & Examples
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