The man who invented the distance formula must have been amazed by distances. And why not? He was a traveler. He was a scientist and a philosopher always seeking the meaning in life. Aside from being educated in Greece, the distance formula inventor traveled other parts of the world to learn from other civilizations. His name was Pythagoras. You may recognize the name, as he also created the Pythagorean theorem. Earlier versions of the Distance Formula were created somewhere around 600 BC. The Pythagorean Theorem was created 10 years earlier.

What is the Concept?

Distance is a numerical description of how far apart objects are. In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and is a concrete way of describing what it means for elements of some space to be "close to" or "far away from" each other. In most cases, "distance from A to B" is interchangeable with "distance between B and A".

Why was the Distance Formula Developed?

The Distance formula was developed to receive accurate measurements of large distances between things such as cities or building locations in cities. This formula can be used in Euclidean Space distances, variational formulation of distance, and generalization to higher-dimensional objects. Keep in mind that each of these measurements require different versions of the Distance Formula.

How Can this Concept be used in real life? (Also, a Sample of the Distance Formula in Use).

This concept can be used to measure very large distances based on coordinates. The formula could, in this sense, be used to find the distance you would need to travel to get from Los Angeles to new york. You would plug in LA's coordinates (X1 and Y1) and New York's coordinates (X2 and Y2) into the distance formula. The formula is strictly based on your current coordinates and the destination's coordinates on a coordinate plane.

**Studying The Distance Formula**

**By Gregory Phifer,**

January 30, 2015

January 30, 2015

What is the Concept? (Cont.)

In analytic geometry, the distance between two points of the xy-plane can be found using the distance formula. The distance between (x1, y1) and (x2, y2) is given by:

d=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.\,

n the Euclidean space Rn, the distance between two points is usually given by the Euclidean distance (2-norm distance). Other distances, based on other norms, are sometimes used instead.

For a point (x1, x2, ...,xn) and a point (y1, y2, ...,yn), the Minkowski distance of order p (p-norm distance) is defined as:

1-norm distance = \sum_{i=1}^n \left| x_i - y_i \right|

2-norm distance = \left( \sum_{i=1}^n \left| x_i - y_i \right|^2 \right)^{1/2}

p-norm distance = \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}

infinity norm distance = \lim_{p \to \infty} \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}

= \max \left(|x_1 - y_1|, |x_2 - y_2|, \ldots, |x_n - y_n| \right).

What is the Concept? (Cont.)

Here are Some Pictures, representing the Distance Formula

A Sample Distance Form. Problem

How Distance is Calculated based on coordinates

Sample Calculations

Find the distance between J and K

K

J

Find the distance between A and B

A

B

Conclusion

We can conclude that the Distance Formula has many uses. It can be used for finding the distance from one point of a circle to another point on the circle, or it could be used to measure the distance from New York to Los Angeles. What will you measure next?