## “Four”: Up, Down, Left, Right – All Four Directions

Two pairs are 4. Number 4 is very common. When you play bridge (a card game), you have four people.

> Look at the identity: 4 = 2 × 2.

Conclusion: 4 is the smallest composite number (a composite number can be written as the product of smaller positive integers). Number 4 is a perfect square; it is also a power of 2.

> Quadrilaterals (4-sided polygons) include many common shapes, like rectangles, parallelograms, etc. Can you give more examples?

> In a plane there are 4 directions: up, down, left and right.

> A tetrahedron (4-faced polyhedron) has 4 vertices, 6 edges and 4 faces.

A tetrahedron – also called a triangular pyramid, is the simplest polyhedron (in terms of number of faces is smallest).

There is a favourite topic in science fiction: 4-dimensional space.

> Number 4 inspired the imagination of mathematicians and their enthusiastic exploration: and yield fruitful results: the most famous are “four-colours theorem” and the “sum-of-four-squares” [this item is a bit hard].

About Four-colours Theorem: On a plane or a sphere, it suffice to employ only four colours to seperate all neighbouring regions apart.

About Sum-of-four-squares theorem: Any integer can be written as the sum of at most four squares.

For example: 5 = 2^{2}+ 1^{2}, 25 = 4^{2}+ 3^{2}[As Sum of two perfect squares]

7 = 2^{2}+ 1^{2}+ 1^{2}+ 1^{2}[As Sum of four perfect squares]

> If we list all powers of 4: 4, 16, 64, 256, 1024, 4096, etc.

Have you noticed: the last digit (units-place) is either 4 or 6.

By the way, the complimentary number of 4 is 6, and 4 × 6 = 24.

Conclusion: ________________ (The reader may summarize by himself).

> The sum of four consecutive integers is an even number, however, it is NEVER a multiple of 4. For example: 1 + 2 + 3 + 4 = 10.

Since ten resembles “perfect”, so as number 4 is loved in some cultures (for example, the Pythagorean scholars in ancient Greece).