Root 2 - Numberphile

Summary

In this engaging conversation, Roger Bowley and James Grime explore the fascinating properties of the square root of 2 (√2), from its appearance in geometry to its role in the dimensions of A4 paper. They discuss how √2 emerges from the Pythagorean theorem and provide a mathematical proof of its irrationality. The script also delves into the historical context of the Pythagoreans, who believed all numbers could be represented as ratios of integers, and the persecution of Hippasus, who first demonstrated that √2 is irrational. This exploration highlights the evolution of mathematical thought and the challenges to conventional wisdom.

Takeaways

• 😀 The square root of 2 (approximately 1.414) is a fundamental irrational number in mathematics.
• 📏 The ratio between the long and short edges of A4 paper is the square root of 2, making it a special and intentional design.
• 📐 When you fold A4 paper in half, the new size, A5, also retains the same ratio of the square root of 2.
• 🔢 The square root of 2 can be derived from the Pythagorean theorem using a right triangle with both legs of length 1.
• 📚 The Pythagoreans believed that all numbers could be represented as ratios of integers, a concept challenged by the square root of 2.
• 💡 The discovery of the square root of 2’s irrationality was so controversial that its discoverer, Hippasus, was allegedly persecuted.
• 🔄 A0 paper is defined as having an area of 1 meter squared and a ratio of square root of 2, highlighting its unique properties.
• ⚖️ The square root of 2 is the only ratio that works for the scaling of paper sizes without distortion.
• ✖️ Proof by contradiction is used to demonstrate that the square root of 2 cannot be written as a fraction of two integers.
• 🚫 The irrationality of the square root of 2 means it cannot be expressed as a simple fraction, unlike rational numbers.
• ⚔️ Throughout history, challenging conventional wisdom has often led to the persecution of those who introduced new ideas, as seen with Hippasus and others like Giordano Bruno.

Q & A

What is the significance of the square root of 2 in the context of A4 and A5 paper?

The square root of 2 is the ratio between the long edge and the short edge of A4 paper. When you fold A4 paper in half to create A5 paper, the ratio between the long and short edges remains the same, i.e., the square root of 2. This property allows the paper sizes to be scaled up or down without changing the proportions.

How is the square root of 2 related to Pythagoras' Theorem?

The square root of 2 comes from Pythagoras' Theorem. If you take a right-angled triangle with two sides of length 1, the length of the hypotenuse will be the square root of 2, because the square of 1 plus the square of 1 equals 2, and the hypotenuse is the square root of that sum.

Why is the ratio of square root of 2 used in A0, A4, and A5 paper sizes?

The ratio of the square root of 2 is used in A0, A4, and A5 paper sizes because it allows for consistent scaling. If you cut a sheet of A0 paper in half, you get two sheets of A1 paper, and this scaling property is maintained all the way down through A4 and A5, without distorting the proportions.

What is the role of Hippasus in the discovery of irrational numbers?

Hippasus, a disciple of Pythagoras, is believed to have discovered that the square root of 2 is an irrational number, meaning it cannot be expressed as a ratio of two integers. This discovery went against the Pythagorean belief that all numbers could be represented as ratios of integers, leading to his persecution.

What is an irrational number, and why was the discovery of the square root of 2 significant?

An irrational number is a number that cannot be expressed as a fraction or ratio of two integers. The discovery of the square root of 2 as irrational was significant because it challenged the Pythagorean belief that all numbers were either integers or ratios of integers.

What historical consequences did Hippasus face for his discovery of irrational numbers?

According to legend, Hippasus was persecuted for his discovery of irrational numbers, particularly the irrationality of the square root of 2. It is said that he was either drowned or left to die on a deserted island by members of the Pythagorean cult who rejected his findings.

What does the concept of mathematical contradiction involve in proving the irrationality of square root of 2?

The proof by contradiction assumes that the square root of 2 can be written as a fraction of two integers. By manipulating the equation and showing that both integers must be even, the proof reveals a contradiction because fractions should be in their simplest form, meaning that both numbers cannot be even.

Why is it important that the numbers a and b are in their smallest possible terms when proving the irrationality of square root of 2?

It is important that the numbers a and b are in their smallest possible terms because if they were not, the fraction could be simplified further, which would invalidate the proof. The proof relies on the assumption that a and b are in their simplest form, and any contradiction shows that the assumption was false.

What is the outcome of the proof by contradiction for the square root of 2?

The proof by contradiction leads to the conclusion that the square root of 2 cannot be written as a fraction of two integers because assuming it can results in both a and b being even, which contradicts the assumption that the fraction is in its simplest form. Therefore, square root of 2 is irrational.

How does the discussion of irrational numbers relate to modern mathematical concepts like complex numbers?

The discussion about irrational numbers parallels modern mathematical concepts like complex numbers. Just as people struggled to accept irrational numbers at first, some today find it difficult to accept complex numbers. Both concepts challenge conventional wisdom but are ultimately valid and essential parts of mathematics.