Arithmetic
Square Roots
The Discovery That Got Someone Killed
The Pythagoreans believed that **everything is a number**, meaning a ratio of whole numbers (a fraction). But when **Hippasus** constructed the diagonal of a unit square and tried to express its length as a fraction, he found it was impossible. √2 is an irrational number.
Everything is a number., Pythagoras (before the discovery of √2)
Hippasus's discovery showed that the number line contains 'holes' between fractions. Filling those holes took millennia, only in the 19th century did mathematicians formalize the real numbers.
The diagonal of a square with side 1 equals √2. This discovery stunned the ancient Greeks: the number exists, it can be constructed, but it cannot be written as a fraction! Square roots open the door to the world of irrational numbers.
- **Geometry:** Pythagorean theorem, diagonals, distances
- **Physics:** kinetic energy, waves, oscillations
- **Statistics:** standard deviation
What is a Square Root
Quake III Arena (1999) computes 1/√x in 4 Newton iterations: the fast inverse square root with the 0x5f3759df magic constant. If 5² = 25, finding the number whose square equals 25 is the reverse problem, **extracting a square root**.
**√a**, the square root of a, is the number whose square equals a. √25 = 5, because 5² = 25 √9 = 3, because 3² = 9 √2 ≈ 1.414...
The square root is written as √ (radical). The number under the radical is called the **radicand**. √a is defined only for a ≥ 0.
What is √144?
Properties of Square Roots
Roots follow their own rules. Some resemble the power rules, but there are important differences!
**Factoring out from under the radical:** √72 = √(36 × 2) = 6√2 Find the largest perfect square that divides the radicand. 72 = 36 × 2, where 36 = 6², a perfect square.
A root is a power of 1/2: √a = a^(1/2). So root rules follow from power rules. But that's a topic for fractional exponents!
Simplify √98:
Estimating Roots
What is √50? The exact value is irrational. Estimation is straightforward: √49 = 7, √64 = 8, so √50 is just above 7.
**Quick estimation:** For √N, if N is close to a perfect square k²: √N ≈ k + (N - k²) / (2k) √50 ≈ 7 + (50-49)/(2×7) = 7 + 1/14 ≈ 7.07
Before calculators, people used tables and approximation methods. Heron's algorithm was invented 2,000 years ago, and is still used in computers today!
Between which two integers does √30 lie?
Irrationality of Roots
√4 = 2, a rational number. What about √2? The ancient Greeks proved that √2 cannot be expressed as a fraction. It is an **irrational number**.
**Historical note:** The discovery of the irrationality of √2 is attributed to the Pythagorean Hippasus (5th century BCE). According to legend, this discovery so shocked the Pythagoreans (who believed that everything is a number = a fraction) that Hippasus was drowned at sea.
Irrational numbers were an important discovery. They showed that between rational numbers there are 'gaps' that fill the number line to make it continuous.
√(a² + b²) = a + b
√(a² + b²) ≠ a + b, roots cannot be 'opened' for sums
√(3² + 4²) = √(9 + 16) = √25 = 5. But 3 + 4 = 7 ≠ 5. The Pythagorean theorem uses exactly √(a² + b²), not a + b. The root of a product factors out; the root of a sum does not!
Which of the following is rational?
Key Ideas
- √a, the number whose square equals a
- √(ab) = √a × √b, but √(a+b) ≠ √a + √b
- √a exists only for a ≥ 0
- √n is rational only when n is a perfect square
Related Topics
Roots are connected to powers and numbers:
- nth Roots — Generalization of the square root
- Irrational Numbers — Numbers that are not fractions
- Pythagorean Theorem — The main application of roots
Вопросы для размышления
- Why is √(a²) = |a|, not simply a?
- How did ancient mathematicians compute roots without calculators?
- Why was the discovery of irrational numbers so shocking?