HOME

David Terr
Ph.D. Math, UC Berkeley

 

Home >> Everyday Math >> 5. Miscellaneous >> 5.3. Important Numbers

>> 5.3.2. The Golden Ratio (φ ≈ 1.618)

 

5.3.1. The Square Root of Two (2 ≈ 1.414)

The square root of two is one of the most important mathematical constants. This number is equal to the length of the diagonal of a square with sides of length 1, as shown in the following diagram:

unit square with diagonal

Figure 5.3.1.1: The Unit Square, With Diagonal of Length √2

 

It is easy to see this from the Pythagorean Theorem. Consider either of the two right triangles shown in the above figure. The lengths of the legs of these right triangles are both equal to 1. Thus, we have a = b = 1. The Pythagorean Theorem then tells us that the square of the length of the hypotenuse c is given by

c2 = a2 + b2

= 12 + 12

= 1 + 1 = 2

whence c = √2 as claimed.

It is easy to show that √2 is an irrational number, i.e. it cannot be expressed as the ratio of two integers. Suppose otherwise, i.e. suppose there exist integers a and b such that √2 = a / b. Squaring both sides, we see that 2 = a2 / b2, whence a2 = 2 b2. Thus we see that a2 is even, whence a must be even. Now in order for the fraction to be reduced, b must be odd (otherwise just divide both a and b by 2). But if a is even, then a2 = 2 b2 must be divisible by 4, implying that b2 is divisible by 2, whence b2 and thus b is even. But since we assumed that b is odd, we get a contradiction. This implies that our original assumption that √2 is rational was wrong. Thus we see that √2 is irrational.

In ancient Greece, Hippasus of Metapontum, one of the Pythagoreans, proved that √2 was irrational. This created such an uproar among the society that legend has it that he was drowned. [1]

The value of the square root of two to 10 decimal places is 1.4142135624. [2]

 

References:

  1. http://en.wikipedia.org/wiki/Irrational_number (Wikipedia page on irrational numbers)
  2. http://en.wikipedia.org/wiki/Square_root_of_2 (Wikipedia page on the square root of two)

 

Home >> Everyday Math >> 5. Miscellaneous >> 5.3. Important Numbers

>> 5.3.2. The Golden Ratio (φ ≈ 1.618)