Be There or be Square!

The square root of 2 is the positive real number that satisfies x² -2=0. It is most simply the number whose square is 2.

It is also the diagonal length of a right triangle who's side lengths are both 1. We can prove this without even using Pythagoras' theorem. Check out the animation below. The area in the middle is C².

Notice that when the triangles move, we can see that the middle area must also be the sum of two squares both of area 1. Thus, C²=2. This means that C must be root 2!

Interesting equalities:

• √2= 2sin(π/4)
• √2= e^.5ln(2)

Root 2 is Discovered!

The idea of the square root of 2 first appeared sometime before 600 BCE in an ancient Indian math text called the Sulbasutras, written by the Vedic Hindu scholars. Root 2 was the first surd (root) ever discovered. They called root 2 “dvi-karani”, the Sanskrit word which literally means “that which produces 2”. The scholars came up with a pretty good estimation of root 2, correct to 5 decimal places:

1 + 1/3 + 1/(3*4) - 1/(3*4*34) = 577/408 ≈ 1.414215686

Later, around 430 BCE, tradition has it that Pythagoras determined that a square, whose sides are length 1, must have a diagonal of length root 2. As a result, root 2 is sometimes referred to as Pythagoras’ constant.

Mad Influence!

Root 2 has expanded our number system past simply adding, subtracting, multiplying, and dividing whole numbers. Before its irrationality was discovered, the Greeks believed that every number could be expressed as a combination of whole numbers with those four operations. Root 2 broke the mold and showed us that there were more numbers than we ever imagined existing.