The square root of 2 | square root of 2
ThePythagoreanphilosophersofancientGreecehadadeepregardfornumbers.Arithmetic[1]andgeometry[2]wereguardedasaformofsecretknowledgetobedisclosedonlytoaprivilegedfew.TheybelievedtheUniverseitselfwasamanifestationofnumbersandgeometry,sothatbyexploringmathematicstheyweregaininginsightsintoadeeperlayerofcosmicreality.SoitcameasaprofoundshockwhenthePythagoreansdiscoveredthatsomenumberslayoutsidetheirtidyintellectualscheme.Asyoungchildren,wefirstlearntocountthewholenumbers,1,2,3…whichmathematiciansca...
The Pythagorean philosophers of ancient Greece had a deep regard for numbers. Arithmetic[1] and geometry[2] were guarded as a form of secret knowledge to be disclosed only to a privileged few.
They believed the Universe itself was a manifestation of numbers and geometry, so that by exploring mathematics they were gaining insights into a deeper layer of cosmic reality. So it came as a profound shock when the Pythagoreans discovered that some numbers lay outside their tidy intellectual scheme.
As young children, we first learn to count the whole numbers, 1, 2, 3 … which mathematicians call the “natural” numbers. We’re then taught fractions, which are expressed as the ratio of natural numbers, such as 2/5 or 1/3. After that come decimals.
Every fraction, we learn, may be expressed as a decimal – for example, 2/5 = 0.4 and 1/3 = 0.33333… (where “…” indicates that the succession of 3s goes on forever). But is the reverse true? Can every dec...