What is dimension of a space (maths)?

In mathematics, dimension of a space is informally defined as the minimum number of coordinates, values, numbers required to specify an exact position in that space.

Lets workout some examples and in process learn dimensions of different type of spaces.

Example 1:

Just imagine below is a city with 6 houses, L1,L2,L3 all three houses are separated by unit distance on the left of fountain and H1,H2,H3 are also separated by unit distance on the right of fountain. Distance between fountain and L1, H1 respectively are also unit distance.

What is the position of Ram’s house? H2 (2 units to the right of fountain)
What is the position of Abhi’s house? L3 (3 units to the left of fountain)

Example 2:

Below is a number line, origin at the centre. Positive numbers are unit distance separated to the right of origin and negative numbers are unit distance separated to the left of origin. Different people are standing on the numbers.

What is the position of Ryan? -2 (2 units to the left of origin)
What is the position of Houdi? 4 (4 units to the right of origin)

In all the above examples if someone just tells us one value we will be able to locate that position exactly. So they are 1-dimensional space.

Example 3:

Below is a classroom where student with roll numbers are sitting in a row and column fashion. Columns are numbered 1 to 7 from left to right. Rows are numbered 1 to 7 from bottom to top.

How can we specify exact position of roll 22 and roll 2 students?

Position of Roll 22: (3,3) 3rd row, 3rd column or 3rd column, 3rd row
Position of Roll 2: (4,2) 4th row, 2nd column or (2,4) 2nd row, 4th column

Another way can be

Position of Roll 22: (r1, theta1) radius r1 and angle theta1 or (theta1, r1) angle theta1 and radius r1
Position of Roll 2: (r2, theta2) radius r2 and angle theta2 OR (theta2, r2) angle theta2 and radius r2

So we need minimum 2 values (radius, angle) or (row number, column number) to specify an exact position in that space. So its a 2-dimensional space.

Example 4:

Lets take one more example where a society has houses in circular fashion. The circle is of radius r and all the 8 houses of separated by unit distance.

What is the position of Ram’s house?

Position of Ram’s house: theta2 or -(360 deg-theta2) w.r.t. horizontal line touching H5

In the above space we needed only one value theta to specify the exact position. So its a 1-dimensional space.

As you saw different types of spaces have different dimensions. These are just few examples you can pick different space like solid or hollow sphere and try finding out their dimension.

Mathematically : line or circle are examples of 1-Dimensional space, plane or surface of sphere are examples of 2-Dimensional space, Solid cube or Solid sphere are examples of 3-Dimensional space.

In case of any doubt or queries or any suggestions please leave a comment below. Would love to help.