Mathematics · Classes 9-10 · Chapter 1

Rational and Irrational Numbers.

√9 wears a root sign, yet it is exactly 3. √13 can never be written out in full. What separates them? One question settles it: can the number be written as p/q?

1. Rational numbers

A number of the form p/q is called a rational number when p and q are integers and q ≠ 0. So 3/1 = 3, 11/2 = 5.5 and 5/3 = 1.666... are all rational. That makes every integer and every fraction a rational number.

Any rational number can also be written as the ratio of two mutually prime numbers. This fact will do real work in the proof further down.

2. Irrational numbers

If a number cannot be written as p/q, with p and q integers and q ≠ 0, it is called an irrational number. No irrational number can be expressed as the ratio of two integers.

The root of a natural number that is not a perfect square, or a fraction of such a root, is an irrational number. For example √2 = 1.414213..., √3 = 1.732... and √5/2 = 1.118... are all irrational.

Write a number as a decimal and its nature shows. Guess first, then watch the digits come.

3. Simplify under the root first

Never judge by the root sign alone. Simplify what is inside first. √9 = 3, √625 = 25, √(4/9) = 2/3, √(16/9) = 4/3, ∛27 = 3, ∛(8/27) = 2/3 and ∛125 = 5 are all rational. But √13, √(25/50) = √(1/2) and ∛7 are irrational. If a = b then √(ab) = √(a²) = a, which is rational too.

For a quotient of roots, bring both under one root. √27/√48 = √(27/48) = √(9/16) = 3/4 is rational. But √7/√21 = 1/√3 and √8/√7 are irrational, and so is 5/√3.

One example: if a = √3 and b = √12, then ab = √36 = 6 is rational and a/b = √(3/12) = 1/2 is rational. But a + b = √3 + 2√3 = 3√3 is real and irrational. Now sort a few yourself.

4. Between two numbers

To find two irrational numbers between √3 = 1.7320508... and 4, the book's Example 1 takes a = √3 + 1 and b = √3 + 2, because √3 < √3 + 1 < √3 + 2 < 4. In fact countless irrational numbers can be built this way.

For an irrational number between two decimals, write an infinite non-repeating decimal between them, such as 0.13010010001... between 0.12 and 0.31. For a rational one, take a finite decimal, such as 0.11 between 0.1 and 0.12, or 1.1001 between 1.1 and 1.11.

To count integers, take √3 ≈ 1.732, and the integers between √3 and 5 are 2, 3, 4: three in all. Bounds come without taking the root too: since 9 < 10 < 16, 3 < √10 < 4. Likewise 1 < √3 < 2 and 5 < √29 < 6. Each sits between two consecutive integers, so none of them is an integer.

5. Why √2 is irrational

Proposition: √2 is an irrational number. Proof: suppose √2 is rational and √2 = p/q, where p and q are mutually prime natural numbers and q > 1. Squaring, 2 = p²/q², and multiplying by q, 2q = p²/q. Clearly 2q is an integer, but p²/q is not, because p and q are mutually prime and q > 1. So 2q ≠ p²/q, that is √2 ≠ p/q. Therefore √2 is irrational. Do not drop the two conditions, “mutually prime” and “q > 1”: the whole proof rests on them.

The same method proves √3, √5, √6, √7, √10 and √29 irrational. Assume √5 = p/q and you get 5q = p²/q: an integer on the left, not on the right. In another question, if n = 2m − 1 and m = 3 then n = 5, so √n = √5 is irrational.

6. Check yourself

Three questions in the exam's style. Start with the first.

In the second, simplify the inside first.

The last follows the same rule.

The whole note on eight cards.

Source: Mathematics, Classes 9-10 (NCTB), Chapter 1 “Real Numbers”, the “Rational and Irrational Numbers” part.