Arithmetic
Irrational Numbers
The Crisis That Changed Mathematics
The Pythagorean school built mathematics on a simple belief: **'Everything is number'**, meaning whole numbers and their ratios. Musical intervals, planetary motion, the harmony of the world, all should be expressible as fractions. The discovery of √2 destroyed that foundation.
The first 'foundational crisis' taught mathematicians an important lesson: intuition can be wrong. Filling the 'gaps' between rational numbers took two thousand years, only in the 19th century did Dedekind and Cantor rigorously define the real numbers.
The diagonal of a unit square exists, it can be drawn. But its length cannot be written as a fraction, it is √2. The discovery of irrational numbers stunned Greek mathematicians: the numbers exist, but they are not fractions! It expanded the concept of number and changed mathematics forever.
- **Geometry:** √2, √3, √5, lengths of diagonals
- **Physics and engineering:** π in formulas for circles and waves
- **Economics and biology:** e in growth models
The Discovery of Irrational Numbers
Around 500 BC the Pythagoreans believed: 'Everything is number', meaning whole numbers and their ratios (fractions). The diagonal of a square with side 1 shattered that belief: √2 cannot be written as p/q.
An **irrational number** is a number that CANNOT be expressed as a fraction p/q. In decimal form: an infinite NON-repeating decimal. √2 = 1.41421356237309504880168872420969807856967187537694... The digits go on without any repeating pattern.
The word 'irrational' literally means 'not a ratio' (not a ratio). These numbers cannot be expressed as a ratio of integers.
What does 'irrational number' mean?
Proof of the Irrationality of √2
The classic proof by contradiction: assume √2 is a fraction, then derive a contradiction.
**Which roots are irrational:** √n is irrational when n is NOT a perfect square. √4 = 2, rational (4 = 2²) √5, irrational (5 ≠ k²) √9 = 3, rational (9 = 3²) √10, irrational
This proof is a model of mathematical rigor. It is over 2,500 years old and remains flawless.
What is the key step in proving the irrationality of √2?
The Number π
π is the most famous irrational number. The ratio of a circle's circumference to its diameter is the same for ALL circles: that's π.
**Mnemonic for π:** 'How I want a drink, alcoholic of course', count the letters: How(3) I(1) want(4) a(1) drink(5) = 3.1415 Or remember: 3.14159265...
The transcendence of π proves the impossibility of 'squaring the circle', constructing a square with the same area as a given circle using only compass and straightedge.
Why is 22/7 NOT the exact value of π?
The Number e
e is the second great irrational number after π. It appears in compound interest, population growth, radioactive decay, anywhere there is exponential growth.
**Mnemonic for e:** e = 2.7 1828 1828 45 90 45... 2.7, then 1828 twice (Tolstoy's birth year), then angles of an isosceles right triangle: 45-90-45.
The numbers e and π are connected more deeply than they appear. Euler's formula e^(iπ) = -1 links them to the imaginary unit i and is considered the most beautiful formula in mathematics.
Irrational numbers are unpredictable and chaotic
Irrational numbers are strictly defined and computable
√2, π, e are not 'random' numbers. They have precise definitions, and any digit after the decimal point can be computed. They are non-repeating, but not chaotic. Non-repetition is a property, not a flaw. Many algorithms efficiently compute billions of digits.
How did the number e arise?
Key Ideas
- Irrational numbers cannot be expressed as p/q
- √n is irrational when n is not a perfect square
- π, ratio of circumference to diameter
- e, limit of (1 + 1/n)ⁿ as n → ∞
Related Topics
Irrational numbers complement rationals:
- Rational Numbers — Fraction-numbers (ℚ)
- Real Numbers — ℝ = ℚ ∪ irrationals
- Square Roots — √n is often irrational
Вопросы для размышления
- Why did the discovery of irrational numbers shock the Pythagoreans?
- How can one prove that √3 is irrational (by analogy with √2)?
- What do π and e have in common besides irrationality?