Distance

Distance is a way to measure how far apart things are, whether they are objects, people, or even ideas. In physics or everyday life, distance often means the actual length between two points, like how far a ball is thrown or how long a road is. Sometimes, people use the word "distance" to talk about how different things are from each other, like how two groups of numbers or pieces of text compare.

In social sciences, distance can also mean how separate people feel from each other, such as when someone feels far away from another person even if they are close by. Many ideas about distance, both real and imaginary, are studied in mathematics using something called a metric space.

The concept of distance helps us understand the world around us and how things relate to one another, whether we are measuring the space between two cities or the feeling between two friends.

Distances in physics and geometry

Main article: Euclidean distance

The distance between two places can mean different things depending on the situation. In everyday life and in physics, distance often means the straight-line length between two points. This is the shortest path and is used in classical physics, like in Newtonian mechanics.

Straight-line distance can be calculated using a math rule called the Euclidean distance. For points on a flat surface, like paper, the distance is found using a simple formula that adds up the squares of how far apart the points are horizontally and vertically, and then takes the square root. This idea can also be used for points in three dimensions, like in space, by adding another part for the up-and-down distance.

There are many ways to measure straight-line distances. You can use a ruler for short distances, radar for longer ones, or special tools like interferometry for very tiny distances. For extremely far distances, like those between stars, scientists use a method called the cosmic distance ladder.

Main articles: Geographic distance and geodesic

When measuring distances on the Earth, the straight line through the ground isn’t practical. Instead, we measure the shortest path along the Earth’s surface, similar to how a bird might fly. This is called the great-circle distance. More generally, the shortest path on any curved surface is called a geodesic.

In the theory of relativity, distances can change depending on how you are moving. For very large distances, like between galaxies, the expansion of the universe also affects how we measure distance.

Unusual definitions of distance can help us understand certain situations. For example, the distance between two points along city streets can be different from the straight-line distance. In chess, the distance a king moves between two squares can also be measured in a special way.

Metaphorical distances

Many ideas in math, science, and engineering use the idea of "distance" to show how different or separate things are. For example, in statistics, we can measure how different two sets of chances or possibilities are. In computer science, we can see how different two words or strings of letters are by counting how many letters need to be changed to make them match.

In graphs, which are like maps of connections, distance can show how many steps it takes to get from one point to another. In everyday life, distance can also describe feelings — like how close or far someone feels from another person in terms of friendship, or how different groups in society might be from each other.

Mathematical formalization

Main article: Metric space

In math, distance can be thought of in a special way called a metric. This means there is a rule or way to measure how far apart two things are. This rule follows a few important ideas:

1. Something is always right next to itself, so the distance is zero.
2. If two things are different, the distance between them is more than zero.
3. The distance from one thing to another is the same as going back the other way.
4. If you go from one place to another and then to a third place, it won't be shorter than going straight from the first to the third place. This means taking a stop in between doesn't save distance.

Distance between sets

There are several ways to measure how far apart objects are when they have more than one point:

• You can measure the distance between important points, like the center of mass. This is used for big distances, such as the Earth–Moon distance.
• You can also measure the distance between the closest points of the two objects. For example, the altitude of an airplane is its distance from the Earth. This idea is also used in geometry to find the distance from a point to a line, distance from a point to a plane, or the perpendicular distance between certain flat spaces called affine subspaces.

This idea can even help us understand the distance between groups of points in a space. The distance between two groups is the smallest distance between any two points — one from each group. However, this way of measuring doesn’t always follow the normal rules for distances, especially when the groups overlap.

• The Hausdorff distance looks at how much two groups differ by measuring how far apart their farthest points are. This method works well for certain groups in space.

Related ideas

Further information: Length

The word distance can also describe ideas that are not just about how far apart things are.

Distance travelled

The distance travelled is how long a path an object takes between two points. For example, it could be how far you walk in a maze or the distance shown on a milepost or an odometer. Sometimes, this distance can loop back to where it started, like a ball thrown up into the air or the Earth moving in its orbit. In math, this is called the arc length.

Distance travelled can also be signed: moving forward is shown as a positive number, and moving backward is shown as a negative number.

Circular distance is how far a point on the edge of a wheel travels. This idea helps when making vehicles or machines. If a wheel has a radius of 1, it travels about 6.28 units (2π) with each turn.

Displacement and directed distance

Main article: Displacement (vector)

In physics, displacement measures how an object's position changes over time. Unlike distance, which only tells how much space is covered, displacement also tells the direction. The directed distance is the difference between two places, showing where you start, where you end, the direction, and how far. For example, the directed distance from the New York City Main Library flag pole to the Statue of Liberty flag pole includes:

• A starting point: library flag pole
• An ending point: statue flag pole
• A direction: -38°
• A distance: 8.72 km