A length with no fraction

The idea

Numbers on the number line that cannot be expressed as a ratio of integers are irrational numbers; the diagonal of a square of side 1 unit has the irrational length √2.

The diagonal

Draw a square whose sides are each exactly 1 unit long. By the Baudhāyana–Pythagoras Theorem, its diagonal d satisfies 1² + 1² = d². So d² = 2.

Can the diagonal d be written as a fraction p/q?

• Yes, some fraction must work
• No, no fraction works
• Not sure

Hunt for the fraction

First build the square on the diagonal. Then try to find a fraction p/q whose square is 2.

Irrational

Numbers on the number line that cannot be expressed as a ratio of integers are called irrational numbers. The diagonal of a square of side 1 unit has the irrational length √2.

d² = 1² + 1² = 2, so d = √2. You got close with fractions like 17/12, but never exactly 2. Baudhāyana met lengths like this in his Śulbasūtra, around 800 BCE.

Notes

Numbers on the number line that cannot be expressed as a ratio of integers are irrational numbers. The diagonal of a square of side 1 unit has the irrational length √2.

Check yourself

A square has side 1. Its diagonal d satisfies d² = 1² + 1². What is d²?

Answer: 2

1² + 1² = 1 + 1 = 2, so d² = 2 and d = √2.

What is an irrational number?

Which of these is an irrational number?

Is every number on the number line a rational number?

• A number with a decimal point. Many numbers with a decimal point are fractions: 0.5 = 1/2.
• A number on the number line that cannot be expressed as a ratio of integers — correct. Yes. √2 is an example.
• A negative number. Negative numbers such as −3/4 are ratios of integers.
• 3/4. 3/4 is a ratio of the integers 3 and 4.
• −5. −5 = −5/1 is a ratio of integers.
• √2 — correct. Yes. The diagonal of a unit square cannot be written as p/q.
• No. Some are irrational, such as √2 — correct. Yes. The length √2 sits on the number line but is not a fraction.
• Yes. Every length is a fraction. For centuries people believed this, but the diagonal of a unit square shows it is false.
• Only the negative ones. Rational and irrational numbers can both be positive or negative.